Abstract
The reported anomaly in deepinelastic scattering at HERA has revived interest in the phenomenology of parity violation. From the theoretical point of view, the existence of violating interactions poses two considerable problems. The first one concerns the flavour structure of the interactions and the origin of an appropriate suppression of flavourchanging neutralcurrent processes and leptonfamily transitions. The second one concerns the way of embedding violating interactions in a grand unified theory (GUT) without introducing unacceptable nucleon decay rates. We show that the second problem can be solved by a mechanism which is purely group theoretical and does not rely on details of the flavour theory. We construct explicit GUT models in which our mechanism can be realized.
CERNTH/9776
hepph/9704339
Parity Violation and Unification
G.F. Giudice^{1}^{1}1On leave of absence from INFN, Sez. di Padova, Italy. and R. Rattazzi
Theory Division, CERN, CH1211, Genève 23, Switzerland
CERNTH/9776
April 1997
1. The anomaly in deepinelastic scattering events reported by H1 [1] and ZEUS [2] and the excess of fourjet events observed by ALEPH [3], but not confirmed by the other LEP experiments [4], have revived interest [5, 6] in the phenomenology of parity violating interactions. Certainly more statistics is required to understand if experiments are really observing some signal of new physics. Nevertheless we believe it is timely and important to investigate what are the consequences of parity violation in our understanding of the theoretical framework of supersymmetric models.
The two main problems arising from parity violation are connected with flavour and with unification. As we emphasize below, we believe that these two problems are quite different both in their quantitative aspect (the unification problem being numerically more acute) and in their conceptual aspect. In this paper we will present a solution to the unification problem which is independent of the flavour structure of the theory, and relies only on GUT symmetry properties. At present, the question of flavour remains unresolved, since we do not know any fully convincing theory which generates the peculiar hierarchy of the Yukawa couplings – let alone the parity violating couplings. We believe that separating the two puzzles, and solving the unification problem in terms of GUTs, leads to important theoretical progress since, as we argue below, it is probably unrealistic to hope that an ultimate flavour theory can provide the right cure.
2. We concentrate on the violating interaction suggested to explain the large data at HERA, which is given by the term in the superpotential
(1) 
with [5] and equal to 2 or 3. Here refer to generation indices and we employ a standard notation for quark and lepton superfields. The flavour problem arises because the generation structure of the operator in eq. (1) is in general not aligned with the generation structure of the Yukawa interactions
(2) 
We work in a basis where and are diagonal, and is a diagonal matrix times the KobayashiMaskawa matrix. Because of the mismatch in flavour space, squarks and sleptons mediate effective fourfermion interactions which lead to flavourchanging neutralcurrent processes and leptonfamily transitions. For instance, measurements of the mixing parameters and imply [7]
(3) 
(4) 
where is the relative phase between the two couplings. Bounds on – conversion imply [7]
(5) 
for any . While limits on a single coupling are weak enough [8] to allow for an important phenomenological rôle of parity breaking, the product of two couplings with different generation indices is severely constrained. A successful theory of flavour and parity violation should explain the origin of this strong hierarchy.
Let us now turn to the unification problem. If the interaction of eq. (1) has to be embedded in a trilinear term arising from a GUT, then the superpotential in general also contains the interactions
(6) 
While the and couplings violate lepton number, violates baryon number. Their simultaneous presence is therefore strongly constrained by nucleondecay searches. For instance the experimental bound on implies
(7) 
Here is the typical supersymmetrybreaking mass parameter in the mass matrix. The presence of the quark mass in eq. (7) is a product of the leftright squark mixing necessary to construct the fourfermion operator. Of course, in the case , it does not amount to any significant suppression.
From eq. (7) we see that the unification problem (or, in other words, the simultaneous presence of baryon and lepton number violation) poses a more severe difficulty than flavour. There can be hope that hierarchies between couplings analogous to those required by eqs. (3)–(5) can be explained in a complete theory of flavour. On the other hand, we prefer to believe that the observed suppression of nucleon decay is caused by the small ratio between the weak and the GUT scale rather than by some broken flavour symmetry, which generates hierarchies as a power expansion of some parameter like the Cabibbo angle. Examples of theories in which the baryonnumber and violating interactions are suppressed by flavour symmetries exist [9], but are not embedded in a GUT. It is not clear how they can be unified and made consistent with the size of couplings suggested by the HERA data. For this reason, we believe that the solution should lie within the GUT dynamics.
3. We now want to embed the parity violating interaction of eq. (1) in a GUT, without running into the problem of baryonnumber violation. To start, we choose the simplest example of and denote the matter content by () and the Higgs superfields by and , respectively a and of . The Yukawa couplings are
(8) 
Here and are functions of the adjoint field , which spontaneously breaks the symmetry. After gets its vacuum expectation value (VEV), they reproduce the ordinary Yukawa couplings , , at low energy.
The first attempt one can try is to include in the superpotential only the bilinear term^{2}^{2}2The phenomenology of the nonGUT version of this term has been considered, for instance, in ref. [10].
(9) 
where are mass parameters smaller than the weak mass scale. The operators in eq. (9) could be generated by some mechanism similar to the one responsible for the Higgsmixing term. By defining appropriate mass eigenstates, we can rotate the term in eq. (9) into some violating trilinear couplings. The states mix with the Higgs triplet contained in and give rise to the baryonnumber violating coupling of eq. (6) with
(10) 
Since the Higgstriplet mass is of the order of the GUT scale, we obtain a considerable suppression of the baryonviolating interaction. An even further suppression exists in models where the doublettriplet splitting is obtained without a direct mass term . The mixing between the lepton superfields and the Higgs doublet generates couplings and which are suppressed only by the ratio , where is the Higgsmixing term of the order of the weak scale. However, in this case, and the value of the violating coupling constant suggested by the HERA data is incompatible with the limit on the electron neutrino mass which implies [5]
(11) 
An interesting possibility, which was first suggested in ref. [7], is that the only parity violation comes from an operator in the superpotential
(12) 
If and (the Yukawa coupling defined in eq. (8)) are simultaneously diagonal in for any and for any index, then the interaction (12) generates nonvanishing , while and identically vanish [7]. This is simply because and are antisymmetric in , while has no symmetry properties. The coupling constants vanish at the GUT scale, but small values are generated by the renormalization to the weak scale. It was also shown in ref. [7] that the renormalization to the weak scale gives only small further violation of flavour in the sector, and therefore the ansatz on the generation structure of specified above can render the breaking interpretation of the HERA data compatible with unification. However this ansatz may seem rather ad hoc. Also it seems to rely on flavour properties, which we find a disturbing aspect, as previously discussed.
We turn now to discuss how, with no reference to the flavour theory, a GUT can lead to parity violating couplings relevant for squark production at HERA together with vanishing baryonnumber violating couplings. In terms of GUT representations, the violating interactions in eqs. (1) and (6) are written as^{3}^{3}3This defines a complete set of operators. This can be understood by counting the number of gauge invariants. For matter generations, the operators listed in eqs. (13)–(16) contain or flavour components, if the two are combined in a or , respectively. This makes a total of invariants and matches the number of invariants of the lowenergy theory, which are described by couplings and couplings and .
(13)  
(14)  
(15)  
(16) 
Here we have specified the contractions of the indices with a pendix denoting the product representation. Operators with more powers of reduce to combinations of the above since, for any , is a linear combination of and the identity. We have also marked explicitly which of the couplings , , or are generated by the various operators after gets its VEV. If we select the operators or , at low energies we retain only the couplings. This can be easily implemented in a GUT, since specific operators can be selected by appropriately choosing the virtual states which generate them. This selection is a consequence of grouptheoretical properties and does not rely on the flavour dynamics.
combines the two in a symmetric state. Therefore it is symmetric in the indices and and its contribution to and identically vanishes. We will give later an example of how this case can be realized in GUTs. also selects just the coupling, although it is antisymmetric in the flavour indices and . Only the coupling survives because is projected only onto , after GUT symmetry breaking.
Solving the problem of baryonnumber and parity violating interactions with GUT dynamics may not be sufficient. We are now concerned with higherdimensional operators suppressed by powers of the Planck mass generated by the unknown dynamics of quantum gravity. These operators may have the most generic structure compatible with the unbroken symmetries, and therefore reintroduce the unwanted couplings in the lowenergy effective theory. These couplings will only be suppressed by some powers of , and therefore bounds like the one in eq. (7) require that these operators should not be present at least up to some high dimensionality.
The nonrenormalization theorems [11] of supersymmetry can protect the GUT theory from this danger. If the operator or is generated only after some stage of symmetry breaking, this same symmetry together with the constraint of holomorphicity of the superpotential can forbid any dangerous higherdimensional operator. This mechanism, in which a broken symmetry protects against the appearence of certain terms in the superpotential at all orders, has also been used in the constructions of flavour theories [12]. Of course the renormalization of the Kähler function is not under control and it can effectively generate new terms in the superpotential, once supersymmetry is broken. The size of these effects, which are proportional to some power of the ratio between the supersymmetrybreaking scale and can be estimated in a given model and then compared with the experimental bound on nucleon decay.
4. Our mechanism is best illustrated by a simple example in . To generate the desired operators and , we introduce some fields in the symmetric product of two fundamentals, , which transform as . The presence of the field together with its conjugate insures the cancellation of anomalies and allows a superheavy mass term. Let us consider the following interaction for the fields and :
(17) 
Here is a gauge singlet, which plays the rôle of a mass parameter, and is somewhat larger than the GUT scale. The effective theory below , obtained by integrating out and , contains the operator . Just below , is generated, but and vanish.
In order to explain the desired structure of breaking interaction, we also have to justify the absence of the renormalizable coupling . As explained above, this is in general not sufficient, because the strong bounds on nucleon decay also require the absence of a large number of higherdimensional operators. The simplest symmetry we can introduce is an abelian flavourindependent . The charge assignment is completely determined by the requirement that the most general superpotential consistent with the symmetry is given by eqs. (8) and (17) together with terms responsible for the GUT symmetry breaking involving , bilinears in , and possibly other fields. We find the following charges: , , , , , , , . This is anomalous, but the GreenSchwarz mechanism [13] can be invoked to cancel the gauge anomalies. It is interesting that an anomalous group usually appears in the effective field theory derived from strings [14]. The effective theory then contains a FayetIliopoulos term, equal to [14]
(18) 
If the signs of and are opposite (and, in our example, this is true at least in the observable sector), then can get a VEV, given by
(19) 
The theory has an accidental discrete symmetry, under which , , , and are odd, while all other chiral superfields are even. This can be identified with the usual parity, and it is broken by at the scale . The size of the lowenergy parity violating coupling is .
We turn now to discuss the suppression of higherdimensional operators. The property of holomorphicity of the superpotential and the symmetry forbid all possible quantumgravity derived operators, which give rise to parity breaking in the lowenergy effective theory. Indeed terms of the generic form
(20) 
cannot appear since holomorphicity requires only positive powers of in the functions and , while invariance requires a negative power of . Of course this property depends on the particular charge assignment and it would not hold if there existed fields which acquire VEVs of the order of the GUT scale and have negative charges.
Planckmass suppressed operators can also affect the dynamics of the fields and . In particular the most general interactions consistent with symmetry have the same form of those in eq. (17), with arbitrary insertions of fields. The key point is that these operators are not going to modify our mechanism, since they can generate , but never or . We can understand this result differently by considering the content of the 15. The 15 does not contain any standard model representation with the correct quantum numbers to mediate effective interactions or . Only a colour triplet, weak doublet can be propagated between the and states. From this point of view, our mechanism is analogous to the missing partner mechanism [15] used to split the masses of the Higgs doublet and triplet belonging to the same GUT representation^{4}^{4}4The use of an anomalous to implement to all orders the missing partner mechanism was studied in ref. [16]..
The Kähler function can contain terms of the kind
(21) 
where is the spurion superfield which parametrizes supersymmetry breaking and has a nonzero VEV of the auxiliary field. The terms in eq. (21) give rise to the lowenergy parameters and , where is the gravitino mass. In theories where the breaking of supersymmetry is communicated to the observable sector by gravity [17] is of the order of the weak scale. Then, a comparison with the bound in eq. (7) shows that a certain degree of suppression coming either from the flavour theory or from quantum gravity is necessary. Lacking much knowledge of either of the two theories, we cannot exclude this case. On the other hand, in theories where supersymmetry breaking is communicated by particles lighter than , is smaller than the weak scale and it can efficiently suppress any Kählerinduced parity violation.
The last comment we wish to make about this model concerns the flavour structure of the parity violation. Although our mechanism does not address the question of flavour, as a byproduct of this model, we obtain a restriction on the generation structure of the couplings which, just below the GUT scale, have the form
(22) 
This is a consequence of our assumption that a single state mediates the effective interaction which generates the operator . Only within a complete theory of flavour can we hope to understand the hierarchical structure of the vector and the matrix .
5. The model discussed in the previous section is certainly not the only possibility to realize our mechanism. Instead of an anomalous symmetry, one could use an symmetry which easily protects against higherdimensional terms. However, in this case, one should specify the whole model, including the GUT symmetry breaking sector, which it was left undetermined in our previous example.
Another possibility is to consider different GUT groups, flipped [18] being a very interesting option. The GUT group is , with matter transforming as , and the usual Higgs doublets embedded in and . The Yukawa couplings are
(23) 
where we have denoted the matter superfields by their content. The GUT symmetry breaking is triggered by the VEVs of the fields and , which allow a simple implementation of the missingpartner mechanism [18] through the interactions
(24) 
An interesting feature of flipped is that renormalizable parity interactions are forbidden by gauge invariance. The lowenergy violating operators in eqs. (1) and (6) are generated by the following GUT operators:
(25)  
(26)  
(27)  
(28) 
This is a complete set of operators. Of course one can rearrange the contractions of the indices or take linear combination of the various operators. By doing this, we can construct an operator which selects only the couplings:
(29) 
This operator is generated by exchange of states with flipped quantum numbers , .
If we select only or , we obtain the violating coupling invoked for an interpretation of the HERA data, but forbid the baryonnumber violating ones. The operator is generated by the virtual exchange of the superfields , transforming as , with interactions
(30) 
where is a gauge singlet such that . However we have to forbid the couplings , which generate the operator and introduce the baryonnumber violating couplings . All unwanted structures are eliminated in renormalizable and nonrenormalizable interactions by a discrete parity and an anomalous . The superfields , , , , and are odd under parity , while all others are even. The charge assignment of the anomalous , up to a linear combination with the of flipped , is , while all other fields are neutral. Both the parity and the anomalous are spontaneously broken the VEVs of and . There are no fields with GUT scale VEV and negative charge , and therefore holomorphicity and symmetry invariance protect against operators with unwanted structures of parity breaking. As in the previous example, higherdimensional operators involving the fields and do not affect the mechanism, since only a single component of the field (a colour triplet, weak doublet) is propagated in the exchange between the and states. We also notice that, in the flipped case, the flavour structure of the violating coupling factorizes between quark and lepton indices
(31) 
in contrast to the case of eq. (22).
Finally notice that the operator can be generated by exchange through the couplings . The simultaneous presence of and interactions is constrained by flavour and lepton violating processes. However, since flipped is not based on a simple group and the unification is not complete, the couplings and are here unrelated. It is therefore possible that some flavour symmetry suppresses without affecting the coupling invoked to explain the HERA anomaly.
In the case of , leftright symmetry forbids violating dimensionfour operators. We can then consider dimensionfive operators of the kind , where describes a generation of matter and is the Higgs representation which breaks into . Since these operators are invariant, they predict . It is possible to construct higherdimensional operators which select only the coupling. For instance
(32) 
which are mediated by a heavy and respectively, can give the right structure of violation, once the heavy is exchanged. The model is obviously rather involved and depends on the unspecified dynamics which characterize the symmetrybreaking pattern.
6. We have shown that the parity violating interaction suggested by the HERA data and the observed absence of fast nucleon decay is compatible with the idea of grand unification. The interactions between matter and the heavy fields which break the GUT symmetry can split the different violating interactions. By choosing an appropriate field content, it is possible to select a preferred pattern of parity breaking. This pattern is protected against higherdimensional operators with different violating structures by the combined effect of a symmetry, broken at the GUT scale, and the holomorphicity of the superpotential.
Our mechanism relies on purely grouptheoretical properties and does not require any assumption about flavour structure. We have shown specific examples of GUT models in which our idea can be realized. Only in the context of a complete theory of flavour will it be possible to address the question of flavourchanging neutral current processes and leptonfamily transitions.
We wish to thank G. Altarelli, G. Dvali, and M. Mangano for useful discussions.
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