On the triplet anti-triplet symmetry in 3-3-1 models
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We present a detailed discussion of the triplet anti-triplet symmetry in 3-3-1 models. The full set of conditions to realize this symmetry is provided, which includes in particular the requirement that the two vacuum expectation values of the two scalar triplets responsible for making the W and Z bosons massive must be interchanged. We apply this new understanding to the calculation of processes that have a \(Z-Z'\) mixing.
Interesting extensions of the Standard Model (SM), based on the local gauge group \(SU(3)_C\otimes SU(3)_L\otimes U(1)_X\) (3-3-1), have been widely studied (see Ref.  and references therein, see also Refs. [2, 3, 4, 5, 6] for similar models but with lepton-number violation). Fermions are typically organized into triplets and anti-triplets of \(SU(3)_L\) in three generations. We therefore have two possible choices, either to put leptons in triplets or anti-triplets.
Using Eq. (1) it is possible to show that the electric-charge spectrum is invariant. Therefore, at first glance, we expect that physics must be the same. This was noted e.g. in Ref.  (see the sentence after Eq. (2.4) therein) and at the end of Section II in Ref. . We think that this is well recognized in the 3-3-1 model community.
We notice, however, one missing ingredient in the above discussion. Namely, the parameter a defined in Eq. (4) changes sign under the triplet anti-triplet transformation. The authors of Ref.  did not see this because they chose a to be an input parameter and kept it unchanged under the transformation. If we instead choose the charged gauge-boson masses as independent input parameters and calculate a from them, then we will see that the value of a changes sign. This is the main point of this letter, which, to the best of our knowledge, has not been noted in the literature. The choice of the charged-gauge boson masses as input parameters is natural as this is directly related to physical observables. Reference  focused on the neutral gauge bosons and did not touch the charged gauge-boson masses, hence this important point was missed out. With this new piece of information, we will see that \(\sin \xi \) can only change sign under the triplet anti-triplet transformation.
In trying to solve this puzzle, we have realized that there exists no detailed discussion of the triplet anti-triplet symmetry in the literature apart from some brief remarks as above noted. Since this is an important issue in 3-3-1 models, we think it can be useful to show in detail how this symmetry works. We have found that the actual implementation of this symmetry in practice requires not only a careful attention to the input parameter scheme as above noted but also possible sign flips in many places in the Feynman rules and in book-keeping parameters. We will also show that the full definition of the triplet anti-triplet transformation is more complicated than Eq. (2) and changing the sign of the parameter a. This is easy to see because the full Lagrangian depends also on many other parameters, which may also flip signs or interchange under the transformation.
There is another issue related to the comparison with Ref. . Indeed, Ref.  provided results for two models called \(F_1(\beta ,a)\) and \(F_2(-\beta ,a)\), related by the transformation Eq. (2). For each model, results for different values \(a = 0,\pm 12/13\) are also given. Numerical results of Ref. , see Fig. 4 and Fig. 5 therein, show that \(F_1(\beta ,a)\) and \(F_2(-\beta ,-a)\) are not the same. This is very surprising to us because we expect them to be identical according to the triplet anti-triplet symmetry. We have discussed this issue with the authors of Ref. , but, unfortunately, no conclusive finding has been reached. Our investigation has led us to the conclusion that there seems to be an issue with the sign of the couplings between the \(Z'\) and the leptons in the model \(F_2\).
The paper is organized as follows. In Sect. 2, we discuss the two models related by the transformation Eq. (2) and provide the full set of conditions for them to be identical. In Sect. 3, we make application to the processes with a \(Z-Z'\) mixing and perform some crosschecks with Ref.  and other papers. Conclusions are given in Sect. 4. In Appendix A we provide details on the calculation of the \(Z-Z'\) mixing and of the couplings between the Z, \(Z'\) gauge bosons to the leptons in the two models with a general sign convention for the \(Z'\) field definition.
2 Two identical models
In this section we consider two 3-3-1 models denoted \(M_1\) and \(M_2\), related by the triplet anti-triplet transformation defined by Eq. (2). We note that Eq. (2) is not enough to make the two models identical, because the physical results depend also on the values of other input parameters such as masses, mixing and coupling parameters. Since we impose here that the two models are identical, there must be relations between the parameters of the two models. These relations can be found by comparing the two Lagrangians.
The parameter \(\beta \) will be denoted \(\beta _1\) and \(\beta _2\) for the two models, respectively. We will use the indices \(m,n = 1,2\) to distinguish the models.
Anomaly cancellation requires that the number of triplets and anti-triplets must be equal. Since quarks come in three colors, this means that one family of quarks must be in anti-triplet and the other two families are in triplets or vice versa. This implies two choices, the leptons are either put in triplets or in anti-triplets. Because Feynman rules for the quarks are similar to those for the leptons, we will ignore the quarks and focus on the leptons in the following.
In summary, the two models \(M_1\), where the leptons are organized in anti-triplets, and \(M_2\), where the leptons are in triplets, are equivalent if, besides identical gauge couplings, the relations Eqs. (7), (12), and (25) are satisfied. The important relation Eq. (23) is a consequence of Eq. (25). We therefore remark that the conditions in Eq. (2) are necessary but not sufficient to realize the triplet anti-triplet symmetry.
3 Application to neutral-current processes
From the above discussion, we can now see clearly that \(F_1(a)\) and \(F_2(a)\) are not equivalent, leading (unsurprisingly) to the fact that the results for \(F_1(a)\) and for \(F_2(a)\) are not the same if \(a\ne 0\).
For model \(F_1\), we agree with Table 1 of Ref. .
For model \(F_2\), we agree with Ref. .
For model \(F_2\), we agree with Ref.  on \(\sin \xi \) and can reproduce the Table 2 of Ref.  if a minus sign is added to the \(llZ'\) couplings.1 However, if the correct sign is used, the results change because of the \(Z-Z'\) interference terms.
To facilitate comparisons, our results for the llZ, \(llZ'\) couplings and for \(\sin \xi \) are provided in Appendix A. All these findings have been communicated to the authors of Ref. .
In this work we have pointed out that the recognized triplet anti-triplet symmetry in 3-3-1 models should include a sign change in the parameter \(a = (v_1^2-v_2^2)/(v_1^2 + v_2^2)\), besides the well-known sign change in the parameter \(\beta \) and changing from triplets to anti-triplets and vice versa. We have shown that the full transformation is more complicated than that and attention has to be paid to the input parameter scheme and also to the parameters of the scalar potential. The transformations of those parameters have been provided.
We have applied the new understanding to the processes with a \(Z-Z'\) mixing and in particular to the calculations of Ref. . We have found a possible sign issue with the couplings between the \(Z'\) and the leptons in the model where the leptons are put in the triplet representation.
We would like to thank Andrzej Buras and Fulvia De Fazio for discussions. We are grateful to Julien Baglio for his careful reading of the manuscript and for his helpful comments. LTH acknowledges the financial support of the International Centre of Physics at the Institute of Physics, Vietnam Academy of Science and Technology. This research is funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant number 103.01-2017.78. LDN acknowledges the support from DAAD to perform the final part of this work at the University of Tübingen under a scholarship. He thanks the members of the Institut für Theoretische Physik for their hospitality.
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