### Abstract

The spectrum of spinless modes in a model with the \(SU{{(2)}_{L}} \times U{{(1)}_{R}}\) symmetrical four-quark-interaction proposed by Miransky, Tanabashi, and Yamawaki is studied. For the sake of simplicity, only four-fermion interactions of top and bottom quarks are considered. The spinless modes result from spontaneous electroweak interaction symmetry breaking and are coupled quark–antiquark states associated with two \(SU\left( 2 \right)\) Higgs doublets. Their dynamics is described by the effective Lagrangian obtained by the Schwinger–DeWitt method. The spectrum is represented by excitations of five types, the mass of each being expressed by the parameters of the model. It is shown that the model yields phenomenologically acceptable values of both the mass of quarks \({{m}_{t}} = 173\,\,{\text{GeV}}\) and \({{m}_{b}} = 4.18\,\,{\text{GeV}}\) and the mass of the standard Higgs state \({{m}_{{{{\chi }_{1}}}}} = 125\,\,{\text{GeV}}\). The masses of the particles that comprise the second Higgs doublet, \({{m}_{{{{h}^{ \pm }}}}} = 275\,\,{\text{GeV}}\), \({{m}_{{{{\chi }_{2}}}}} = 346\,\,{\text{GeV}}\), and \({{m}_{{{{\phi }_{0}}}}} = 125\,\,{\text{GeV}}\), have been calculated. The Nambu sum rule and the conditions for satisfying it in the theories with the broken \(~U{{(1)}_{A}}\) symmetry are discussed

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## REFERENCES

- 1
H. Terazawa, Y. Chikashige, and K. Akama, “Unified model of the Nambu-Jona-Lasinio type for all elementary-particle forces,” Phys. Rev. D: Part. Fields

**15**, 480–487 (1977). - 2
H. Terazawa, “Subquark model of leptons and quarks,” Phys. Rev. D: Part. Fields

**22**, 184–199 (1980). - 3
H. Terazawa, “t-quark mass predicted from a sum rule for lepton and quark masses,” Phys. Rev. D: Part. Fields

**22**, 2921 (1980). - 4
Y. Nambu, “Quasisupersymmetry, bootstrap symmetry breaking and fermion masses,” in

*Proceedings of the International Workshop New Trends in Strong Coupling Gauge Theories, Nagoya, Japan, Aug., 24–27,**1988,*Ed. by M. Bando, T. Muta, and K. Yamawaki (World Scientific, Singapore, 1989); EFI Report No. 89-08 (1989, unpublished). - 5
Y. Nambu, “New theories in physics,” in

*Proceedings of the 11th International Symposium on Elementary Particle Physics, Kazimierz, Poland,**1989,*Ed. by Z. Ajduk, S. Pokorski, and A. Trautman (World Scientific, Singapore, 1989), pp. 1–10. - 6
V. A. Miransky, M. Tanabashi, and K. Yamawaki, “Dynamical electroweak symmetry breaking with large anomalous dimension and t quark condensate,” Phys. Lett. B

**221**, 177–183 (1989). - 7
V. A. Miransky, M. Tanabashi, and K. Yamawaki, “Is the t quark responsible for the mass of W and Z bosons?,” Mod. Phys. Lett. A

**4**, 1043–1053 (1989). - 8
W. A. Bardeen, C. T. Hill, and M. Lindner, “Minimal dynamical symmetry breaking of the standard model,” Phys. Rev. D: Part. Fields

**41**, 1647–1660 (1990). - 9
G. Cvetic, “Top quark condensation,” Rev. Mod. Phys.

**71**, 513–574 (1999). - 10
C. T. Hill and E. H. Simmons, “Strong dynamics and electroweak symmetry breaking,” Phys. Rep.

**381**, 235–402 (2003). - 11
Y. Nambu and G. Jona-Lasinio, “Dynamical model of elementary particles based on an analogy with superconductivity. I,” Phys. Rev.

**122**, 345–358 (1961). - 12
Y. Nambu, “Fermion-boson relations in BCS-type theories,” Phys. D (Amsterdam, Neth.)

**15**, 147–151 (1985). - 13
G. E. Volovik and M. A. Zubkov, “Nambu sum rule and the relation between the masses of composite Higgs bosons,” Phys. Rev. D: Part. Fields

**87**, 075016 (2013). - 14
G. E. Volovik and M. A. Zubkov, “Nambu sum rule in the NJL models: From superfluidity to top quark condensation,” JETP Lett.

**97**, 301 (2013); arXiv:1302.2360 [hep-ph]. - 15
M. A. Luty, “Dynamical electroweak symmetry breaking with two composite Higgs doublets,” Phys. Rev. D: Part. Fields

**41**, 2893–2902 (1990). - 16
M. Suzuki, “Composite Higgs bosons in the Nambu-Jona-Lasinio model,” Phys. Rev. D: Part. Fields

**41**, 3457–3463 (1990). - 17
M. Harada and N. Kitazawa, “Vacuum alignment in the top quark condensation,” Phys. Lett. B

**257**, 383–387 (1991). - 18
A. A. Osipov and M. M. Khalifa, “Catalysis of the \(\left\langle {\bar {b}b} \right\rangle \) condensate in the composite Higgs model,” JETP Lett.

**110**, 387–393 (2019). - 19
A. A. Osipov, B. Hiller, A. H. Blin, F. Palanca, J. Moreira, and M. Sampaio, “Top condensation model: A step towards the correct prediction of the Higgs mass,” arXiv:1906.09579 [hep-ph] (2019).

- 20
J. Schwinger, “On Gauge invariance and vacuum polarization,” Phys. Rev.

**82**, 664–679 (1951). - 21
B. S. DeWitt,

*Dynamical Theory of Groups and Fields*(Gordon and Breach, New York, 1965). - 22
B. S. DeWitt, “Quantum field theory in curved spacetime,” Phys. Rep.

**19**, 295–357 (1975). - 23
R. D. Ball, “Chiral Gauge theory,” Phys. Rep.

**182**, 1–186 (1989). - 24
G. C. Branco, P. M. Ferreira, L. Lavoura, M. N. Rebelo, M. Sher, and J. P. Silva, “Theory and phenomenology of two-Higgs-doublet models,” Phys. Rep.

**516**, 1–102 (2012). - 25
R. D. Peccei and H. R. Quinn, “CP conservation in the presence of pseudoparticles,” Phys. Rev. Lett.

**38**, 1440–1442 (1977). - 26
R. D. Peccei and H. R. Quinn, “Constraints imposed by CP conservation in the presence of pseudoparticles,” Phys. Rev. D: Part. Fields

**16**, 1791–1797 (1977). - 27
E. Witten, “Current algebra theorems for the U(1) Goldstone boson,” Nucl. Phys. B

**156**, 269–283 (1979).

## ACKNOWLEDGMENTS

A.A. Osipov would like to thank C.T. Hill for his interest in this study and their correspondence, as well as expresses gratitude to the European Cooperation in Science and Technology for support provided within the COST Action CA16201 project.

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Translated by O. Lotova

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Osipov, A.A., Khalifa, M.M. The Nambu Sum Rule in the Composite Two Higgs Doublet Model.
*Phys. Part. Nuclei Lett.* **17, **296–302 (2020). https://doi.org/10.1134/S1547477120030127

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