A group of theoretical physicists from the Joint Institute for Nuclear Research and two Russian universities has obtained important results in the study of low-energy implications of the superstring theory by methods of supersymmetric field theory. The key achievements of the scientists were the formulation of new methods for constructing effective actions with preservation of explicit and hidden symmetries at all stages of calculations and their subsequent application to various models of classical and quantum fields. The development of research in the field of higher spins in the harmonic superspace was presented in a recently published work.

During the study, the research group including Head of the Supersymmetry Sector of the Laboratory of Theoretical Physics at JINR Evgeny Ivanov and his colleagues from Tomsk State Pedagogical University, Iosif Buchbinder and Boris Merzlikin, as well as Konstantin Stepanyants from Moscow State University, applied quantum field theory methods to investigate the fundamental problem of describing low-energy implications of superstring theory for physics beyond the Standard Model.

“We studied the quantum structure of supersymmetric theories in harmonic superspace, which follow from superstring theory in the low-energy limit, and obtained a number of new results along the way”, Evgeny Ivanov said.

Superstring theory was developed in the 70s. This is a supersymmetric generalisation of string theory, in which physicists study the dynamics of the interaction of particles as one-dimensional extended objects, thr so-called quantum strings. Superstring theory is self-consistent, that is, it does not contain any intrinsic contradictions and is now considered as the most advanced version of the unified theory of all fields and particles. It is the first and so far the only example of a finite (i.e., non-divergent) theory of quantum gravity, no alternatives have been proposed to it yet. Superstring theory includes known quantum field theories as its low-energy limits.

Superstring theory is based on supersymmetry, a hypothetical symmetry that connects fermions and bosons and introduced as a mathematical concept in the 1960s and 70s. There are two types of particles in nature: bosons (with integer spin) and fermions (with half-integer spin). They have radically different properties. In particular, according to the Pauli’s principle, two fermions cannot be in the same quantum state, they must necessarily have different quantum numbers. Therefore, new particles cannot be composed from identical fermions, unlike bosons. All other known kinds of symmetries are implemented separately on bosons and on fermions. Fields and particles with the same symmetry are unified in multiplets (groups), and all interactions of states within the given multiplet are identical. Such is the symmetry of the Poincaré group, symmetry with respect to rotations and shifts in a four-dimensional Minkowski spacetime characterised by vector coordinates (three spatial and one timelike). Supersymmetry unites bosons together with fermions into single multiplets. According to superstring theory, all known fermions should have hypothetical superpartners, bosons, and each boson should have a partner fermion. Since there is no mass degeneracy of fermions and bosons in nature, supersymmetry must be broken, and the search of appropriate mechanisms for such a symmetry violation is an urgent task.

“Superstring theory is regarded as a unified theory that reproduces all known theories in the low energy limit. The energies that are currently achievable at accelerators are considered to be quite small from the point of view of superstring theory. Unfortunately, supersymmetry is unlikely to be confirmed experimentally in the near future”, the scientist explained. Some theoretical predictions suggest that superpartners may have masses far greater than those of already discovered particles, and to detect them in accelerators, we would need energies that are beyond the reach of today’s facilities (and perhaps even the next-generation of accelerators). However, supersymmetry has profound theoretical implications that make it an indispensable concept. In particular, it provides the self-consistency of superstring theory. An important prediction of supersymmetry is the existence of supersymmetric extension of the theory of gravity, supergravity, and supersymmetric partner of the graviton, gravitino, a particle with 3/2 spin. All these implications and their consistency should be checked theoretically. “These studies contribute to the development of a self-consistent theory of all the forces of nature. Confirmation of the hypothesis that superstrings describe all fundamental interactions is a careful and long-term work”, Evgeny Ivanov stressed.

Supersymmetry in theory is realised in a superspace where additional fermionic dimensions, the so-called Grassmann coordinates, are added to the Minkowski space. The Grassmann coordinates have no physical interpretation. Each of them squared gives zero. Thus, the superspace is a speculative auxiliary structure which makes it possible to implement supersymmetry on the Minkowski space as simply and clearly as possible. There are also theories with real (bosonic) extra dimensions, corresponding to superspaces with 10 bosonic coordinates, and even more complicated theories with 11-dimensional space. These additional bosonic dimensions, which are not observed at the currently achievable energies, are necessary for the consistency of superstring theory at the quantum level.

Functions defined in the superspace (superfields) in the expansion by Grassmannian variables automatically give all the fields that unite into supermultiplets. Soon after the discovery of supersymmetry, it became clear that simple superspaces do not fully correspond to superstring theory and its low-energy limits, and it is necessary to introduce extended superspaces, where the Grassmannian coordinates have an internal index, and therefore are also transformed by internal symmetry. In order to describe such extended superspaces in the most natural and simple way, it is necessary, in addition to spatial coordinates and Grassmannian variables, to introduce additional coordinates, namely the so-called harmonic coordinates, which are coordinates associated with internal symmetry. Harmonic superspace was discovered in Dubna by a team of researchers. For this discovery, theorists of BLTP, Alexander Galperin, Evgeny Ivanov, Victor Ogievetsky, and Emery Sokatchev, received the First Prize of the 1987 JINR Award competition.

Nowadays, the concept of harmonic superspace has become generally accepted in mathematical physics. It turned out to be indispensable for studying supersymmetric gauge theories, and especially their quantum properties, in spaces with a different number of dimensions (from four to ten). To study the structure of superstrings, it is necessary to fully understand all the field-theoretic limits of this theory. Since the late 90s and up to the present, the above-mentioned team has been carrying out scientific research in this area: Evgeny Ivanov (BLTP at JINR), Joseph Buchbinder (Tomsk State Pedagogical University (TSPU), BLTP at JINR), Boris Merzlikin (TSPU), and Konstantin Stepanyants (Moscow State University). “We have developed new powerful methods of quantum computing, which have made it possible to significantly advance the harmonic superspace method in the quantum domain. Some stage of the work has been completed, but now there are many new challenges that we continue to address. The results of specific calculations within the framework of superstring theory will finally make it possible to find connections between the observable coupling constants in nature”, Evgeny Ivanov concluded.

The authors’ works have a high citation rate. Their results are used and actively participated in their further development by many scientific groups in the world: in Australia, Germany, USA, France and other countries.

The main results of the research were presented by the scientists themselves:

“The series of current research carried out during the last seven years is directed to the development of explicitly covariant and explicitly supersymmetric methods for the construction of effective actions of gauge field theories with extended supersymmetry in different dimensions. The research is based on the universal harmonic superspace approach previously proposed at the JINR BLTP. The general motivation and goals of the works included in the series are related to the study of low-energy limits of superstring theory by supersymmetric field theory methods:

• A method of superfield realisation of hidden supersymmetries in N = 4, 4D and N = 2, 5D supersymmetric Yang-Mills theories formulated in terms of harmonic superfields has been developed. It has been shown that this method makes it possible to reproduce all the known superfield invariants corresponding to such theories in a unified way and to construct new superinvariants.
• A new bi-harmonic superfield formulation of the N = 4, 4D supersymmetric Yang-Mills theory has been proposed and an approach for constructing N = 4 supersymmetric effective actions has been developed on its basis.
• A harmonic superfield approach to the N= 2, 4D supersymmetric theory of higher spin gauge fields has been developed. Explicit N = 2 supersymmetric free action of higher-spin gauge superfields and the cubic vertex of the interaction of such superfields with a hypermultiplet have been constructed.
• A method for studying the quantum effective action in N = (1, 0), 6D and N = (1, 1), 6D supersymmetric gauge theories has been developed. A superfield background field method has been developed that provides the explicit gauge invariance and explicit N = (1, 0) supersymmetry in calculating the effective action.
• A method of studying the structure of the one-loop and two-loop divergences in the six-dimensional theories considered has been worked out. It has been shown that, although all these theories are non-renormalizable, the quantum N = (1, 1) theory can be completely finite in the one-loop approximation.
• A superfield proper-time method has been developed for the explicitly supersymmetric and gauge invariant calculation of the one-loop effective action in the N = 2, 5D and N = (1, 0), 6D supergauge theories.
• A one-loop N = (1, 1), 6D supersymmetric low-energy effective action (a generalisation of the effective Heisenberg-Euler action for a constant electromagnetic field) has been constructed that depend on all fields of the N = (1, 1), 6D gauge supermultiplet”.

The results were published in 22 articles, mainly in Physics Letters B, Nuclear Physics B, and Journal of High Energy Physics, and presented in 15 plenary reports at international conferences. The series of works “New methods in classical and quantum field theory with extended supersymmetry” received the First Prize in the category of theoretical physics research of the JINR 2022 Awards.

Publications based on research results:

1. I.L. Buchbinder, E.A. Ivanov, B.S. Merzlikin, K.V.Stepanyantz, One-loop divergences in the 6D, N = (1, 0) Abelian gauge theory, Physics Letters, Vol. B763, pp. 375 – 381, 2016.
2. I.L. Buchbinder, E.A. Ivanov, B.S. Merzlikin, K.V. Stepanyantz, One-loop divergences in the 6D, N = (1, 0) and N = (1, 1) SYM theory, Journal of High Energy Physics, 01 (2017), 128.
3. I.L. Buchbinder, E.A. Ivanov, B.S. Merzlikin, K.V. Stepanyantz, Supergraph analysis of the one-loop divergences in 6D, N = (1, 0) and N = (1, 1) gauge theories, Nuclear Physics, Vol. B 921, No 1, pp. 127 – 158, 2017.
4. I.L. Buchbinder, E.A. Ivanov, B.S. Merzlikin, K.V. Stepanyantz, On the two-loop divergences of the 2-point hypermultiplet supergraphs for 6D, N = (1, 1) SYM theory, Physics Letters, Vol. B.778, No 1, pp. 252 – 255, 2018.
5. I.L. Buchbinder, E.A. Ivanov, B.S. Merzlikin, Leading low-energy effective action in 6D, N = (1, 1) SYM theory, Journal of High Energy Physics, 09 (2018) 039.
6. I.L. Buchbinder, E.A. Ivanov, B.S. Merzlikin, K.V. Stepanyantz, Gauge dependence of the one-loop divergences in 6D, N = (1, 0) Abelian theory, Nuclear Physics, Vol. B936, pp. 638 – 660, 2018.
7. I.L. Buchbinder, E.A. Ivanov, I.B. Samsonov, Low-energy effective action in 5D, N = 2 supersymmetric gauge theory, Nuclear Physics, Vol. B940, No 1, pp. 54 – 62, 2019.
8. I.L. Buchbinder, E.A. Ivanov, B.S. Merzlikin, K.V. Stepanyantz, On gauge dependence of the one-loop divergences in 6D, N = (1, 0) and N = (1, 1) SYM theories, Physics Letters, Vol. B798, No 10 (2019) 134957.
9. I.L. Buchbinder, E.A. Ivanov, B.S. Merzlikin, Quantum calculation of the low-energy effective action in 5D, N = 2 SYM theory, Physics Letters Vol. B802 (2020) 135218.
10. I.L. Buchbinder, E.A. Ivanov, B.S. Merzlikin, Low-energy 6D, N = (1, 1) SYM effective action beyond the leading approximation, Nuclear Physics, Vol. B 954 (2020) 114995.
11. I.L. Buchbinder, E.A. Ivanov, V.A. Ivanovskiy, Superfield realization of hidden R symmetry in extended supersymmetric gauge theories and its applications, Journal of High Energy Physics 04 (2020) 124.
12. I.L. Buchbinder, E.A. Ivanov, Hidden supersymmetry as a key to constructing low energy superfield effective actions, Proceedings of the Steklov Institute of Mathematics, Vol. 309 (2020) pp. 57-77.
13. I.L. Buchbinder, E.A. Ivanov, B.S. Merzlikin, K.V. Stepanyantz, Supergraph calculation of one-loop divergences in higher-derivative 6D SYM theory, Journal of High Energy Physics, 08 (2020) 169.
14. I.L. Buchbinder, E.A. Ivanov, B.S. Merzlikin, K.V. Stepanyantz, The renormalization structure of 6D, N = (1, 0) supersymmetric higher-derivative gauge theory, Nuclear Physics, Vol. B 961 (2020) 115249.
15. I.L. Buchbinder, E.A. Ivanov, V.A. Ivanovskiy, New bi-harmonic superspace formulation of 4D, N = 4 SYM theory, Journal of High Energy Physics, 04 (2021) 010.
16. I.L. Buchbinder, E.A. Ivanov, B.S. Merzlikin, K.V. Stepanyantz, On the two-loop divergences in 6D, N = (1, 1) SYM theory, Physics Letters, Vol. B 820 (2021) 136516.
17. S. Buyucli, E. Ivanov, Higher-dimensional invariants in 6D super Yang-Mills theory, Journal of High Energy Physics, 07 (2021) 190.
18. I.L. Buchbinder, E. Ivanov, N. Zaigraev, Unconstrained off-shell superfield formulation of 4D, N = 2 supersymmetric higher spins, Journal of High Energy Physics, 12 (2021) 016.
19. I.L. Buchbinder, E. Ivanov, N. Zaigraev, Off-shell cubic hypermultiplet couplings to N = 2 higher spin gauge superfields, Journal of High Energy Physics, 05 (2022) 104.
20. I.L. Buchbinder, E.A. Ivanov, N.G. Pletnev, Superfield approach to the construction of effective action in quantum field theory with extended supersymmetry, Physics of Particles and Nuclei, Vol. 47, No 3, pp. 291 – 369, 2016.
21. I.L. Buchbinder, E.A. Ivanov, I.B. Samsonov, The low-energy N = 4 SYM effective action in diverse harmonic superspaces, Physics of Particles and Nuclei, Vol. 48, No 3, pp. 333 – 388, 2017.
22. I.L. Buchbinder, E.A. Ivanov, B.S. Merzlikin, K.V. Stepanyantz, Harmonic Superspace Approach to the Effective Action in Six-Dimensional Supersymmetric Gauge Theories, Symmetry, Vol. 11, No 1, pp. 1 – 29, 2019.