The standard model (SM) of elementary particles, based on gauge symmetry, is a magnificent creation by Murray Gell-Mann, Sheldon Glashow, Steven Weinberg, Abdus Salam and a whole galaxy of brilliant scientists. SM perfectly describes the interactions between quarks and leptons at distances of the order of 10–17 m (1% of the proton diameter), which can be studied using modern accelerators. However, it begins to slip already at distances of 10−18 m and, moreover, does not provide progress to the treasured Planck scale of 10−35 m.

It is believed that it is there that all fundamental interactions merge in quantum unity. SM will someday be replaced by a more complete theory, which, most likely, will not be the last and final either. Scientists are trying to find a replacement for the Standard Model. Many believe that the new theory will be built by expanding the list of symmetries that form the foundation of the SM. One of the most promising approaches to solving this problem was laid not only outside the context of SM problems, but even before its creation.

## A mixture of opposites

*At the end of the 1960s, Yuri Golfand, senior researcher at the Physics Department of the Lebedev Physical Institute, proposed to his graduate student Yevgeny Likhtman to generalize the mathematical apparatus used to describe the symmetries of four-dimensional space-time of the special theory of relativity (Minkowski space).*

Lichtman found that these symmetries can be combined with the internal symmetries of quantum fields with nonzero spins. In this case, families (multiplets) are formed, combining particles with the same mass, having a whole and half-integer spin (in other words, bosons and fermions). This was both new and incomprehensible, since both of them obey different types of quantum statistics. Bosons can accumulate in the same state, and fermions follow the Pauli principle, which strictly prohibits even paired unions of this kind. Therefore, the appearance of boson-fermion multiplets looked like mathematical exotics, not related to real physics. So this was accepted at the LPI. Later in his Memoirs, Andrei Sakharov called the unification of bosons and fermions a great idea, but at that time it did not seem interesting to him.

## Beyond the standard

Where are the boundaries of the SM? “The standard model is consistent with almost all the data obtained at high-energy accelerators. - explains Sergey Troitsky, a leading researcher at the Institute for Nuclear Research of the Russian Academy of Sciences. - However, the experimental results testifying to the presence of mass in two types of neutrinos, and possibly in all three, do not quite fit into its framework. This fact means that the SM needs to be expanded, and in which, no one really knows. Astrophysical data also indicate incomplete CM. Dark matter, and it accounts for more than a fifth of the mass of the Universe, consists of heavy particles that do not fit into SM. By the way, it would be more accurate to call this matter not dark, but transparent, since it not only does not emit light, but also does not absorb it. In addition, SM does not explain the almost complete absence of antimatter in the observable Universe. ”

There are also objections to the aesthetic order. As Sergey Troitsky notes, the SM is arranged very ugly. It contains 19 numerical parameters, which are determined by experiment and, from the point of view of common sense, take on very exotic values. For example, the Higgs vacuum average of the field, which is responsible for the masses of elementary particles, is 240 GeV. It is not clear why this parameter is 1017 times smaller than the parameter determining the gravitational interaction. I would like to have a more complete theory, which will make it possible to determine this relationship from some general principles.

The SM does not explain the huge difference between the masses of the lightest quarks, of which protons and neutrons are composed, and the mass of the top quark exceeding 170 GeV (in all other respects, it does not differ from the u quark, which is almost 10 thousand times lighter). Where it seems to come from the same particles with such different masses is not yet clear.

Lichtman defended his dissertation in 1971, and then went to VINITI and almost abandoned theorophysics. Golfand was fired from the LPI to reduce staff, and for a long time he could not find work. However, the staff of the Ukrainian Institute of Physics and Technology Dmitry Volkov and Vladimir Akulov also discovered the symmetry between bosons and fermions and even used it to describe neutrinos. True, neither Muscovites nor Kharkiv found any laurels at that time. Only in 1989, Golfand and Lichtman received the Prize of the USSR Academy of Sciences in Theoretical Physics named after I.E. Tamm. In 2009, Vladimir Akulov (now a professor of physics at the Technical College of the City University of New York) and Dmitry Volkov (posthumously) were awarded the National Prize of Ukraine for scientific research.

## The birth of supersymmetry

*In the West, mixtures of bosonic and fermionic states first appeared in the nascent theory, which represents elementary particles not as point objects, but as vibrations of one-dimensional quantum strings.*

In 1971, a model was built in which a pair of fermion vibrations was combined with each vibration of the bosonic type. True, this model did not work in the four-dimensional Minkowski space, but in the two-dimensional space-time of string theories. However, already in 1973, the Austrian Julius Wess and the Italian Bruno Zumino reported to CERN (and a year later published an article) about a four-dimensional supersymmetric model with one boson and one fermion. It did not pretend to describe elementary particles, but demonstrated the capabilities of supersymmetry using a clear and extremely physical example. Soon, these same scientists proved that the symmetry they discovered was an extended version of the symmetry of Golfand and Lichtmann. So it turned out that within three years, three pairs of physicists independently discovered supersymmetry in Minkowski space.

The results of Wess and Zumino pushed the development of theories with boson-fermion mixtures. Since these theories associate gauge symmetries with space-time symmetries, they were called super-gauge, and then supersymmetric. They predict the existence of many particles, not one of which has yet been discovered. So the supersymmetry of the real world is still hypothetical. But even if it exists, it cannot be strict, otherwise the electrons would have charged bosonic relatives with exactly the same mass, which could easily be detected. It remains to be assumed that the supersymmetric partners of known particles are extremely massive, and this is only possible if supersymmetry is broken.

Supersymmetric ideology came into force in the mid-1970s, when the Standard Model already existed. Naturally, physicists began to build its supersymmetric extensions, in other words, introduce symmetries between bosons and fermions into it. The first realistic version of the supersymmetric SM called the Minimal Supersymmetric Standard Model (MSSM) was proposed by Howard Georgie and Savas Dimopoulos in 1981. In fact, this is the same Standard Model with all its symmetries, but a partner is added to each particle, whose spin differs from its spin by ½, a boson to a fermion and a fermion to a boson.

Therefore, all SM interactions remain in place, but are enriched by the interactions of new particles with old ones and with each other. Later, more complex supersymmetric versions of the SM arose. All of them correlate particles already known to the same partners, but in different ways explain the supersymmetry breaking.

## Particles and Super Particles

*The names of fermion super partners are built using the prefix "c" - electron, smuon, squark.* *Superpartners of bosons acquire the ending “foreign”: photon - photino, gluon - gluino, Z-boson - zino, W-boson - wine, Higgs boson - Higgsino.*

The spin of the superpartner of any particle (with the exception of the Higgs boson) is always ½ less than its own spin. Consequently, the partners of the electron, quarks, and other fermions (as well as, of course, their antiparticles) have a zero spin, and the partners of a photon and vector bosons with a single spin have a half. This is due to the fact that the greater the number of states of a particle, the greater its spin. Therefore, replacing the subtraction with addition would lead to the appearance of redundant super partners.

Take for example an electron. It can be in two states - in one its spin is directed parallel to the momentum, in the other - antiparallel. From the point of view of SM, these are different particles, since they do not participate in weak interactions in the same way. A particle with a single spin and nonzero mass can exist in three different states (as physicists say, it has three degrees of freedom) and therefore is not suitable as an electron partner. The only way out will be to assign to each of the states of the electron one superpartner with zero spin and consider these electrons as different particles.

The superpartners of the bosons of the Standard Model are somewhat trickier. Since the mass of the photon is equal to zero, then with a single spin it has not three, but two degrees of freedom. Therefore, he is easily matched with a photino, a superpartner with a half spin, which, like an electron, has two degrees of freedom. Gluino arises in the same way. With Higgs, the situation is more complicated. In MSSM there are two doubles of Higgs bosons, which correspond to four super partners - two neutral and two oppositely charged Higgsino. Neutrals are mixed in different ways with fotino and zino and form four physically observable particles with the common name neutralino. Similar mixtures with the strange name for the Russian ear, Chardzhino (in English - chargino) form the superpartners of the positive and negative W-bosons and a pair of charged Higgs.

The situation with neutrino super partners has its own specifics. If this particle had no mass, its spin would always be directed opposite to momentum. Therefore, a massless neutrino could be expected to have a single scalar partner. However, real neutrinos are still not massless. It is possible that there are also neutrinos with parallel momenta and spins, but they are very heavy and have not yet been detected. If this is true, then each type of neutrino has its own superpartner.

According to Gordon Kane, a professor of physics at the University of Michigan, the most universal mechanism of supersymmetry breaking is associated with gravity.

However, the magnitude of its contribution to the masses of superparticles has not yet been clarified, and the theorists' estimates are contradictory. In addition, it is hardly the only one. So, the Next-to-Minimal Supersymmetric Standard Model, NMSSM, introduces two more Higgs bosons that add their additives to the mass of superparticles (and also increases the number of neutralos from four to five). Such a situation, Kane notes, dramatically multiplies the number of parameters embedded in supersymmetric theories.

Even a minimal extension of the Standard Model requires about a hundred additional parameters. This should not be surprising, since all these theories introduce many new particles. As more complete and consistent models appear, the number of parameters should decrease. As soon as the detectors of the Large Hadron Collider catch superparticles, new models will not keep you waiting.

## Particle hierarchy

*Supersymmetric theories eliminate a number of weak points in the Standard Model.* *Professor Kane puts the riddle associated with the Higgs boson, which is called the hierarchy problem, in the first place* .

This particle gains mass in the course of interaction with leptons and quarks (similar to how they themselves gain mass when interacting with a Higgs field). In SM, the contributions from these particles are represented by diverging series with infinite sums. True, the contributions of bosons and fermions have different signs and, in principle, can almost completely cancel each other out. However, such a repayment should be practically ideal, since the Higgs mass, as is now known, is only 125 GeV. This is not impossible, but extremely unlikely.

For supersymmetric theories, this is not a big deal. With exact supersymmetry, the contributions of ordinary particles and their super partners must completely compensate for each other. Since supersymmetry is broken, the compensation is incomplete, and the Higgs boson acquires a finite and, most important, calculated mass. If the superpartner masses are not too large, it should be measured by one or two hundred GeV, which is true. As Kane emphasizes, physicists began to take supersymmetry seriously when it was shown that it solves the problem of hierarchy.

The possibilities of supersymmetry do not end there. It follows from the SM that in the region of very high energies the strong, weak and electromagnetic interactions, although they have approximately the same strength, are never combined. And in supersymmetric models at energies of the order of 1016 GeV, such a combination takes place, and this looks much more natural. These models also offer a solution to the problem of dark matter. Superparticles during decays give rise to both superparticles and ordinary particles, naturally, of a lower mass. However, supersymmetry, unlike SM, allows rapid decay of a proton, which, fortunately, does not actually occur.

The proton, and with it the entire surrounding world, can be saved by assuming that in processes involving superparticles, the quantum number of R-parity is preserved, which is equal to unity for ordinary particles, and minus one for superpartners. In this case, the lightest superparticle should be completely stable (and electrically neutral). By definition, it cannot decay into superparticles, and the conservation of R parity prevents it from decaying into particles. Dark matter can consist of just such particles that arose immediately after the Big Bang and avoided mutual annihilation.

## Waiting for experiments

*“Shortly before the discovery of the Higgs boson on the basis of M-theory (the most advanced version of string theory), its mass was predicted with an error of only two percent!* *- says Professor Kane.* *- We also calculated the masses of electrons, smuons and squarks, which turned out to be too large for modern accelerators - of the order of several tens of TeV.* *Superpartners of the photon, gluon and other calibration bosons are much easier, and therefore there is a chance to detect them on the LHC. ”*

Of course, the accuracy of these calculations is not guaranteed by anything: M-theory is a delicate matter. And yet, is it possible to detect traces of superparticles on accelerators? “Massive superparticles must disintegrate immediately after birth. These decays occur against the background of decays of ordinary particles, and it is very difficult to single out them unequivocally, ”explains Dmitry Kazakov, chief researcher at the JINR Laboratory of Theoretical Physics in Dubna. - It would be ideal if the superparticles showed themselves in a unique way that cannot be confused with anything else, but the theory does not predict this.

We have to analyze many different processes and look among them for those that are not fully explained by the Standard Model. These searches have not yet been crowned with success, but we already have restrictions on the mass of super partners. Those of them that participate in strong interactions should pull at least 1 TeV, while the masses of other superparticles can vary between tens and hundreds of GeV.

In November 2012, at the Kyoto symposium, the results of experiments on the LHC were reported, during which for the first time it was possible to reliably register a very rare decay of the Bs meson into the muon and antimuon. Its probability is approximately three billionths, which is in good agreement with the predictions of SM. Since the expected probability of this decay, calculated on the basis of MSSM, may turn out to be several times greater, some people decided that supersymmetry is over.

Однако эта вероятность зависит от нескольких неизвестных параметров, которые могут давать как большой, так и малый вклад в конечный результат, здесь еще много неясного. Поэтому ничего страшного не произошло, и слухи о кончине MSSM сильно преувеличены. Но из этого вовсе не следует, что она неуязвима. БАК пока не работает на полную мощность, он выйдет на нее лишь через два года, когда энергию протонов доведут до 14 ТэВ. И вот если тогда не найдется никаких проявлений суперчастиц, то MSSM, скорее всего, умрет естественной смертью и настанет время новых суперсимметричных моделей.

## Числа Грассмана и супергравитация

Еще до создания MSSM суперсимметрию объединили с гравитацией. Неоднократное применение преобразований, связывающих бозоны и фермионы, перемещает частицу в пространстве-времени. Это позволяет связать суперсимметрии и деформации пространственно-временной метрики, которые, согласно общей теории относительности, и есть причина тяготения. Когда физики это поняли, они начали строить суперсимметричные обобщения ОТО, которые называются супергравитацией. Эта область теоретической физики активно развивается и сейчас.

Тогда же выяснилось, что суперсимметричным теориям необходимы экзотические числа, придуманные в XIX столетии немецким математиком Германом Гюнтером Грассманом. Их можно складывать и вычитать как обычные, но произведение таких чисел изменяет знак при перестановке сомножителей (поэтому квадрат и вообще любая целая степень грассманова числа равна нулю). Естественно, что функции от таких чисел нельзя дифференцировать и интегрировать по стандартным правилам математического анализа, нужны совершенно другие приемы. И они, к счастью для суперсимметричных теорий, уже были найдены. Их придумал в 1960-е годы выдающийся советский математик из МГУ Феликс Березин, который создал новое направление — суперматематику.

Однако есть и другая стратегия, не связанная с БАК. Пока в ЦЕРН работал электронно-позитронный коллайдер LEP, на нем искали наиболее легкие из заряженных суперчастиц, чьи распады должны порождать наилегчайших суперпартнеров. Эти частицы-предшественники легче зарегистрировать, поскольку они заряжены, а легчайший суперпартнер нейтрален. Эксперименты на LEP показали, что масса таких частиц не превышает 104 ГэВ. Это не так уж много, но их трудно обнаружить на БАК из-за высокого фона. Поэтому сейчас началось движение за постройку для их поиска сверхмощного электрон-позитронного коллайдера. Но это очень дорогая машина, в скором времени ее уж точно не построят».

## Закрытия и открытия

*Однако, как считает профессор теоретической физики Университета Миннесоты Михаил Шифман, измеренная масса бозона Хиггса слишком велика для MSSM, и эта модель, скорее всего, уже закрыта:*

«Правда, ее пытаются спасти с помощью различных надстроек, но они столь неизящны, что имеют малые шансы на успех. Возможно, что другие расширения сработают, но когда и как, пока неизвестно. Но этот вопрос выходит за рамки чистой науки. Нынешнее финансирование физики высоких энергий держится на надежде обнаружить на БАК что-то действительно новое. Если этого не произойдет, финансирование урежут, и денег не хватит для строительства ускорителей нового поколения, без которых эта наука не сможет реально развиваться». Так что суперсимметричные теории по‑прежнему подают надежды, но ждут не дождутся вердикта экспериментаторов.

Статья «Больше, чем симметрия» опубликована в журнале «Популярная механика» (№2, Февраль 2013).**Do you like the article?**

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