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ПРЕГЛЕДНИ РАДОВИ ОБЗОРНЫЕ СТАТЬИ REVIEW PAPERS
Supersymmetry
Nicola Fabiano
University of Belgrade, "Vinca" Institute of Nuclear Sciences - National Institute of the Republic of Serbia, Belgrade, Republic of Serbia, e-mail: nicola.fabiano@gmail.com, ORCID iD: ©https://orcid.org/0000-0003-1645-2071
DOI: 10.5937/vojtehg71-40268;https://doi.org/10.5937/vojtehg71-40268 FIELD: physics
ARTICLE TYPE: review paper Abstract:
Introduction/purpose: Supersymmetry is a symmetry of the Lagrangian that goes beyond Lie groups. It allows the exchange of bosons and fermions. The most important model is the Minimal Supersymmetric Standard Model, or MSSM.
Methods: Supercharge algebra, superfields, Grassmann numbers, Berezin integral.
Results: Supersymmetric transformations are global, they do not depend on spacetime coordinates. In the case of Supergravity, they are local. Conclusion: Supersymmetric models, and MSSM in particular, could describe more physics and more particles beyond the Standard Model. Key words: supersymmetry, minimal supersymmetric standard sodel.
Supersymmetry
Supersymmetry - or SUSY for short - is a symmetry that interchanges bosons with fermions (Gervais & Sakita, 1971; Volkov & Akulov, 1972, 1973, 1974; Ramond, 1971). It is one of the best candidates for physics beyond the Standard Model, with the so-called Minimal Supersymmetric Standard Model, MSSM for short (Fayet, 1975, 1976, 1977; Fayet & Ferrara, 1977). In this extension to the Standard Model, each particle has its corresponding superpartner with same mass and other quantum numbers, but with different spin by one half. For instance, the electron has a selec-tron, a bosonic superpartner, while photon, Z and gluon have fermionic superpartners called photino, zino and gluino respectively. Names are as-
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signed in following manner: bosonic superpartners gain the s- prefix, while fermionic superpartners gain the -ino suffix.
As we easily distinguish between bosons and fermions at current energies, this symmetry has to be spontaneously broken. After supersymmetry breaking superpartners masses may differ.
Supersymmetry algebra
A supersymmetric transformation brings a scalar to a fermion and viceversa. The generator Qa of the transformation, known as the supercharge, brings from the boson field 0 to the Weyl spinor 0a.
In order to describe supersymmetry transformation it is better to operate with Weyl spinors rather than Dirac ones, and to change usual notation a bit. Weyl spinors transform under Lorentz group as
0L,R(x) ^ 0l,R(x') = AL,R0L,R(x)
(1)
where the transformations KL, R are given by
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x
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Al , R = exp ( 2a • (ü ^ iv)
a being the three Pauli matrices
ai
0 1 1 0
a2
0 -i i0
, a3
10 01
(2)
(3)
w the three real rotational parameters, and V the three real boost parameters. The transformations KL and KR are related by
A-1 = AR .
(4)
The four component Dirac spinor ^ is written as follows:
equivalent to
* = № • <5>
* = (X i. (6)
Often, it is found that Weyl spinors are written starting from Dirac spinors in the following way, as projections, see for instance (Fabiano, 2021):
^L,R = 1(1 ± 75)^ •
(7)
In this formalism we have the Weyl representation of the Dirac matrices
7'
, —
0
0
(8)
where = (1,ai) = (a№)al3 and = (1, -ai) = (a№)al = . The matrices and mix dotted and undotted indices, that is left and right spinor indices. The dot in the notation is similar to the covariant and con-travariant - or upper and lower indices in general and special relativity. We always contract an upper with a lower index, and we have the additional rule that only dotted or undotted indices can be contracted together, not mixed dotted-undotted.
The generator Qa transforms as a Weyl spinor, that means
[J,v, Qa] = -i(a№V)lQi , (9)
where Jis the generator of Lorentz group, while
_ - avand a,v _ ~(a№av - ava№) . (10)
4 ' 4
Qa is independent from spacetime coordinates, then
[P№,Qa]=0. (11)
Denoting the conjugate of Qa as Qa we have
[J,vQ] = -i(a,v)aQ • (12)
The conjugation of spinors works as follows:
(rr = T, and (^ay = ^ • (13)
The supersymmetry algebra is given by the anticommutator
{Qa,Q$} = 2(a,)JlP, . (14)
How to justify (14)? By inspection, the rhs of the expression should carry indices a and /3. The simplest object that carries those two indices is ,
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which also carries a Lorentz index. The latter has to be contracted with a vector index, that is PM, the generator of translations. The factor of 2 is for normalisation.
By the same line of reasoning we should have {Qa,Ql3} = ^ c1(a^v)iJnV + c2bi, where c1 and c2 are constants. Commuting with PA
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results that ci = 0. Since we have that Qa = eapQp where e01 = 1, we obtain {Qa, Qp} = c2eap, but because lhs is symmetric in a and / we de-^ duce that c2 = 0. The complete supersymmetry algebra is therefore given
o o
by relations:
< {Qa,Q p} =2K)cf P, (15)
[Qa,Qp } = 0 (16)
[Qà,Qp} = 0 (17)
| [QJ ] = 2K)cf Qn (18)
[Qa J ] = - 2 Qp (^v )t (19)
[Qa, PJ = 0 (20)
§ [QaP]=0 . (21)
y There is an immediate physical result coming from eq. (14): when contracted with (a)&a one has
4Pv = (a)la{Qa, Qp} ■ (22)
The first component P0 is the Hamiltonian, so
4H = Y,{Qa,Qp} = T,{Qa, Ql} = E(QaQl + QaQa) , (23)
a
because the conjugate of Qa is Q^. It is clear that H is non negative definite (the value 0 is admitted), so in supersymmetric theory any physical state \S) has a non negative energy:
(S \H\S) = 1 EElSlQllS)!2 > 0 . (24)
a S'
Superspace and superfields
From the relation (14), that could be also written as
{Qa,Ql} = 2(a№)aaP, ,
(25)
we could construct a so-called superspace, an abstract space with both bosonic and fermionic coordinates. The fermionic coordinates should be constructed with the aid of Grassmann numbers (Grassmann, 1844). Those are anticommuting numbers 9i that commute with ordinary numbers
x:
It is clear that
didj = —djdi , and dix = xdi .
i)2 = 0 because didi = —didi
(26)
(27)
and therefore any function defined on Grassmann numbers has at most a constant and a linear term: f (9) = a + b9, where a, b are ordinary numbers. It is also easy to find out that a product of two Grassmann numbers, that is 9i9j, obeys to Bose-Fermi statistics.
The integral over Grassmann functions is called Berezin integral (Berezin, 1966). It is a sort of integral over fermionic variables, and can be determined by asking basic properties of ordinal integration like linearity
I'd9 [af (9) + bg(9)] = a I'dO f (9) + b I'dO g(9) (28)
translational invariance
ddf (d + d') = j ddf (d) ,
and partial integration formula
dd
d
dddf (d)
0.
(29)
(30)
Starting from formula (29), and the fact that any function of Grassmann numbers is linear one finds out that
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dd(a + be + be') = j de(a + be)
a j de + b J dee + be' J de = a J de + b J dee providing the result
J de = 0 .
This leads to very simple integration rules:
and
de 1 = 0,
de e = 1,
(31)
(32)
(33)
(34)
(35)
the latter formula is a matter of convention for normalisation, used originally by Berezin.
Formally, the polynomials constructed by n Grassmann variables ei,...,en, form the Grassmann algebra Gn. The Grassmann algebra uses the wedge product as multiplication, being anticommutative and associative, which is similar to the more familiar cross product of two vectors. The Berezin integral on Gn is defined as a linear functional having the following properties:
[ deeu...,en = 1, (36)
and
' Gn
Ldef=°''=1
.n ;
(37)
it could be shown that Berezin integral is the only possible functional with the above mentioned properties.
The superspace is formed from bosonic and fermionic coordinates,
[x^,ea,eß}. Here ea is a two component left handed Weyl spinor
(e(0),e(1)), eß a two component right handed Weyl spinor (0(o),0(1)). We have a total of 4 bosonic dimensions and 4 Grassmann dimensions. This combination of ordinary coordinates x^ and one couple of Weyl spinors
{ea,eß} is called N = 1 supersymmetric space.
The basic relation (14) for supersymmetry generators shows that the application of two consecutive transformations leads to P», a translation in 1 ordinary bosonic space. So we expect that operators Qa and Q generate ^
-0 ^ a translation in superspace, respectively in Qa and Q Grassmann coordinates. The supercharges are explicitly represented by ¡=
E
_ d ■( ß\ Qa _ dQa - %(a )ai
and
These expressions satisfy the anticommutation relation (14). Notice that
the "naive" guess Qa = d/d6a and Q = -d/dd3 would give 0 in the anticommutator, thus not satisfying (14).
A superfield is a function defined on superspace: $(x^,6a,63). An infinitesimal supersymmetric transformation acts on $ in the following manner
$ - $ = 5$ = i(£aQa + l.Qa)$ , (40)
where £ and £ are Grassmann variables. Looking at the explicit form of the supersymmetric generators in eqs. (38)-(39), we could define also another set of independent generators:
= JQa + H^a Qa
and
Da _ öQä + i(^)aaOadß (41)
_ d
D $ _ -— - Q(v») ßßd» • (42)
ödp
The D set is "orthogonal" to the Q set: in fact Da and D ^ anticommute with Qa and Q. So if we impose the condition D^$ _ 0 because of (40) also Dp_ 0 holds true. Such superfields are called chiral superfields. An analogy in the ordinary R2 space is the following: find a function f (x, y) such that [x(d/dy) - y(d/dx)] f (x,y) _ 0. Defining r = (x2 + y2)1/2 any f (r) will satisfy this condition.
> CO
Qa _ dQa - ^ß)aaQ (38) *
iß
_ d
Qß _ -—^ + Qv) ßfA ■ (39) s
ödp
(M <D
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Now if we call y. = (x. + %9a(a.)aa9a), which is a bosonic variable as x. and 99' are such, the application of D^ gives
> D ß y" =
CM
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--T xß - %-T
d9ß d9ß
f)
--j - %9ß K) ßßdv
d9
(x. + %9a(a.)aàr ) = 9a(a.)aà9a - %9ß(av) ßdx*+
o e1 K)ea(^Uea =
< 0 + iea(anali -e(*%j + 0 = 0 . (43)
x A chiral superfield depends only on a two component spinor beyond the
w bosonic coordinate, $(y, e). Because of that and of Grassmann numbers
^ properties we can form an object with at most two powers of e, that is ee.
ft Upper powers of e vanish, so for a chiral superfield we have
$(y, e) = 0(y) + V2e^(y) + eeF(y), (44)
where $ and F are complex scalar fields, 0 a Weyl spinor. Expanding with
u Taylor (44) around x we obtain
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$(y, 9) = 0(x) + V290(x) + 99F(y) +
b %9aß9dß^(x) - 19a^99av9dßdv0(x) + V29%9a»9d^(x) . (45)
The conjugate of a chiral superfield, will satisfy the relation
Da$ = 0 , (46)
and is called antichiral superfield.
The most general Lagrangian with two derivatives constructed from chiral superfields is written as
L = K+ W+ W($i)\r , (47)
where the subscripts denote the coefficients in power expansion of 9 and 9. K is a real function, the Kahler potential (Kahler, 1933), W is a holomor-phic function, the superpotential. We have the relations of integration over
Grassmann coordinates
and
a
d2d d2 0K = K (48) §
_ ——2 —
k = m , w = — $2 + -, (50)
' 2 3 7 v '
where parameters — and - are real. Plugging in the expansion of the su-perfield (45) we have
l = I d2e d2e $$ + I d2e ( — $2 + -$3 ) +
obtaining
2 3
(
23
Wm_ + A, (51)
__mm _ _2___
L = FF + m^F + A02F - y^ - A^ + m^F + AfF - y
+ derivative terms acting on 0 and f . (52)
The F field can be eliminated from equation of motion which reads
dL ^ dL dL — , , ,2
dF - d* d[dPF) =°= dF = F + m0 + ■ (53)
So we obtain two equations
F = -m0 - X02 , F = -m0 - X02 . (54)
Substituting this expressions for F and F into eq. (52) we eventually obtain
_ ___2 _2
L = -d„ w0 + i^d^ -- (f2 + f) - \0f2 - w -
<u
Jd26W = W\d2 . (49) I
The Lagrangian (47) is renormalisable if and only if K is quadratic and W is at most a cubic function. ^
A standard example of a supersymmetric model is the Wess-Zumino model (Wess & Zumino, 1974), the first known supersymmetric interacting model. It is defined as
<d
\m$ - \4>2\2 . (55)
This is the Lagrange for theory of a massive complex scalar coupled with Yukawa interactions to a massive two component Weyl spinor 0. As > we have already discussed, a consequence of supersymmetry is that the scalar and spinor masses are equal.
Supersymmetry transformations we have seen are global. When they
o
yy are combined with general relativity and become local, we have supergrav-
on
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8 & Likhtman, 1971; Nath & Arnowitt, 1975; Gol'fand & Likhtman, 1989). < This theory has become less popular after meeting various shortcomings, among others having an unrealistically large cosmological constant and o gauge anomalies (Fabiano, 2022).
There are many existing supersymmetric theories. I n particle physics, dc the MSSM has been extensively searched for, in particular at Cern with the Large Hadron Collider - LHC - for years, without even a hint of new observed phenomena. The lack of experimental evidence, united to the absence of indications for a new energy scale beyond the electroweak model w to search for, have ruled out some supersymmetric extensions to the Stan-^ dard Model, and, albeit supersymmetry has not been completely excluded S2 as a theory, those facts are responsible for the decline towards the interest in the subject.
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Berezin, F.A. 1966. The Method of Second Quantization, 1st edition. New York, London: Academic press. ISBN-13: 978-0120894505.
Fabiano, N. 2022. Anomalies in quantum field theories. Vojnotehnicki glas-nik/Military Technical Courier, 71(1), pp.100-112. Available at: https://doi.org/10.5937/vojtehg71-38164.
Fabiano, N. 2021. Quantum electrodynamics divergencies. Vojnotehnicki glas-nik/Military Technical Courier, 69(3), pp.656-675. Available at: https://doi.org/10.5937/vojtehg69-30366.
Fayet, P. 1975. Supergauge invariant extension of the Higgs mechanism and a model for the electron and its neutrino. Nuclear Physics B, 90, pp.104-124. Available at: https://doi.org/10.1016/0550-3213(75)90636-7.
Fayet, P. 1977. Spontaneously broken supersymmetric theories of weak, electromagnetic and strong interactions. Physics Letters B, 69(4), pp.489-494. Available at: https://doi.org/10.1016/0370-2693(77)90852-8.
References
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Fayet, P. 1976. Supersymmetry and weak, electromagnetic and strong interactions. Physics Letters B Volume 64, Issue 2, Pages 159-162. Available at: https://doi.org/10.1016/0370-2693(76)90319-1.
Fayet, P. & Ferrara, S. 1977. Supersymmetry. Physics Reports, 32(5), pp.249334. Available at: https://doi.org/10.1016/0370-1573(77)90066-7.
Gervais, J.-L. & Sakita, B. 1971. Field theory interpretation of supergauges in dual models. Nuclear Physics B, 34(2), pp.632-639. Available at: https://doi.org/10.1016/0550-3213(71)90351-8.
Gol'fand, Yu.A. & Likhtman, E.P. 1971. Extension of the algebra of the Poincare group generators and violation of P invariance. JETP Letters, 13(8), pp.452-455 (in Russian) [online]. Available at: http://jetpletters.ru/ps/717/article_11110.sht ml [Accessed: 20 September 2022]. (In the original: Гольфанд Ю.А. и Лихтман Е.П. 1971. Расширение алгебры генераторов группы Пуанкаре и нарушение Р-инвериантности. Письма вЖЭТФ, 13(8), стр.452-455 [онлайн]. Доступно на: http://jetpletters.ru/ps/717/article_11110.shtml [Дата обращения: 20 Сентябрь 2022].)
Gol'fand, Yu.A. & Likhtman, E.P. 1989. Extension of the algebra of Poincaré group generators and violation of p invariance. In: Salam, A. & Sezgin, E. (Eds.) Supergravities in Diverse Dimensions Commentary and Reprints (In 2 Volumes). Singapore: World Scientific. Available at: https://doi.org/10.1142/9789814542340_0001.
Grassmann, H. 1844. Die Lineale Ausdehnungslehre - Ein neuer Zweig der Mathematik (in German). Leipzig: Verlag von Otto Wigand [online]. Available at: https://gdz.sub.uni-goettingen.de/id/PPN534901565 [Accessed: 20 September 2022].
Kähler, E. 1933. Über eine bemerkenswerte Hermitesche Metrik. Abh.Math.Semin.Univ.Hambg., 9, pp.173-186. Available at: https://doi.org/10.1007/BF02940642.
Nath, P. & Arnowitt, R. 1975. Generalized super-gauge symmetry as a new framework for unified gauge theories. Physics Letters B, 56(2), pp.177-180. Available at: https://doi.org/10.1016/0370-2693(75)90297-X.
Ramond, P. 1971. Dual Theory for Free Fermions. Physical Review D, 3(10), pp.2415-2418. Available at: https://doi.org/10.1103/PhysRevD.3.2415.
Volkov, D.V. & Akulov, V.P. 1972. Possible Universal Neutrino Interaction. JETP Letters, 16(11), pp.438-440 [online]. Available at:
http://jetpletters.ru/ps/1766/article_26864.shtml [Accessed: 20 September 2022].
Volkov, D.V. & Akulov, V.P. 1973. Is the neutrino a goldstone particle? Physics Letters B, 46(1), pp.109-110. Available at: https://doi.org/10.1016/0370-2693(73)90490-5.
Volkov, D.V. & Akulov, V.P. 1974. Goldstone fields with a spin one half. Teor. Mat. Fiz., 18(1), pp.39-50 (in Russian).
LT) ^
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£± cp
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<u cp
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ф Volkov, D.V. & Soroka, V.A. 1973. Higgs effect for Goldstone particles with spin
1/2. JETP Letters, 18(8), pp.529-532 [online]. Available at: http://jetpletters.ru/ps/1568/article_24038.shtml [Accessed: 20 September 2022]. Wess, J. & Zumino, B. 1974. Supergauge transformations in four dimensions. > Nuclear Physics B, 70(1), pp.39-50. Available at: CO https://doi.org/10.1016/0550-3213(74)90355-1.
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О Никола Фабиано
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Суперсимметрия
< Белградский университет, Институт ядерных исследований
° «Винча» - Институт государственного значения для Республики
х Сербия, г. Белград, Республика Сербия
ш
РУБРИКА ГРНТИ: 29.05.03 Математические методы
^ теоретической физики,
¡< 29.05.23 Релятивистская квантовая теория.
Квантовая теория поля 29.05.33 Электромагнитное взаимодействие ВИД СТАТЬИ: обзорная статья
Резюме:
о
— Введение/цель: Суперсимметрия — это симметрия
>о лагранжиана, выходящая за пределы Группы Ли. Это
позволяет обмениваться бозонами и фермионами. ^ Большинство важной моделью является минимальная
о суперсимметричная стандартная модель или MSSM.
^ Методы: Алгебра суперзарядов, суперполя, числа Гоассма-
на, интеграл Березина.
Результаты: Суперсимметричные преобразования глобальны, они не зависят от координат пространства-времени. В случае Супергравитации они локальны.
Выводы: Суперсимметричные модели и, в частности, МSSМ могли бы описывать больше физики. и больше частиц за пределами Стандартной модели.
Ключевые слова: суперсимметрия, минимальная суперсимметричная стандартная модель.
Суперсиметри]а
Никола Фабиано
Универзитет у Београду, Институт за нуклеарне науке "Винча"-Институт од националног знача]а за Републику Срби]у, Београд, Република Срби]а
ОБЛАСТ: физика
КАТЕГОРИJА (ТИП) ЧЛАНКА: прегледни рад Сажетак:
Увод/цил>: Суперсиметри]а }е симетри]а Лагранжиана ко}а у опису симетри]а иде даъе од Ли}евих група. Суперсиме-три]а омогуЬава размену бозона и фермиона. На]важни]и модел }е минимални суперсиметрични стандардни модел, или МSSМ.
Методе: Алгебра супернабо}а, суперпоъа, Грасманови бро]еви, интеграл Березина.
Резултати: Суперсиметричне трансформаци]е су глобал-не, не зависе од просторно-временских координата. У слу-ча]у супергравитаци}е, оне су локалне.
Закъучак: Суперсиметрични модели, а посебно МSSМ, могли би унапредити опис физике честица у односу на стан-дардни модел.
Къучне речи: суперсиметри]а, минимални суперсиметрич-ни стандардни модел.
Paper received on / Дата получения работы / Датум приема чланка: 13.09.2022. Manuscript corrections submitted on / Дата получения исправленной версии работы / Датум достав^а^а исправки рукописа: 25. 03. 2023.
Paper accepted for publishing on / Дата окончательного согласования работы / Датум коначног прихвата^а чланка за об]ав^ива^е: 27. 03. 2023.
© 2023 The Authors. Published by Vojnotehnicki glasnik/Military Technical Courier (http://vtg.mod.gov.rs, http://втг.мо.упр.срб). This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/rs/).
© 2023 Авторы. Опубликовано в "Военно-технический вестник / Vojnotehnicki glasnik / Military Technical Courier" (http://vtg.mod.gov.rs, http://втг.мо.упр.срб). Данная статья в открытом доступе и распространяется в соответствии с лицензией "Creative Commons" (http://creativecommons.org/licenses/by/3.0/rs/).
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