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Extended Akulov-Volkov superfield theory
Volume 174, number 3
EXTENDED AKULOV-VOLKOV
PHYSICS LETTERS B
10 July 1986
SUPERFIELD THEORY
E.S. K A N D E L A K I S Institut far Theoretische Physik, Universitgu Karlsruhe, D-7500 Karlsruhe, Fed. Rep. Germany Received 15 April 1986
The superfield description, given by Samuel and Wess. of the N = 1 non-linear Akulov-Volkov theory is generalized in the case of N = 2 and N = 4 extended supersymmetry. The interacting lagrangian is also presented.
Supersymmetry [11 ,a gives an equal status to fermions and bosons. The extended SUSY provides in addition the non-trivial mixing of SUSY, represented by the spinor generators Q/, ~g&i (i = 1,..., N ) , and internal symmetry, represented by the group generators T a [21:
[Qia, Tal = iDi/O/a~,-~ =/: O. A second theorem [3] determines the spectrum of the extended super Yang-Mills theories: a characteristic property is that the photons are accompanied among others also by spinor fields, X/(x), the photinos. It is a common belief that SUSY should be broken at low energy, because no mass degeneracy between fermions and bosons is observed. By the spontaneous breaking of the global SUSY in N = 1 [4,1 ], that photinos combine with the matter fields and Goldstone spinors appear, the so-called Akulov-Volkov ( A - V ) fields, denoted also by X~(x). They transform inhomogeneously and non-linearly and they are massless. The same effect is expected in the extended SUSY: by a known argument [5], if one SUSY is broken, then all of them must automatically be broken by the same amount. Correspondingly N Goldstone fields X/(x) will appear. We describe their transformation and their lagrangian and most importantly we extend [6] successfully the superfield formulation of Samuel and Wess ( S - W ) [7, 8] known till now only for N = 1. We present below, in some detail, the essential content of our work.
One can first convince oneself that the results of the N = 1 A - V theory, in components, apply for every N. One must only add an index i to X~(x) and repeat the theory, for example as in ref. [1]. Compare also ref. [9]. We write them below, but in the new representation of S-W: Transformation law: 6~. X~(x) = (l/K) ~'~ -- 2iK X(x) orn~o m Xt(x ). Closure property: (~5~- -- ~-~r/) ~t~(x) : --2i(r/om (--fomr?-) Om )tt(x)" Lagrangian: ~?= 1/2K 2 +~i[X(x) om3m X(x) - O m X(x) am X(x)] + Interactions. In the constraint-like superfield formulation of the A - V theory by S-W, one creates a superfield A 7 from the field X~(x) and proves that it satisfies the following defining constraints [10] : s Dt~Ai
=
-. a = - 2 i ~ A ~ju m~S m A t • D~iAi The superfield is created in the usual (linear [1] )way: A?(x, 0, 0) - e ~ X X?(x), A X X~'(x)= (l/x)O 7 - 2 i g X(x)amOOm Xt(x ). Notice the crucial points, that
,1 We follow the notation of ref. [1].
(O~', Dfis} A 7 = - 2 i g 6fomOm /¢ g/J A t 301
0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 174, number 3
PHYSICS LETTERS B
10 July 1986
and that
Constructing its superfield lagrangian
8A~ = (~'D + ~D)A~ = ~'~(1/K) - 2 i t A/3..o~flJarn A~,
L--eA X~0,
which follow from the constraints. Our aim is to generalize t h e N = 1 lagrangian [7]
-e 0 = - -~ a m C(x)
1
am
C(x),
we can prove that ~N=I-
~f
d 2 0 d 2 0 A2A2
to the extended case. It turns out that the generalization to N = 2 is K6 ~-
"~N---2 -
fd4Od40 A4A 4,
where
=
f d2NOd2N0
(L + c.c.)
= "Qo + K2i(k°nam ~ - a m k ° n ~ ) Tmn + "'" for N = 2, 4. The proof is again direct and technical [6]. We notice that the N = 2 extended supersymmetry has been broken down to the N = 1, by using the N = 1 formulation, in ref. [12].
A4 = A , A o, fl A i / = A c~ i A/earl,
AiJ = A k l e k i el].
For the proof one first defines consistently the N = 2 0-algebra and makes in a technical way the calculation. The N = 4 S - W lagrangian, direct generalization of those given before f o r N = 1 and 2, is K14 - 8 f2N= 4 = -- --~--Jd 0 d80 m8m 8, where A 8 = A4
i~<4.1.),
A4 = A2a# A276 (kl) eoL,,te36, A2e0 = A~A~lei/kl. The proof follows the same lines as in N = 2, but now one has first to define the N = 4 0-algebra and its identities. One can now repeat the calculation of ref. [8] and find again the coupling with the e n e r g y - m o m e n t u m tensor and study the phenomenological aspects, as in N = 1 [11]. Namely, if a standard superfield C is constructed in the usual way
C(x, o, ~)- e" x C(x) A X c(x) = --2ixomO~m c(x) one [10] can prove its constraints: D i c = 0,
~iC= 302
m - 2 1 A• oi oodam C.
The author wishes to thank Professor J. Wess for invaluable support and Dr. W. Lang for many very useful discussions.
Refer en ces [ 1] J. Wess and J. Bagger, Supersymmetry and supergravity (Princeton U.P., Princeton, NJ, 1983). [2] R. Haag, J. Lopuszanski and M. Sohnius, Nucl. Phys. B88 (1975) 207. [3] A. Salam and J. Strathdee, Nucl. Phys. B80 (1974) 449; S. Ferrara, in: Supergravity '81, Proc. (Trieste, 1981) (Cambridge U.P., Cambridge, 1982). [4] P. Fayet and J. llliopoulos, Phys. Lett. B51 (1974) 461; V.P. Akulov and D.V. Volkov, Phys. Lett. B46 (1973) 109; M. Ro~ek, Phys. Rev. Lett. 41 (1978) 451; E. Ivanov and A. Kapustnikov, J. Phys. A l l (1978) 2375; J. Phys. G8 (1982) 167. [5] E. Witten, Nucl. Phys. B188 (1981) 513; B. Zumino, preprint UCB-PTH-83/2. [6] E.S. Kandelakis, Doktorarbeit, University of Karlsruhe (1984). [7] S. Samuel and J. Wess, Nucl. Phys. B221 (1983) 53; B226 (1983) 289. [8] J. Wess, Karlsruhe preprint (December 1982); Karlsruhe preprint, Bonn lectures (1983). [9] W. Bardeen and V. Visnjic, Nucl. Phys. B194 (1982) 422; T. Uematsu and C. Zachos, Nucl. Phys. B201 (1982) 250; S. Ferrara, L. Maiani and P.C. West, Z. Phys. 19 (1983) 267; S. Ferrara, CERN preprint TH. 3514 (1983). [10] J. Wess, Karlsruhe preprint (December 1983). [ 11] J. Wess, Karlsruhe preprint KA-THEP 84-1; O. Nachtmann and M. Wirbel, Heidelberg preprint HDTHEP 83-13; W. Banzhaf and E. Diegele, Kaxlsruhe preprint KA-THEP 84-2. [12] J. Bagger and J. Wess, preprint SLAC-PUB-3255 (1983).
EXTENDED AKULOV-VOLKOV
PHYSICS LETTERS B
10 July 1986
SUPERFIELD THEORY
E.S. K A N D E L A K I S Institut far Theoretische Physik, Universitgu Karlsruhe, D-7500 Karlsruhe, Fed. Rep. Germany Received 15 April 1986
The superfield description, given by Samuel and Wess. of the N = 1 non-linear Akulov-Volkov theory is generalized in the case of N = 2 and N = 4 extended supersymmetry. The interacting lagrangian is also presented.
Supersymmetry [11 ,a gives an equal status to fermions and bosons. The extended SUSY provides in addition the non-trivial mixing of SUSY, represented by the spinor generators Q/, ~g&i (i = 1,..., N ) , and internal symmetry, represented by the group generators T a [21:
[Qia, Tal = iDi/O/a~,-~ =/: O. A second theorem [3] determines the spectrum of the extended super Yang-Mills theories: a characteristic property is that the photons are accompanied among others also by spinor fields, X/(x), the photinos. It is a common belief that SUSY should be broken at low energy, because no mass degeneracy between fermions and bosons is observed. By the spontaneous breaking of the global SUSY in N = 1 [4,1 ], that photinos combine with the matter fields and Goldstone spinors appear, the so-called Akulov-Volkov ( A - V ) fields, denoted also by X~(x). They transform inhomogeneously and non-linearly and they are massless. The same effect is expected in the extended SUSY: by a known argument [5], if one SUSY is broken, then all of them must automatically be broken by the same amount. Correspondingly N Goldstone fields X/(x) will appear. We describe their transformation and their lagrangian and most importantly we extend [6] successfully the superfield formulation of Samuel and Wess ( S - W ) [7, 8] known till now only for N = 1. We present below, in some detail, the essential content of our work.
One can first convince oneself that the results of the N = 1 A - V theory, in components, apply for every N. One must only add an index i to X~(x) and repeat the theory, for example as in ref. [1]. Compare also ref. [9]. We write them below, but in the new representation of S-W: Transformation law: 6~. X~(x) = (l/K) ~'~ -- 2iK X(x) orn~o m Xt(x ). Closure property: (~5~- -- ~-~r/) ~t~(x) : --2i(r/om (--fomr?-) Om )tt(x)" Lagrangian: ~?= 1/2K 2 +~i[X(x) om3m X(x) - O m X(x) am X(x)] + Interactions. In the constraint-like superfield formulation of the A - V theory by S-W, one creates a superfield A 7 from the field X~(x) and proves that it satisfies the following defining constraints [10] : s Dt~Ai
=
-. a = - 2 i ~ A ~ju m~S m A t • D~iAi The superfield is created in the usual (linear [1] )way: A?(x, 0, 0) - e ~ X X?(x), A X X~'(x)= (l/x)O 7 - 2 i g X(x)amOOm Xt(x ). Notice the crucial points, that
,1 We follow the notation of ref. [1].
(O~', Dfis} A 7 = - 2 i g 6fomOm /¢ g/J A t 301
0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 174, number 3
PHYSICS LETTERS B
10 July 1986
and that
Constructing its superfield lagrangian
8A~ = (~'D + ~D)A~ = ~'~(1/K) - 2 i t A/3..o~flJarn A~,
L--eA X~0,
which follow from the constraints. Our aim is to generalize t h e N = 1 lagrangian [7]
-e 0 = - -~ a m C(x)
1
am
C(x),
we can prove that ~N=I-
~f
d 2 0 d 2 0 A2A2
to the extended case. It turns out that the generalization to N = 2 is K6 ~-
"~N---2 -
fd4Od40 A4A 4,
where
=
f d2NOd2N0
(L + c.c.)
= "Qo + K2i(k°nam ~ - a m k ° n ~ ) Tmn + "'" for N = 2, 4. The proof is again direct and technical [6]. We notice that the N = 2 extended supersymmetry has been broken down to the N = 1, by using the N = 1 formulation, in ref. [12].
A4 = A , A o, fl A i / = A c~ i A/earl,
AiJ = A k l e k i el].
For the proof one first defines consistently the N = 2 0-algebra and makes in a technical way the calculation. The N = 4 S - W lagrangian, direct generalization of those given before f o r N = 1 and 2, is K14 - 8 f2N= 4 = -- --~--Jd 0 d80 m8m 8, where A 8 = A4
i~<4.1.)
A4
C(x, o, ~)- e" x C(x) A X c(x) = --2ixomO~m c(x) one [10] can prove its constraints: D i c = 0,
~iC= 302
m - 2 1 A• oi oodam C.
The author wishes to thank Professor J. Wess for invaluable support and Dr. W. Lang for many very useful discussions.
Refer en ces [ 1] J. Wess and J. Bagger, Supersymmetry and supergravity (Princeton U.P., Princeton, NJ, 1983). [2] R. Haag, J. Lopuszanski and M. Sohnius, Nucl. Phys. B88 (1975) 207. [3] A. Salam and J. Strathdee, Nucl. Phys. B80 (1974) 449; S. Ferrara, in: Supergravity '81, Proc. (Trieste, 1981) (Cambridge U.P., Cambridge, 1982). [4] P. Fayet and J. llliopoulos, Phys. Lett. B51 (1974) 461; V.P. Akulov and D.V. Volkov, Phys. Lett. B46 (1973) 109; M. Ro~ek, Phys. Rev. Lett. 41 (1978) 451; E. Ivanov and A. Kapustnikov, J. Phys. A l l (1978) 2375; J. Phys. G8 (1982) 167. [5] E. Witten, Nucl. Phys. B188 (1981) 513; B. Zumino, preprint UCB-PTH-83/2. [6] E.S. Kandelakis, Doktorarbeit, University of Karlsruhe (1984). [7] S. Samuel and J. Wess, Nucl. Phys. B221 (1983) 53; B226 (1983) 289. [8] J. Wess, Karlsruhe preprint (December 1982); Karlsruhe preprint, Bonn lectures (1983). [9] W. Bardeen and V. Visnjic, Nucl. Phys. B194 (1982) 422; T. Uematsu and C. Zachos, Nucl. Phys. B201 (1982) 250; S. Ferrara, L. Maiani and P.C. West, Z. Phys. 19 (1983) 267; S. Ferrara, CERN preprint TH. 3514 (1983). [10] J. Wess, Karlsruhe preprint (December 1983). [ 11] J. Wess, Karlsruhe preprint KA-THEP 84-1; O. Nachtmann and M. Wirbel, Heidelberg preprint HDTHEP 83-13; W. Banzhaf and E. Diegele, Kaxlsruhe preprint KA-THEP 84-2. [12] J. Bagger and J. Wess, preprint SLAC-PUB-3255 (1983).