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On off-shell supermultiplets
Volume 105B, number 4
ON OFF-SHELL
PHYSICS LETTERS
8 October 1981
SUPERMULTIPLETS
W. SIEGEL 1 California Institute of Technology, Pasadena, CA 91125, USA and
M. ROEEI,: 2 DAMTP, University of Cambridge, Cambridge CB3 9EW, England
Reveived 20 February 1981
Applying a simple counting argument to all supermultiplets, we find that for the N = 4 super Yang-Mills theory the auxiliary field problem cannot have a solution within any previouslyknown framework. We propose alternatives.
Auxiliary fields (non-propagating fields appearing in the lagrangian) are important in supersymmetry for three reasons: (1) Construction of interacting lagrangians. (2) Linear off-shell representations of supersymmetry algebras, essential for the application of Ward identities, e.g., to prove renormalizability. (3) Manifestly supersymmetric quantization through the use of supersymmetric gauge fixing and supergraphs. In gauge theories, (2) and (3) require, in addition to physical and auxiliary degrees of freedom, gauge components which are removed by the supersymmetric partners of the gauge parameter. These fields appear in the transformation laws, but not in the lagrangian. The auxiliary fields are known for all representations o f N = 1 (simple)supersymmetry of physical interest. For N = 2 (extended) supersymmetry, they are known for the vector [1 ], tensor [2], and (Poincar6 and Weyl) supergravity [3] multiplets, but not for the complete scalar multiplet. (The existing formulation of the scalar multiplet [4] contains central charges 1 Supported in part by the UK Science Research Council. 2 Work supported in part by the US Department of Energy under Contract No. DE-AC0379ER0068 and by the Fleisehman Foundation.
which do not vanish off-sheU when they do on-shell, and as a result its supersymmetry transformations become nonlinear and coupling-constant dependent when it is coupled to a gauge multiplet even in supersymmetric gauges [5], violating condition (2) above. This indicates that additional auxiliary fields have been elim. inated by their field equations. The same remarks apply to the formulations of extended super YangMills of ref. [6], which furthermore have nonlocal lagrangians and supersymmetry transformations due to the self-interactions [5] .) For highar N, only the N = 4 (and N = 3 by truncation) Weyl supergravity multiplet [7] is known off-shell. We now consider general arguments which restrict the structure of possible auxiliary fields. An arbitrary N-extended superfield in D dimensions can be expressed as the tensor product of a representation of SO (D - 1, 1) ® SO (N) with the smallest superfield (a real scalar superfield), which has dirnensionality 2 d (chiral superfields have lower dimensionality, but they should be viewed as scalar superfields with some transverse gauge components removed). Here d is the number of supersyrmnetry generators, and is N times the dimensionality of the smallest SO (D - 1, 1) spinor. Similarly, an arbitrary linear, massless, off-shell representation of supersymmetry without central charge can be expressed as the product of an SO(D - 1)® SO(N) representation with the smallest off-shell supersym275
Volume 105B, number 4
metry representation, which has dimensionality 2 d/2. This can be seen by noting that an off-shell massless theory hasp 2 4= 0, and thus behaves like an on-shell massive theory. Finally, a massless on-shell representation is the product of an SO(D - 2)® SO(N) representation with the smallest on-shell supersymmetry representation [8 ], with dimensionality 2 d/4 . The on-shell representation contains only physical degrees of freedom, the off-shell representation contains physical and auxiliary, and the superfield contains physical, auxiliary, and gauge. Thus, any off-shell representation of O(N) supersymmetry in four dimensions has an integral multiple of 22N components, and so an integral multiple of 22N- 1 Fermi or Bose components. For even N in D = 4, all off-shell representations of supersymmetry for multiplets of physical interest contain a physical Bose field with an even number of internal symmetry SU(N) indices. [In D = 4, the SO(N) vector indices can be considered as SU(N) spinor indices.] Since the supersymmetry generator has one spinor and one internal symmetry index, all Fermi fields have an odd number of spinor and an odd number of internal symmetry indices (this is not true for odd N, where the N-index totally antisymmetric tensor can convert an even number of internal symmetry indices into an odd number). Thus, every Fermi field is in a spinor representation of both the spacetime and the internal symmetry groups; however all, spinor representations of SU(2n) [and of SO(3) ~ SU(2)] contain an integral multiple of the number of components of the smallest spinor representation, and hence each Fermi field has an integral multiple of d components. Since all spinor auxiliary fields occur in pairs (one as the Lagrange multiplier for the other), the total Fermi dimensionality of the off-shell representation is thus determined modulo 2d( = 8N) by the total dimensionality of the physical Fermi fields. Similar results can be obtained by dimensional reduction from D > 4 [E.g., for N = 4 super Yang-Mills, consideration o f N = 1 in D = 10 gives the same result without the need to consider SO(N) indices.] The compatibility of our first and second conditions on the total dimensionality of off-shell Fermi components thus gives a further restriction on the theory: for even N, the total number of physical + auxiliary degrees of freedom contained in the physical spinor fields (i.e., spinor fields which contain physical, as well as auxiliary, degrees of freedom) is an integral multiple of 276
8 October 1981
PHYSICS LETTERS
Table 1 (physical Fermi fields) = (integer) X 2d/2-1 rood 2d for even N(D = 4).
N
2d = 8N
2
16
4 6 8
32 48 64
2d/2-1 mod 2d
Super Yang-Mills
8--,8
128--,0 2048 ~ 32 32768 --, 0
8
[]
Supergravity 8--,8
32~0 128 ~ 32 256 --, 0
2 d / 2 - 1 mod 2d. In table 1 we have listed the relevant numbers for D = 4 and all even N for the super YangMills and Poincar6 supergravity multiplets. (The N = 2 scalar and tensor multiplets each contain one physical spinor of smallest dimensionality, and thus satisfy the compatibility condition easily.) The physical Fermi fields of super Yang-MiUs contain 4
N+ [(1) (\3/Jll- e-,
components for N < Nmax = 4 (2 all4+ 1 for arbitrary D i> 4), and ~ 2 N+ 1 = 16 for N = Nmax due to selfconjugacy under CPT. For supergravity, we have
413(N]+(N]+ N N 1! \31 (5)+3(7)]
=2N+lm°d8N
for e v e n N < N m a x = 8, and 1 9_ N + l = 256 mod 64
for N = 8 (note that each spin 3/2 field contributes 3 times as many components as a spin 1[2 field). The two counts of Fermi components are thus consistent for all theories except N = 4 super Y a n g - ~ l l s (and for N = Nmax super Yang-Mills for any D). Note that even this theory would give consistent counts if the CPT self-conjugacy condition were not applied. This inconsistency may imply a similar problem for all other'supersymmetric theories for which auxiliary fields are not yet known: the N = 2 scalar multiplet, since it may combine with the known N = 2 super Yang-Mills to give N = 4 super Yang-Mills; N = 3 super Yang-Mills, which is identical to N = 4 on-shell and may also be so off-shell;N= 3 or 4 Poincar6 supergravity, which may contain super Yang-Mflls as a compensating multiplet;W > 4 Poincar6 supergravity, which contains N -- 4 supergravity as a submultiplet.
Volume 105B, number 4
PHYSICS LETTERS
The following are alternatives to the usual auxiliary field formalisms which might solve this counting problem in N = 4 super Yang-Mills: (1) The fields might form a nonlinear realization of supersymmetry [9]. However, this would cause difficulties in quantization due to violation of conditions (2) and (3) discussed above. (2) We might alter the algebra b y either allowing the introduction of central charges which act only on irreducible submultiplets containing no physical fields (central charges acting on gauge fields [6] introduce nonlocalities into the lagrangian and supersymmetry transformations), or assuming the four supersymmetry generators do not form a representation of SO(4) [in analogy to D = 3, where this theory has eight supersymmetries, but the seven scalars can only form a representation of SO (7), as expected from reduction from D = 10]. Neither case could result from reduction from D = 10 (or solve the auxiliary field problem there), where the algebra is uniquely determined, and thus SU(4) would be broken off-shell. In the former case, conformal invariance would also be violated off-shell. (3) Spinor auxiliary fields could appear without doubling if they were gauge fields with the same form as their field strengths. (E.g., for odd D, terms of the form 1a~ 1 ...a(D -1 )/2 Fal"''a (D --1 )/2, with gauge transformation / 1-.. o (D-l)/2 =
ataa x!a2...a(D--1)/2]
and field strength F : l "'a(D - 1)/2 = eax ""aD a "a Aa t (O+l)/2 (O+3)/2...ao] i , contain no physical degrees of freedom, where i can be an internal and/or Lorentz index.) However, even if such fields exist, there may be difficulties in making such gauge invariances consistent with the nonabelian Yang-Mills supergauge invariance. (4) Undoubled spinor fields which appear to be propagating at the linearized level might either decoupie at the nonlinearized level or be constrained to vanish by a nonlinear Lagrange multiplier term. In this
8 October 1981
regard, it is interesting to rtote that the on-shell conditions for the superfield strengths of the covariant derivatives, minus the self-duality condition qbab _- ~1 ~~abcd~ Wcd on the scalars, imply the self-duality condition at the nonlinear level through a Bianchi identity which takes the form i[~ab,~cd] =~l~hd] b]
for some hab = hb a [10]. This follows from an identity of the form, suppressing indices on X7a~, and
[iv, v}, iv, v}] = iv, IV,iv, v}]) = {v, [~, (v, ~)] }, and contracting all Weyl spinor indices, while using the identities - ¢+b,
fVa
,
-
8
We thank S. MacDowell for pointing out the triviality of the counting argument for odd N in D = 4. M.R. thanks the Caltech High Energy Physics Group for its hospitality. References
[1 ] J. Wess, Fermi-Bose supersymmetry, in: Trends in elementary particle theory (Bonn, 1974) (Springer, Berlin, 1975) p. 352. [2] J. Wess, Acta Phys. Austtiaea 41 (1975) 409. [3] B. de Wit and J.W. van Holten, Nucl. Phys. B155 (1979) 530;
E.S. Fradkin and M.A. Vasiliev, Nuovo Cimento Lett. 25 (1979) 79. [4] P. Fayet, Nucl. Phys. Bl13 (1976) 135 ; M.F. Sohnius, Nucl. Phys. B138 (1978) 109. [5 ] W. Siegel, Nucl. Phys. B173 (1980) 51. [6] M. Sohnius, K.S. Stelle and P.C. West, Phys. Lett. 92B (1980) 123; Nuel. Phys. B173 (1980) 127. [7] E. Bergshoeff, M. de Roe and B. De Wit, Extended conformal supergravity NIKHEF preprint, NIKHEF -1-1/8007 (November 1980). [8] P. Fayet and S. Ferrara, Phys. Rep. 32C (1977) 249. [9] DN. Volkov and V.P. Akulov, Phys. Lett. 46B (1973) 109. [10] W. Siegel, Nucl. Phys. B177 (1981) 325.
277
ON OFF-SHELL
PHYSICS LETTERS
8 October 1981
SUPERMULTIPLETS
W. SIEGEL 1 California Institute of Technology, Pasadena, CA 91125, USA and
M. ROEEI,: 2 DAMTP, University of Cambridge, Cambridge CB3 9EW, England
Reveived 20 February 1981
Applying a simple counting argument to all supermultiplets, we find that for the N = 4 super Yang-Mills theory the auxiliary field problem cannot have a solution within any previouslyknown framework. We propose alternatives.
Auxiliary fields (non-propagating fields appearing in the lagrangian) are important in supersymmetry for three reasons: (1) Construction of interacting lagrangians. (2) Linear off-shell representations of supersymmetry algebras, essential for the application of Ward identities, e.g., to prove renormalizability. (3) Manifestly supersymmetric quantization through the use of supersymmetric gauge fixing and supergraphs. In gauge theories, (2) and (3) require, in addition to physical and auxiliary degrees of freedom, gauge components which are removed by the supersymmetric partners of the gauge parameter. These fields appear in the transformation laws, but not in the lagrangian. The auxiliary fields are known for all representations o f N = 1 (simple)supersymmetry of physical interest. For N = 2 (extended) supersymmetry, they are known for the vector [1 ], tensor [2], and (Poincar6 and Weyl) supergravity [3] multiplets, but not for the complete scalar multiplet. (The existing formulation of the scalar multiplet [4] contains central charges 1 Supported in part by the UK Science Research Council. 2 Work supported in part by the US Department of Energy under Contract No. DE-AC0379ER0068 and by the Fleisehman Foundation.
which do not vanish off-sheU when they do on-shell, and as a result its supersymmetry transformations become nonlinear and coupling-constant dependent when it is coupled to a gauge multiplet even in supersymmetric gauges [5], violating condition (2) above. This indicates that additional auxiliary fields have been elim. inated by their field equations. The same remarks apply to the formulations of extended super YangMills of ref. [6], which furthermore have nonlocal lagrangians and supersymmetry transformations due to the self-interactions [5] .) For highar N, only the N = 4 (and N = 3 by truncation) Weyl supergravity multiplet [7] is known off-shell. We now consider general arguments which restrict the structure of possible auxiliary fields. An arbitrary N-extended superfield in D dimensions can be expressed as the tensor product of a representation of SO (D - 1, 1) ® SO (N) with the smallest superfield (a real scalar superfield), which has dirnensionality 2 d (chiral superfields have lower dimensionality, but they should be viewed as scalar superfields with some transverse gauge components removed). Here d is the number of supersyrmnetry generators, and is N times the dimensionality of the smallest SO (D - 1, 1) spinor. Similarly, an arbitrary linear, massless, off-shell representation of supersymmetry without central charge can be expressed as the product of an SO(D - 1)® SO(N) representation with the smallest off-shell supersym275
Volume 105B, number 4
metry representation, which has dimensionality 2 d/2. This can be seen by noting that an off-shell massless theory hasp 2 4= 0, and thus behaves like an on-shell massive theory. Finally, a massless on-shell representation is the product of an SO(D - 2)® SO(N) representation with the smallest on-shell supersymmetry representation [8 ], with dimensionality 2 d/4 . The on-shell representation contains only physical degrees of freedom, the off-shell representation contains physical and auxiliary, and the superfield contains physical, auxiliary, and gauge. Thus, any off-shell representation of O(N) supersymmetry in four dimensions has an integral multiple of 22N components, and so an integral multiple of 22N- 1 Fermi or Bose components. For even N in D = 4, all off-shell representations of supersymmetry for multiplets of physical interest contain a physical Bose field with an even number of internal symmetry SU(N) indices. [In D = 4, the SO(N) vector indices can be considered as SU(N) spinor indices.] Since the supersymmetry generator has one spinor and one internal symmetry index, all Fermi fields have an odd number of spinor and an odd number of internal symmetry indices (this is not true for odd N, where the N-index totally antisymmetric tensor can convert an even number of internal symmetry indices into an odd number). Thus, every Fermi field is in a spinor representation of both the spacetime and the internal symmetry groups; however all, spinor representations of SU(2n) [and of SO(3) ~ SU(2)] contain an integral multiple of the number of components of the smallest spinor representation, and hence each Fermi field has an integral multiple of d components. Since all spinor auxiliary fields occur in pairs (one as the Lagrange multiplier for the other), the total Fermi dimensionality of the off-shell representation is thus determined modulo 2d( = 8N) by the total dimensionality of the physical Fermi fields. Similar results can be obtained by dimensional reduction from D > 4 [E.g., for N = 4 super Yang-Mills, consideration o f N = 1 in D = 10 gives the same result without the need to consider SO(N) indices.] The compatibility of our first and second conditions on the total dimensionality of off-shell Fermi components thus gives a further restriction on the theory: for even N, the total number of physical + auxiliary degrees of freedom contained in the physical spinor fields (i.e., spinor fields which contain physical, as well as auxiliary, degrees of freedom) is an integral multiple of 276
8 October 1981
PHYSICS LETTERS
Table 1 (physical Fermi fields) = (integer) X 2d/2-1 rood 2d for even N(D = 4).
N
2d = 8N
2
16
4 6 8
32 48 64
2d/2-1 mod 2d
Super Yang-Mills
8--,8
128--,0 2048 ~ 32 32768 --, 0
8
[]
Supergravity 8--,8
32~0 128 ~ 32 256 --, 0
2 d / 2 - 1 mod 2d. In table 1 we have listed the relevant numbers for D = 4 and all even N for the super YangMills and Poincar6 supergravity multiplets. (The N = 2 scalar and tensor multiplets each contain one physical spinor of smallest dimensionality, and thus satisfy the compatibility condition easily.) The physical Fermi fields of super Yang-MiUs contain 4
N+ [(1) (\3/Jll- e-,
components for N < Nmax = 4 (2 all4+ 1 for arbitrary D i> 4), and ~ 2 N+ 1 = 16 for N = Nmax due to selfconjugacy under CPT. For supergravity, we have
413(N]+(N]+ N N 1! \31 (5)+3(7)]
=2N+lm°d8N
for e v e n N < N m a x = 8, and 1 9_ N + l = 256 mod 64
for N = 8 (note that each spin 3/2 field contributes 3 times as many components as a spin 1[2 field). The two counts of Fermi components are thus consistent for all theories except N = 4 super Y a n g - ~ l l s (and for N = Nmax super Yang-Mills for any D). Note that even this theory would give consistent counts if the CPT self-conjugacy condition were not applied. This inconsistency may imply a similar problem for all other'supersymmetric theories for which auxiliary fields are not yet known: the N = 2 scalar multiplet, since it may combine with the known N = 2 super Yang-Mills to give N = 4 super Yang-Mills; N = 3 super Yang-Mills, which is identical to N = 4 on-shell and may also be so off-shell;N= 3 or 4 Poincar6 supergravity, which may contain super Yang-Mflls as a compensating multiplet;W > 4 Poincar6 supergravity, which contains N -- 4 supergravity as a submultiplet.
Volume 105B, number 4
PHYSICS LETTERS
The following are alternatives to the usual auxiliary field formalisms which might solve this counting problem in N = 4 super Yang-Mills: (1) The fields might form a nonlinear realization of supersymmetry [9]. However, this would cause difficulties in quantization due to violation of conditions (2) and (3) discussed above. (2) We might alter the algebra b y either allowing the introduction of central charges which act only on irreducible submultiplets containing no physical fields (central charges acting on gauge fields [6] introduce nonlocalities into the lagrangian and supersymmetry transformations), or assuming the four supersymmetry generators do not form a representation of SO(4) [in analogy to D = 3, where this theory has eight supersymmetries, but the seven scalars can only form a representation of SO (7), as expected from reduction from D = 10]. Neither case could result from reduction from D = 10 (or solve the auxiliary field problem there), where the algebra is uniquely determined, and thus SU(4) would be broken off-shell. In the former case, conformal invariance would also be violated off-shell. (3) Spinor auxiliary fields could appear without doubling if they were gauge fields with the same form as their field strengths. (E.g., for odd D, terms of the form 1a~ 1 ...a(D -1 )/2 Fal"''a (D --1 )/2, with gauge transformation / 1-.. o (D-l)/2 =
ataa x!a2...a(D--1)/2]
and field strength F : l "'a(D - 1)/2 = eax ""aD a "a Aa t (O+l)/2 (O+3)/2...ao] i , contain no physical degrees of freedom, where i can be an internal and/or Lorentz index.) However, even if such fields exist, there may be difficulties in making such gauge invariances consistent with the nonabelian Yang-Mills supergauge invariance. (4) Undoubled spinor fields which appear to be propagating at the linearized level might either decoupie at the nonlinearized level or be constrained to vanish by a nonlinear Lagrange multiplier term. In this
8 October 1981
regard, it is interesting to rtote that the on-shell conditions for the superfield strengths of the covariant derivatives, minus the self-duality condition qbab _- ~1 ~~abcd~ Wcd on the scalars, imply the self-duality condition at the nonlinear level through a Bianchi identity which takes the form i[~ab,~cd] =~l~hd] b]
for some hab = hb a [10]. This follows from an identity of the form, suppressing indices on X7a~, and
[iv, v}, iv, v}] = iv, IV,iv, v}]) = {v, [~, (v, ~)] }, and contracting all Weyl spinor indices, while using the identities - ¢+b,
fVa
,
-
8
We thank S. MacDowell for pointing out the triviality of the counting argument for odd N in D = 4. M.R. thanks the Caltech High Energy Physics Group for its hospitality. References
[1 ] J. Wess, Fermi-Bose supersymmetry, in: Trends in elementary particle theory (Bonn, 1974) (Springer, Berlin, 1975) p. 352. [2] J. Wess, Acta Phys. Austtiaea 41 (1975) 409. [3] B. de Wit and J.W. van Holten, Nucl. Phys. B155 (1979) 530;
E.S. Fradkin and M.A. Vasiliev, Nuovo Cimento Lett. 25 (1979) 79. [4] P. Fayet, Nucl. Phys. Bl13 (1976) 135 ; M.F. Sohnius, Nucl. Phys. B138 (1978) 109. [5 ] W. Siegel, Nucl. Phys. B173 (1980) 51. [6] M. Sohnius, K.S. Stelle and P.C. West, Phys. Lett. 92B (1980) 123; Nuel. Phys. B173 (1980) 127. [7] E. Bergshoeff, M. de Roe and B. De Wit, Extended conformal supergravity NIKHEF preprint, NIKHEF -1-1/8007 (November 1980). [8] P. Fayet and S. Ferrara, Phys. Rep. 32C (1977) 249. [9] DN. Volkov and V.P. Akulov, Phys. Lett. 46B (1973) 109. [10] W. Siegel, Nucl. Phys. B177 (1981) 325.
277