SUPERGRAVITY P. van NIEUWENHUIZEN To Joel Scherk 0370 ...

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248 P. <strong>van</strong> Nieuwenhuizen. SupergravityConsider as an example how the scheme works, quantum electrodynamics. The action is given by= —~9,.,A,.— ~9,.A~) 2 — ~y’~(9,~— ieA~)A. (1)We now continue only coordinates and momenta to n dimensions where n 4. However, the indices j~of A,., remain 4-dimensional, while also A remains a four-component spinor. Thus one can write theaction as= —~(t9,A1 3~4.) ieA 2— ~y’(3~— 2+ iety”AA~. (2)1)A — ~(o,A~)In other words, we have here normal QED in n-dimensions, plus extra scalars coupled to the spinors bya Yukawa coupling. Since 0 i n but n
P. <strong>van</strong> Nieuwenhuizen, Supergravity 249If one manipulates the Dirac algebras as follows=yJt’y,=(2—n)J( (7)= —2Xone finds that (6) is satisfied at the one-loop level to order if one uses either dimensionalregularization or dimensional reduction, but to order °only the latter scheme preserves (6) [556].In ordinary renormalizable models, this scheme respects renormalizability only if these theories aresupersymmetric. For example, the two coupling constants e in (2) are rescaled by different Z-factors.Thus for supersymmetric theories this scheme makes sense, but for non-supersymmetric ones it doesnot. One can also use it for supergravity, where the issue of renormalizability does not exist. One canuse this scheme to calculate anomalies.There exist other regularization schemes which are claimed to preserve global (and sometimes local)supersymmetry. One such scheme is the higher derivative scheme of Slavnov. For example, inYang—Mills theory one adds to the usual Yang—Mills action a term A2(DAG’4,.” )2 where A ultimatelytends to infinity. Propagators now fall off like k4 and this regularizes all loops except single loops. Forthose one sabstracts infinities by adding to the action gauge-invariant Pauli—Villars terms~ (62~1(A,.,,A )/ôA~6A,.)B’,.,B~. — m~(B~..)2 (8)(satisfying, as usual, ~ a, = ~ a,mcorresponding to D~(A)B’4’1= 0= 0) as well as anticommuting ghosts c’ for these new gauge fields B,.,.2? = D~(A)E’ D,,(A) c~— ~ (9)For global supersymmetry one can use superfields and find globally supersymmetric Yang—Mills-gaugefixing terms. For supergravity it is not clear whether this scheme can be applied.There is also the BPHZ-scheme, as applied to supersymmetry by Clark, Piguet, Sybold, where oneexpands Green’s functions in terms of the external momenta. Each coefficient turns out to besupersymmetric, but to prove this requires detailed analysis.2.8. BRST invariance and open algebrasSupergravity without auxiliary fields has an open gauge algebra, by which we mean that it closes onlyon the classical mass shell. On the other hand, for quantum calculations it is much easier to workwithout auxiliary fields. Thus the problem arises what new features arise at the quantum level for gaugetheories with open algebras. In particular, unitarity and gauge invariance of the S-matrix can be derivedfrom Ward identities, and to obtain these Ward identities one needs BRST invariance of the quantumaction. In this subsection we see how to modify the transformation rules and action to obtain BRSTinvariance if the gauge algebra is open [304, 1621. For closed gauge algebras we refer back to subsection2.
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In memoriam Joel ScherkJoel Scherk
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