Manifestly covariant rules and higher-loop finiteness

Nuclear PhysicsB206 (1982) 496-497 © North-HollandPublishingCompany

Errata

A.G. Liparteliani, V.A. Monich, Yu.P. Nikitin and G.G. Volkov, Neutral mesons with heavy quarks and mixing angles in the six-quark model, Nucl. Phys. B195 (1982) 425. In eqs. (13)-(15) a multiple 4 has been omitted. In eqs. (29), (30), (36), (37) [(~-31n(y3/zz))~+(~+31n(y3/z3))] replaced by {l[ln (y3/z3)- 9]}. 1 In eqs. (26)-(28) and (33)-(35) the coefficients ~ should read 5.

must be

R.. D'Auria and P. Fr6, Geometric supergravity in D = 11 and its hidden supergroup, Nucl. Phys. B201 (1982) 101. In eq. (5.10) the ~ should be replaced by 81i. In eq. (5.11) - 3 should read - 6 and 3 should read 43. In table 3 for the part concerning the on-shell solution for the curvatures the in the expression for p should again be replaced by ~i and in the expression for R ab, _7 should be replaced by +1 and +2~6 by + ~4. Also in table 3, in propagation equation (iii) the same correction should be made as in eq. (5.11). M.T, Grisaru and W. Siegel, Supergraphity (II). Manifestly covariant rules and higher-loop finiteness, Nucl. Phys. B201 (1982) 292. The "doubling" trick of sect. 4 cannot be applied covariantly in the case where the scalar multiplet is a complex representation of the Yang-Mills group. (However, it can be applied as described to supergravity, and to real representations of the Yang-Mills group.) This is due to the fact that ~z~ is then not in the same representation as r/, so the operator O is not representation-preserving. As a result, one must use rules at one loop which are not expre~ed manifestly in terms of FA. Explicitly, in terms of fields r~(~) and sources J ( J ) which are chiral (antichiral) with respect to/9a(D~), we have the following equations of motion in the presence of external super-Yang-Mills:

o0+( )0 o(oo 0eV/ If we had used covariantly chiral fields, we would have r / = e r/, 496

Errata

497

but ~72rl = e-W*/~ 2 e w* e

r/,

which does not satisfy the same type of chirality condition due to its belonging to the complex conjugate representation. (The hermitian conjugate IV equals minus the complex conjugate - W * only for real representations.) The non-covariant object 0 can now be squared, since it preserves (/ga) chirality: 0 • 02=(D2eVo D2ev D2eVl~2eV ). The action S; obtained from 0 2 again gives a contribution equal to that of its hermitian conjugate. The propagator is as before, but the vertex is now

O2(eV*D 2 e v - O

2) .

Note that, for real representations, V* = - I7" = - V, so this vertex is just/} 2(V2--O 2), and the improved rules of sect. 4 can be obtained. In general, for a group containing factors for which ~/ is a real representation, we can write V = Va+ V2, where V* = - V 1 , but V2* ¢ - I / 2 ([V1, Vz] = 0), and write the vertex as

l~2(eV;~TzleV2-D2),

~1~ = e

VlD,~ e vl =D~+FI~.

Then V1 appears in the rules only as Fla. The net result is that the effective action is always expressed manifestly in terms Of VA for supergravity, and always for Yang-Mills factors which occur coupled only to real representations, and always for higher-loop contributions to all types of gauge fields. However, at one loop, and only for Yang-Mills factors coupled to complex representations, we have a situation similar to that occurring in extended theories: The one-loop contribution must be calculated in a way where the covariance is not manifest. In particular, this means that the vanishing of the one-loop Fayet-Iliopoulos term is not automatic, although it still vanishes when a supersymmetric, gauge-invariant regularization procedure is used (as described in subsect. 5.1). S.P. Luttrell and S. Wada, The current product expansion of the two-quark process at the twist-four level, Nucl. Phys. B197 (1982) 290. Eq. (27) should read

-4ie~q~p~x-"+ll{ 2 n odd (even)

q

+

~

•

(B,+2(1, i ) - B , + 2 ( 2 , i))

3~i~n+2

(B.+2(1,

i)-B.+2(i, n + 2 ) ) } .

2~
The statement following the equation is based on this correct formula and remains unchanged.