Anomalies in supergravity

Volume 169B, number 2,3

PHYSICS LETTERS

27 March 1986

A N O M A L I E S IN SUPERGRAVITY P. V A L T A N C O L I Dipartimento di Fisica, Universith di Firenze, 1-50125 Florence, Italy

and L. C A N E S C H I Laboratoire de Physique Th~orique et Hautes Energies I, Universitb de Paris-Sud, Bhtiment 211, F-91405 Orsay, France and Dipartimento di Fisica, Universith de Pisa, 1-56100 Pisa, Italy Received 23 April 1985

The anomaly of the supercurrent of an abelian vector multiplet interacting with external supergravity is considered, and it is shown that it is equivalent to a gravitational anomaly. The local counterterms that reinstate the conservation of the supercurrent are exhibited.

Abbott, Grisaru and Schnitzer [1] (AGS) have shown that a model in which an abelian supersymmetric vector multiplet interacts with the (external) multiplet of supergravity exhibits a supercurrent anomaly. In four dimensions in the notations of AGS the spinor supercurrent Su = ( I /x/~) Fa~ 0 ~ 7u X is classically conserved (via the equations of motion):

8uSU = 0

(1)

and satisfies the condition

7 u S u = 0.

(2)

The lowest order perturbative calculation of the matrix element of the quantum supercurrent Su between a one graviton state of momentum - k and a one gravitino state o f momentum q


=

is performed by AGS with the standard technique of decomposing this amplitude in 17 invariant amplitudes. Two of them A 4 and A 5 are finite, receive con1 Laboratoire associ6 au Centre National de la Recherche Scientifique. 0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishin~ Division)

tribution only from the triangle diagrams and are of the usual form (p + k) - 2 . Imposing conditions (1) and (2) on the amplitude, AGS find t h a t A 4 a n d A 5 should vanish: this is inconsistent with the result of the explicit and unambiguous perturbative calculation, hence (1) or (2) are anomalous. The calculation of AGS takes for granted that the classical invariances for general coordinate transformations (Einstein) and for local Lorentz transformations (Lorentz) are maintained at the quantum level. We know now [2] that these invariances are in fact lost in some theories, and we want to reexamine the problem without imposing them a priori. In this case the decomposition of the same process in invariant amplitudes contains thirteen further terms, that we define by splitting the invariants of AGS that contain two terms related by the a ~,/3 symmetry (A 2, A 3, A 6, A 7, A 9, AlO, A11, A12, A13, A14, A15, A16, A17). Our more general decomposition contains, for instance, a term A 2 Pl~kv TaP# + A2 P~ kv 7#Pa etc. The ffmite terms A 4 and A 5 are obviously not affected, and the reconstruction of the amplitude can be carfled out as usual starting from their unambiguous value. Imposing on the amplitude the constraints of Ein197

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stein invariance

(3)

p Suua~ = 0 and of Lorentz invariance

27 March 1986

Conditions (3) and (4) impose the further sets of constraints

(iii) A9 + (P'k)AI =A9 =Zl0 =A15 + (P'k)A2

(4)

S w ~ = S~

leads us back to the situations studied in AGS, in which either (1) or (2) fails. However if we give up either (3) or (4) it is possible to impose (1) and (2), and therefore to have no supercurrent anomaly. In fact imposing (1) + (2) on our more general decomposition, we obtain a system of 27 constraints that are all compatible among themselves and with the known values ofA 4 and A 5 . They are respectively

= A 2 +A13 + (P'k)A 4

=A 3 +A 7 +A12 + 2All + (p'k)A 5 =A6 +A14 + (P'k)'~3 ='z~6 +'416 + (P'k)J~7 =~z~14 +a16 + 2~z~17+ (P'k)-412

(i)

=A17 + (p'k)~Z~ll= AI0 + (p'k)A 8

A9 +'d9 + (P'k)A1 =A10 +AI5 + (P'k)A2

=/~15 + (p'k)+x~13 = 0,

=al0 +A15 + (p'k),42

= A14 + (P "k)A3

(iv)

Ai =Ai" =A14 + (p'k)ff 3 =A13 +'413 +A8 + (P'k)A4 =A12 +J~12 + (P'k)A5 = A6 = "A6 =A16 + (P'k)A7 = "416 + (p" k).4 7

It turns out that (1) + (2) + (3) and (1) + (2) + (4) are both consistent. In the former case we obtain

A4=A5=c/(p'k),

A1 =-c=al2,

A2 ="42 =~c=-A7 =-AT' ='417 + (p'k)All =A17 + (p'k).'~ll A10 = "410 = A8 = A6 = "46 = A12 = if9 = 0, =A7 +A7 +A8 +A1 + (P'k) (A4 +A5) ='46 +A10 + (P'k) (A2 +A3 + 2All)

A13 = - ~ c,

- = !2c , A13

A 9=c(p'k),

A15 = '415 = -½ c(p'k) = -A16 = -A16' =Z6 +'410 + (P°k) ('42 +'z~3 + 2~z~11)

,i1 :A 1, Z14=AI+, i17=A,7,

=A9 +'~16 + (P'k)(A12 +A13)

='49 +A16 + (P'k)(A12 +'513) = 2A17 +A14 +A15 = 2"/~17 +A14 +"z~15 = 0,

0i) 2A 1 +A 2 +.,42 + (p'k)A 4 =A 3 +2All - A 7 = A3 + 2"T11 - A7 = 2A9 + AlO + A15 + (P'k)A13 = 2+,T9 +Alo +.'T15 + (P'k)A13 =A16 - 2A17 -A14 =A16 - 2A17 -A14 =0. 198

A14 = - ( p ' k ) A 3 ,

A17 = - ( p - k ) A l l

and finally A 3 + 2All = - ½ c . Therefore the amplitude is not completely determined by our constraints since we can still choose the value of, e.g., A 3- However the Lorentz anomaly is due to the fact that A9 =/::"49' A12 =/='712' A13 4:'413 and is completely determined in terms of c:

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PHYSICS LETTERS

AA(1 ÷72) = ~ c [(ku% ~ - ICgua) (kvP # - (p'k)gv~)

St~v,,, ~ - Suv#c L = e [~(k - 2p) u + (p-k)'},u] (gvap# - gvOp,,, ). In the latter case (1) + (2) + (4) impose the constraints

Ai=Ai'

27 March 1986

A1 = A 6 = A s = A 9

=0,

+ (a'* t3)l. For the other cases a possible solution for AA(2 ~ 3) is:

AA(2 ~ 3) = - ~ c [ ( k k - g.~(p-k)) (%p~ + p ~ ) A2=-½c=A13=A12

,

Aao=~e(p'k), + (p'k) (gve,?t~ + gva'~a) (p - k ) , l ,

A 7 =-c,

A15 = - A I 6 = - c ( p - k ) ,

A17 = - ( p . k ) A l l ,

A14.=-(P'k)A3,

and again the solution has one degree of freedom, because A 3 and A 11 are only constrained by

whereas for the counterterm that shifts the amplitude with Einstein into the amplitude with the Lorentz anomaly we obtain: AA(3 ~ 4) = ½c [ - 2 ?u kvpc~P~ + ((2p + k)u k v

A 3+2All =-c.

- 3(p'k)guv ) ('yap~ + "Y~Pa)

Also in this case the Einstein anomaly is completely defined, since it only depends on A 3 + 2 A l l , and we find

+ (guaP~ +gl~fjPa)~kv + (gvaP~ +gv~Pa) X ( p ' k ) ? u + (gva?t3 + g v t f f a ) ( p ' k ) ( p - k) u

kaSuvc¢ - (¢(p'k)(g~agv~ +gu~gva) = - ~ c [(k u k v - (p'k)guv)po tc + (p" k) 7~ + (gvaP~ - gvt~pa)[~(k - 2p) u + (p'k)Tu]. X [pukv - (p.k)guv ] + (p.k)tcgv#(p - k)u ] . In the case of the mixed gauge and gravitational chiral anomalies the equivalence of the schemes in which one imposes two of the three classical invariances can be shown by using the index theorem [3] to construct the local counterterms that shift one anomaly into another [4]. Also in the present case we can check that the amplitudes that are reconstructed using three of the four invariances (1), (2), (3) and (4) differ (at this order of perturbative theory) only by local polynomials, i.e., their difference does not have poles at (p.k) = 0. Therefore we can construct regular counterterms that shift one form of the amplitude into another, and these regular counterterms can be deduced from additional local polynomials in the definition of the generating functional. For instance the counterterm that relates the two possible forms of the supercurrent anomaly studied by AGS, namely the difference between the amplitudes obtained by imposing (1) + (3) + (4) or (2) + (3) + (4) is

In this lengthy expression in practice only the last term violates Einstein and Lorentz invariances and is responsible for the shift of the anomaly. In conclusion we have seen that the supercurrent anomaly at this order in perturbation theory can be eliminated at the price of introducing counterterms that do not respect the gravitational invariances ,1. This does not mean that there must be a gravitational anomaly in this theory, as it happens for instance in the case of a chiral fermion interacting only with gravfty in 4n + 2 dimensions. In the latter case Einstein and local Lorentz invariances impose incompatible constraints, whereas we have shown that here eqs. (3) and (4) are compatible. We confirm therefore that this theory does not contain intrinsic gravitational anomalies and can be regularized by a procedure that

,1 This might not be the case in the whole theory: For instance the four gravitinos Green function in this model has the right features to produce a supercurrent anomaly that could not be removed into a gravitational one, since its decomposition in invariant amplitudes contains terms with poles at q2 = 0.

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respects both. The gravitational anomaly that we obtain is completely contained in the counterterms that we introduce in order to reinstate supersymmetry, in analogy with what happens in the case of gauge invariance [4]. In all anomalous problems studied so far the clash among the requirements o f the set o f N classical invariances of the problem allowed N solutions that respected any subset of ( N - 1) invariances. One wonders whether this is a general feature.

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27 March 1986

References [1 ] L.F. Abbott, M.T. Grisaru and H.J. Schnitzer, Phys. Lett. 73B (1978) 71. [2] L. Alvarez-Gaum~ and E. Witten, Nucl. Phys. B234 (1983) 269. [3] L. Alvarez-Gaum~ and P. Ginsparg, Nucl. Phys. B243 (1984) 449; Ann. Phys., to be published. [4] L. Caneschi and P. Valtancoli, Phys. Lett. 156B (1985) 93; Nucl. Phys. B258 (1985) 540.