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

More documents

Recommendations

Info

256 P. van Nieuwenhuizen, Supergravit’yseems unlikely. Secondly, other N = 8 formulations might exist, notably formulations involvingantisymmetric tensor fields. However, it is known that different field representations for a given spin dolead to the same S-matrices (although they lead to different anomalies [560]).It should, however, not beoverlooked, that using antisymmetric tensor fields, the trace anomalies cancel “miraculously” in theN = 8 model [609].Something is going on here.Extra symmetries usually help to eliminate possible invariants. For N = 8, the maximal on-shellsymmetry group is U(8) (see subsection 3.3), and W,Jk, is manifestly SU(8) covariant, while a U(1)invariance would violate the self duality property. Thus there seem to be no on-shell extra symmetriesleft. (The chiral-dual symmetries first discovered in “miraculous” cancellations were later shown to bepart of the U(N) invariance. In addition, the N = 8 model has a self-duality, see section 6.)In the presence of central charges, the integration measure is no longer d320 since in that case thehighest component of a multiplet varies into a total derivative plus something more. However, centralcharges usually vanish on-shell (section 6).Yet another approach is to pin one’s hope on the spontaneously broken version of the N = 8 model(section 6). However, mimicking in superspace the dimensional reduction from the massless d = 5,N =8 theory to the massive d =4, N =8, one would expect to end up with a dangerous invariant £2? ind = 4 dimensions. In fact, the reverse might happen: it might be that massive d = 4, N = 8 is not evenone-loop finite. This is presently unknown and under study.Of course, miracles do sometimes happen, but they would have to happen at every ioop order. Itseems better to accept a fundamental property of gravitons: that no nearby massless theory exist (thevan Dam—Veltman theorem) and to use nonperturbative methods. Perhaps this is the way quantumsupergravity should go in the future, but it is easier to say that one should do nonperturbativecalculations than to do them.2.10. Explicit calculations2.10.1. Green’s functions are not finiteAs a first application of the Feynman rules derived in the last section we calculate the gravitonenergy in simple supergravity at the one-loop level. This yields at once an important result of quantumsupergravity: Green’s functions are not finite in general [510].Later we shall see that S-matrix elementsare sometimes finite.The general amputated two point function receives contributions from a spin 2 loop, a generalcoordinate (spin 1) ghost, a spin 3/2 loop and a supersymmetry (spin ~) ghost, see fig. 1. The divergent partsare local and the general form is (using dimensional regularization)D~,.,,,0.= —~-—~ [Ap~p,.p~p,, + (2B6~~6,.,, + ~ + (2Dp~p,.6,.,~+ 4Ep~pp6,.c,.)p 2]where one should still symmetrize in (j~ii),in (po) and under (~~v) ~ (,po). As regularization scheme we~graviton spin 1 ghost gravitino spin ~ghostFig. 1. Graviton self energy graphs.
P. van Nieuwenhuizen. Supergravirv 257may use either dimensional regularization or dimensional reduction for the one-loop divergences. Thelatter scheme should be used in general (see subsection 7).For the results one findsspin 2 spin 1 spin 3/2 spin 1/2A 13/15 —1/6 —3/40 1/12B 71/240 —1/24 23/480 0C 27/80 —7/48 —7/80 0D —19/60 1/8 7/80 0E —11/40 1/4.8 —13/480 —1/48We have added an extra minus sign to the spin 1 (ghost) and spin 3/2 (physical) loops, but not to thespin ~(ghost).Two conclusions can be drawn from these results:(i) The supersymmetry ghostfield commutes with itself, since only then is the Ward identity~L7)1’ ~Lvpuj-’satisfied. It is of course satisfied separately in the purely gravitational and in the extra half-integer-spinsector.(ii) Green’s functions are not finite. For example, the B coefficients do not sum up to zero. InS-matrix elements the two point functions vanish since p2 =0 and polarization tensors are transverse. Insupergravity, the S-matrix elements of higher point vertices are finite if one adds self energies, triangles,boxes. However, the sum of self energy graphs is separately not zero. In fact one findsB(photon) = , B(photon ghost) =0B(real electron) =1B(real1scalar)=Clearly, all B are positive as one might expect from a unitarity argument and no magical combination offields of different spins can yield zero for the graviton self energy. We have restricted our attentionhere to B because if one couples the gravitons to a traceless conserved source (photons) only Bcontributes.As a natural counterpart one may consider the gravitino self energy. Again the four-gravitinocouplings do not contribute, nor do the four-ghost couplings due to elimination of 5, F, Am from thequantum action. One finds the results in fig. 2. In the gauge y. 4’ the ghost loop is zero separately andthe self energy is again transverse. In more general gauges, the self energy graphs of the gravitino aretransverse independently of whether the supersymmetry and coordinate ghost commute or anticommutewith each other. This freedom has been explained theoretically [553].2.10.2. Matter couplings are not finiteSince Green’s functions are divergent, the next question is whether S-matrix elements are finite. We
Page 3 and 4:
In memoriam Joel ScherkJoel Scherk
Page 6 and 7:
194 P. van Nieuwenhuizen, Supergrav
Page 8 and 9:
Page 10 and 11:
Page 12 and 13:
200 P. van Nieuwenhuizen. Supergrav
Page 14 and 15:
202 P. tan Nieuwenhuizen, Supergrav
Page 16 and 17:
Page 18 and 19: 206 P. van Nieuwenhuizen. Supergrav
Page 20 and 21: 208 P. van Nieuwenhuizen. Supergrar
Page 26 and 27: 214 P. van Nieuwenhuizen, Supergrav
Page 32 and 33: 220 P. ran Nieuwenhuizen. Supergrar
Page 62 and 63: 250 P. ran Nieuwenhuizen, Supergrav
Page 80 and 81: 268 P. Lan Nieuwenhui:en. SupergraL
Page 90 and 91: 278 P. t’an Nieuwenhuizen, Superg
Page 118 and 119:
Page 120 and 121:
Page 122 and 123:
Page 124 and 125:
Page 126 and 127:
(A V,.”)00 = ~ s*a = (D,. + ~ A,.
Page 128 and 129:
Page 130 and 131:
Page 132 and 133:
Page 134 and 135:
Page 136 and 137:
Page 138 and 139:
Page 140 and 141:
328 P. van Nicuwenhuizen, Supergrav
Page 142 and 143:
330 P. van Nieuwenhuizen Supergravi
Page 144 and 145:
Page 146 and 147:
Page 148 and 149:
Page 150 and 151:
Page 152 and 153:
Page 154 and 155:
Page 156 and 157:
Page 158 and 159:
Page 160 and 161:
Page 162 and 163:
Page 164 and 165:
Page 166 and 167:
354 P. van Nieuwenhuizen Supergravi
Page 168 and 169:
Page 170 and 171:
Page 172 and 173:
Page 174 and 175:
Page 176 and 177:
Page 178 and 179:
Page 180 and 181:
Page 182 and 183:
Page 184 and 185:
Page 186 and 187:
Page 188 and 189:
Page 190 and 191:
Page 192 and 193:
Page 194 and 195:
Page 196 and 197:
Page 198 and 199:
Page 200 and 201:
Page 202 and 203:
Page 204 and 205:
392 P. van Nieuwenhui’zen, Superg
Page 206 and 207:
Page 208 and 209:
Page 210:
show all

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

Create successful ePaper yourself

Delete template?

Save as template?