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Divergences and counterterms in N=1 and N=2 supergravities in D=11 and D=10 dimensions
UCLEAR PHYSICE
PROCEEDINGS SUPPLEMENTS ELSEVIER
Nuclear Physics B (Proc. Suppl.) 104 (2002) 204-207
www.elsevier.com/locate/npe
Divergences and Counterterms in N=I and N=2 Supergravities in D = l l and D=10 dimensions Domenico Seminara a* aDipartimento di Fisica, Universith di Firenze, L.go Enrico Fermi 2, and INFN, Sezione di Firenze, 55125 Firenze, Italy After reviewing previous computations for the two-loop divergencies in N = 1 supergravity in D = 11, we evaluate the one-loop divergences of D ----10, N = 1 supergravity. We study the tensor structure of the counterterms appearing in D = 10 and compare these to expressions previously found in the low energy expansion of string theory. The infinities have the primitive Yang-Mills tree amplitude as a common factor.
1. I n t r o d u c t i o n Extended supergravities, although they were displaced as favored candidates for a "final theo r y " b y superstrings and subsequently by (quantum) M theory, keep on playing a central role in understanding q u a n t u m gravity. In fact, they can be considered the effective theories which describe quantum gravity at energies less t h a n the (colossally large) Planck scale. Of the extended supergravities two are naturally singled out. Firstly, there is the D = 11, N = 1 maximally extended theory [1] which reduces to N = 8 in D = 4 [2]. In some ways this theory is the ultimate conventional point-particle field theory. The one-loop amplittide is potentially infinite for D > 8 although in dimensional regularisation it is only infinite for D -- 8 (in the dimensional regularisation ~rescription one-loop amplitudes in odd dimensions are finite and the D = 10 infinity vanishes onshell). At two-loops infinities have been calculated in the amplitudes for D > 7 [3] (including the D = 11 case. The eleven dimensional counterterm was subsequently evaluated in Ref. [4]). Secondly, there is the D = 10, N = 1 theory and its dimensional reduction descendants which include the D = 4, N = 4 supergravity theory (with a specific m a t t e r content). This supergravity is the gravitational sector of the low energy *Research carried out in collaboration with D. Dunbar, Bernard Julia, and Mario Triggiante.
limit of b o t h type I and heterotic string theories [5]. Here we shall consider the one-loop for the D = 10, N = 1 supergravity together with arbitrary m a t t e r multiplets and we shall derive the explicit form of the gravitational counterterms needed to make the amplitudes finite. 2. U n i t a r i t y :
from trees to loops
Our approach is to reconstruct the physical, onshell S-matrix combining its analytic behavior with its s y m m e t r y properties. 4+
',
3+
±~
1In fact, a key p r o p e r t y of the S - m a t r i x is unitarity. Also within dimensional regularisation the amplitude is analytic in the dimension. The optical theorem, a consequence of unitarity, states
2ImT = TiT. In perturbation theory comparing both sides order by order relates, for example, the imaginary part of a one-loop amplitude to the product of tree amplitudes. In practical t e r m s the imaginary part of a one loop amplitude is just the coefficient of a logarithm (or di- or polylogarithms) since ln(s,j) = ln(Is q I) + iTrO(sq), where sij is one of the m o m e n t u m invariants. Naively, the optical theorem only determines part
0920-5632/02/$ - see front matter © 2002 ElsevierScience B.V. All rights reserved. Pll S0920-5632(01)01615-2
D. Seminara/Nudear Physics B (Proc. Suppl.) 104 (2002) 204-207 of the one-loop amplitude since the amplitude may contain rational functions f ( s o ) which have no imaginary part. However we can, by using dimensional regularisation, determine these rational parts also. Within dimensional regularisation the one-loop amplitude has a momentum weight of - 2 e (since dDp ~ dD-2e). This implies that the rational functions must be replaced by terms such as f'(sij)(si5) -2~. Since (sij) -2~ = 1 - 2e in(sij) the amplitude will pick up imaginary parts at O(e). We thus deduce knowledge of the cuts to all powers in e will enable us to determine the amplitude Of course this can be a painful computational burden in many circumstances. (Although in practice it is not always necessary to determine the cuts to all orders in e.) To see how this works consider the cut in a one-loop amplitude. Then the optical theorem states that [7] Disc MI-I°°p(1, 2, 3, 4) ~ - ~ t =
if
(1)
The one-loop amplitude coincides with that computed by Green, Schwarz and Brink [6] for the lower maximal supergravities, namely
o o_,oo 4
e ms)
~---
where Z4(s, t) is the D-dimensional scalar box integral. The one-loop amplitude is finite in D = 11 due to an accidental properties of odd dimensions where one loop divergences are fobidden at oneloop level. Using the technology described previously, the two-loop four-graviton amplitude [3] was obtained in a remarkably simple form
J~4 = (2)6stuMtree(s2~P4 (s,t)-[-S2~P4 (S,U) +s2ZNP(s,t)+s2ZNP(s,u)Tcyclic)
(2)
where 27~ and ZNP are two-loop scalar planar and non planar box integrals. The two-loop amplitude has an ultraviolet divergence in D = 11,
j~D=111 4
u t r e e ( - g ~ , l , 2 , g~')utree(-g~',3,4, g~) i
205
:
Ipole
1 48e (4r) n ×
(a)
l
particles
where the integral is over on-shell g~. We must use this carefully within dimensional regularisation if we wish to determine the LHS correctly to all orders in e. The RHS contains tree amplitudes. Normally we do not regard these as depending upon e, however the momenta gi must match the loop momenta in the LHS. These are in D - 2 e dimensions so that the tree amplitudes should have the momenta g~ in D - 2e and the others in D dimensions. The analysis here is naturally merely indicative and the reader must be referred to elsewhere for the details of how this works and how it may be applied [7,8,10]. 3.
D = 11 S u p e r g r a v i t y A m p l i t u d e s
Eleven dimensional supergravity [2,1] is a fascinating theory whose ultraviolet behavior was suspected but until the last few years has defied definite calculation. We shall review it.
where ~- = (~/2) 6 x s t u M l ree. For four-graviton external states, the linearized counterterms take the form of derivatives acting on
t8tsR 4
=_ t~u2""ust~2"'~8 R.~,2~2
x
R . 3 . , ~ , Ru~.~5~ 6 R,,us~,~8,
(4)
plus the appropriate supersymmetry completion [4]. The D = 11 counterterm is a linear combination of the two tensors
TA
=
tsts • O~,RO~VnROz~pRO~6PR
TB
=
t s t s . O~nRO~nRO~vpRO~PR.
(5)
In each case the indices on the curvatures are contracted with the t8 tensors and the indices on the derivative are contracted with each other. The D = 11 counterterm is 1 --
7r
(2575TA
48e (47r) 11 × 579150-----O\ - i - 2
_~ +
) TB
"
D. Seminara/Nuclear Physics B (Proc. Suppl.) 104 (2002) 204-207
206
\ k 1} . . . .
4. N----1 S u p e r g r a v i t y in D = 10
It is interesting to compare the structures found between types II and type I supergravity (and their lower dimensional descendants.) We have examined the one-loop structures for dimensions 4 _< D < 10 [12]however here we shall restrict presentation to the features of the D = 10 case. The results of calculating the infinities are firstly M N=s'D=I° : 0, as expected. The infinities have two powers more of momentum as compared to D = 8. However we still find that all infinities factorise with one factor of K~. The remaining factor L~ contains the extra two powers of momentum. Specifically we calculate M N = 4 , D = ~ o l ( 2_) 4×= e ×
e
--i 1 X L] (4~r)~ 60480 K~ 1
(47r)a 1440 K~
X Le
where L~ = (e~. e2)(E3, e 4 ) s ( l S u 2 + 41tu + 18t 2) + . . . + 2(el •e2) ( - t z ( l S e 3 . k 4 e 4 . k a + e z . k l e 4 . k 2 ) - u 2 (18~3 • k4e4. k3 q- e3- k2Q. kl) - tu(40e3 • k4e4- ks + ca. kl ¢4" k2 + e3" k2e4. k l ) ) + . ' . + 4 (te~. k3e2. k~ e3. k2 ¢a" k~ +
/
4
(-~E1.k2~2.kzE3,k2E4.k1-u~1-k3~2"k3E 3"
k I E4 • k 2 --I--~E 1 • k3E 2 • k3E 3 • k2E 4 • k I - u~ 1" k2£2" k l E3" k2E4. k2 - U~l, k2£ 2 • k3E3. k l E4-kl - 2Uel"k2E2" k l £3" k l E4' k l - 2UEI" k3E2" k l E3" klE4" k2 -~-~£I" k3E2" kl~3" k2£4" k2 - u~ 1 • k2~ 2 • klE 3 • k2E 4 • k l -~-u£ 1 • k2E 2 • k3E 3 • k l E4' k2 -~"2uE1" k3~2" k l ~3" k2 E4" k l - ~'U~l" k2 ~2" ]gl ~3" k I E4 • k 2 - 2~£ 1 • k2E2 • k 1E3 • k 2 E4 • k I - 2tEl. k2E2- klE3" k2E4. k2 -4- tE 1 • k2E 2 • k3£ 3 • klE 4 • k l -}- 2rE 1 • k2E 2 • k3~ 3 • klE4"k2 - 2 t E l ' k 2 ~ 2 - k 3 £ 3 " k 2 E 4 " k l - t E l " k 2 £ 2 " k l ~ 3 k l E 4 , k 2 - t E I .k2E2.kl~3.k1£4-k I - t E 1 .k3~2-kl£3-kl~ 4. k2-~-tEl.k3E2.k3£3, k1£4-k2--tE1, k3~2.k3~3.k2E4- k l -~tel
m
k3~2" klE3" k2E4" k l - ~£1" k3£2" k1~3" k2£4" k 2 )
The infinities can be cancelled by specific combinations of 0 2 R 4 counterterms. Once more we find local Lorentz invariant counterterms to cancel the infinities. As a working hypothesis we assumed factorisability to make this tractable. The strategy is to calculate arbitrary onshell scalars of the form 0 2 F 4 times the previous t s F 4, to deduce from the resulting three-parameter expression for the amplitude the values of these coefficients and finally to replace the two polynomial solutions in the F ' s by the corresponding invariants in the fourth order in the Riemann tensor. (OnsheU equivalent to the Weyl tensor at this order.) We may choose to express the counterterms in terms of the following set of tensors:
F
5 t E l . k3E 2 • k l e3" k2e4' k2 -I-6tel" k3e2 • k3e3" k1£4"k l -~-
sl = ( OoRp,q,r,sO.P ,q, ,s ) ( Rp,,q,,r,,s, Rp,,q,,r,,s, )
tel. k3e2, kse3. kl e4" k2 - 17tel •k3e2- k3e3. k2~4- k~ + 6tel. k3e2- kae3. k2¢4" k2 - 18te~ • k2e2" kle3- k2~4" k l - 18te~-k2e2.kle3-k2e4,k2 + t e l . k 2 e 2 . k 3 e 3 . k l £ 4 kl-4tel.k2e2.k3e3.kle4.k2-18te~.k2e2.kae3.k2~-4 kl - 23t~1. k2e2' kle3. kle4. k2 - 17te~ • k2e2. kle3 kle4"kl +5tel'k3E2"kl E3" klQ'k2 +6&l'k3E2" klea k~ Q . kl - 18ue~ . kze2 . kl e3 . kl e4 . k~ - 18Uel. k3e2 k~ea. k~ea- k2 q-6Uel, k3e2 . k3e3 . k~ e4 . kl - 23uel k2e2" k l e 3 " k2e4" kl - 17Uel *'k2e2" klE3" k2e4" k2 + 5Uel-k2e2" kae3. kle4" kl + uel. k2e2. k3e3. kl e4" k2 + 5UEI" k2e2" k3E3" k2E4-kl+6Uel-k2E2" k3e3" k2E4" k2 1Tues. k3E2"k3e3" klQ- k2 -~-u£1 •k3£2"k3£3"k2£4" kl + 6ue~. k3£2"k3 £3"k2e4. k2 - 4ue~. k3 £2" k1¢3. k2£4" k~ T U£ 1 • k3£2" k1£3. k2£4" k2 - 18u£~. k~£2. k1£3" klea. k2) and L2 = - (£~. £~) (£3" £4)s (2u 2 + 3tu + 2t 2) + - - . + 2 (£1.
$2 = (O, Rp,q,r, sO~Rp,q,~,t)(Rp,,q,,r,,sRp,,q, #,,t)
£2 ) ( t 2 (2£4"kz£3"k4 +£3"k~ £4"k2)q -u2 (2£34"k4£4"k3 q%
£3 . k2£4 . k ~) + tu( 2e3 . k4 £4 "k3 + e3 "k l e4 "k2 + e3 "k2£4 "
$3 = (OaRp,q,r,sRp,q,r,t)(OaRv,q,,r,,sRV,q,,r,,t) $4 = OaRp,q,r,sO" Rp,q,t,uRe,mv,wRr,s,v,w $5 = OaRp,q,~,~O~ Rp,q,e,uRr,t,~,~Rs,~,v,~ $7 = OaRp,q,r, sOa Rp,t,r,uRt,v,u,wRq,v,s,w Ss : cOaRp,q,r,sOa Rp,t,r,uRt,v,q,wRu,v,s,w $9 = I~,b,~,d R~,~,~,aO,~ R~,f ,~,h O~ R¢,I,~,h
$10 = OpR~,~,t,,~OqR~,~,~,~Rq,d,t,~Rp,d,,~,~. For $1 to $8 the derivatives are contracted with each other and these Si's are related to derivatives acting upon the Ti's of the previous sections. Tensors So and Sm however have the derivatives contracted into the Riemann tensors. Of course many tensors of the form c?2R 4 vanish onshell since they produce amplitudes of the form
D. Seminara/Nuclear Physics B (Proc. Suppl.) 104 (2002) 204-207
~- (s + t + u) x tensor --- 0. In terms of the 5"/'s the infinities are cancelled by the counterterms
4. 5.
c
(4~) ~
'
and __l_(2)4
6.
i_._~cN=4,D=lO (4~) 5
'
where cg=6,D=lO _ I 8~6
1
4.720
($1 \
-
7.
12S2 - 4S3 + 25'4 -t-
+ 16S7 + 8Ss),
cg=4,D=lO _
-1
4.6048
(9S1
\
-
76,-92 + 44S3 + 30S4
8.
-56S5 + 88S6 + 16S7 - 88Ss + 24S9 - 95S10). It is far from obvious that such counterterms lead to infinities which factorise, however we can manipulate them to do so. In fact it is possible to express both tensors in the form
9.
t l O t 8 0 2 R 2 -- tala2aaaaasaeaTaaagal°t 10 8blb2b3bab~ bebvbs Oa D blb2fQ D b3b,. bsbel:~ bTbs (6) 1.L~a~as va4 ~ Lasa6 ~rata8 " ~ a 9 alO "
10.
where we have chosen to contract the derivatives into the tl0 tensor. The specific form of these tensor can be found in Ref. [12]. The counterterms for D = 10, N = 1 supergravities axe given by linear combinations of the vaxious M D = l ° and in fact the infinities are given by K1 × ~ i ciLi since the various infinities factorise in this way. At the level of the counterterms, all type I supergravity divergences contain the factor ts. REFERENCES
1. E. Cremmer, B. Julia and J. Scherk, Phys. Lett. B76 (1978) 409. 2. E. Cremmer and B. Julia, Phys. Lett. B80 (1978) 48; Nucl. Phys. B159 (1979) 141. 3. Z. Bern, L. Dixon, D.C. Dunbar, M. Perelstein and J.S. Rozowsky, Nucl. Phys. B530 (1998) 401, hel>-th/9802162; Class. Quant. Grav. 17 (2000) 979, hep-th/9911194; Z. Bern, L. Dixon, D.C. Dunbar, A.K. Grant,
11.
12.
13.
207
M. Perelstein and J.S. Rozowsky, Nucl. Phys. Proc. Suppl. B88 (2000) 194, hep-th/0002078. S. Deser and D. Seminara, Phys. Rev. Lett. 82 (1999) 2435, hep-th/9812136. M.B. Green, J.H. Schwarz and E. Witten, Superstring Theory, Cambridge University Press, 1987. M.B. Green, J.H. Schwarz and L. Brink, Nucl Phys. B198 (1982) 474. L.D. Landau, Nucl. Phys. B13 (1959) 181; S. Mandelstam, Phys. Rev. 112 (1958) 1344; Phys. Rev. 115 (1959) 1741; R.E. Cutkosky, J. Math. Phys. 1 (1960) 429; Z. Bern, L. Dixon, D.C. Dunbar and D.A. Kosower, Nucl. Phys. B425 (1994) 217; Nucl. Phys. B435 (1995) 59. Z. Bern and D.A. Kosower, Nucl. Phys. B379 (1992) 451; D.C. Dunbar and T. Shimada, Phys. Lett. B312 (1993) 277; D.C. Dunbar and P.S. Norridge, Nucl. Phys. B433 (1995) 181; Z. Bern and D.C. Dunbar, Nucl. Phys. B379 (1992) 562; Z. Bern, Phys. Lett. B296 (1992) 85. I. Robinson, unpublished; L. Bel, Acad. Sci. Paris, Comptes Rend. 247 (1958) 1094 and 248 (1959) 1297. Z. Bern and A.G. Morgan, Nucl. Phys. B467 (1996) 479; Z. Bern, G. Chalmers, L. Dixon and D.A. Kosower, Phys. Rev. Lett. 72 (1994) 2134; Z. Bern and G. Chalmers, Nucl. Phys. B447 (1995) 465. M.B. Green, M. Gutperle and P. Vanhove, Phys. Lett. B409 (1997) 177; J.G. Russo and A.A. Tseytlin, Nucl. Phys. B508 (1997) 245. D.C. Dunbar, B. Julia, D. Seminara and M. Trigiante, J. High Ener. Phys. 0001 (2000) 046. S.A. Fulling, R.C. King, B.G. Wybourne and C.J. Cummins, Class. Quant. Grav. 9 (1992) 1151.
PROCEEDINGS SUPPLEMENTS ELSEVIER
Nuclear Physics B (Proc. Suppl.) 104 (2002) 204-207
www.elsevier.com/locate/npe
Divergences and Counterterms in N=I and N=2 Supergravities in D = l l and D=10 dimensions Domenico Seminara a* aDipartimento di Fisica, Universith di Firenze, L.go Enrico Fermi 2, and INFN, Sezione di Firenze, 55125 Firenze, Italy After reviewing previous computations for the two-loop divergencies in N = 1 supergravity in D = 11, we evaluate the one-loop divergences of D ----10, N = 1 supergravity. We study the tensor structure of the counterterms appearing in D = 10 and compare these to expressions previously found in the low energy expansion of string theory. The infinities have the primitive Yang-Mills tree amplitude as a common factor.
1. I n t r o d u c t i o n Extended supergravities, although they were displaced as favored candidates for a "final theo r y " b y superstrings and subsequently by (quantum) M theory, keep on playing a central role in understanding q u a n t u m gravity. In fact, they can be considered the effective theories which describe quantum gravity at energies less t h a n the (colossally large) Planck scale. Of the extended supergravities two are naturally singled out. Firstly, there is the D = 11, N = 1 maximally extended theory [1] which reduces to N = 8 in D = 4 [2]. In some ways this theory is the ultimate conventional point-particle field theory. The one-loop amplittide is potentially infinite for D > 8 although in dimensional regularisation it is only infinite for D -- 8 (in the dimensional regularisation ~rescription one-loop amplitudes in odd dimensions are finite and the D = 10 infinity vanishes onshell). At two-loops infinities have been calculated in the amplitudes for D > 7 [3] (including the D = 11 case. The eleven dimensional counterterm was subsequently evaluated in Ref. [4]). Secondly, there is the D = 10, N = 1 theory and its dimensional reduction descendants which include the D = 4, N = 4 supergravity theory (with a specific m a t t e r content). This supergravity is the gravitational sector of the low energy *Research carried out in collaboration with D. Dunbar, Bernard Julia, and Mario Triggiante.
limit of b o t h type I and heterotic string theories [5]. Here we shall consider the one-loop for the D = 10, N = 1 supergravity together with arbitrary m a t t e r multiplets and we shall derive the explicit form of the gravitational counterterms needed to make the amplitudes finite. 2. U n i t a r i t y :
from trees to loops
Our approach is to reconstruct the physical, onshell S-matrix combining its analytic behavior with its s y m m e t r y properties. 4+
',
3+
±~
1In fact, a key p r o p e r t y of the S - m a t r i x is unitarity. Also within dimensional regularisation the amplitude is analytic in the dimension. The optical theorem, a consequence of unitarity, states
2ImT = TiT. In perturbation theory comparing both sides order by order relates, for example, the imaginary part of a one-loop amplitude to the product of tree amplitudes. In practical t e r m s the imaginary part of a one loop amplitude is just the coefficient of a logarithm (or di- or polylogarithms) since ln(s,j) = ln(Is q I) + iTrO(sq), where sij is one of the m o m e n t u m invariants. Naively, the optical theorem only determines part
0920-5632/02/$ - see front matter © 2002 ElsevierScience B.V. All rights reserved. Pll S0920-5632(01)01615-2
D. Seminara/Nudear Physics B (Proc. Suppl.) 104 (2002) 204-207 of the one-loop amplitude since the amplitude may contain rational functions f ( s o ) which have no imaginary part. However we can, by using dimensional regularisation, determine these rational parts also. Within dimensional regularisation the one-loop amplitude has a momentum weight of - 2 e (since dDp ~ dD-2e). This implies that the rational functions must be replaced by terms such as f'(sij)(si5) -2~. Since (sij) -2~ = 1 - 2e in(sij) the amplitude will pick up imaginary parts at O(e). We thus deduce knowledge of the cuts to all powers in e will enable us to determine the amplitude Of course this can be a painful computational burden in many circumstances. (Although in practice it is not always necessary to determine the cuts to all orders in e.) To see how this works consider the cut in a one-loop amplitude. Then the optical theorem states that [7] Disc MI-I°°p(1, 2, 3, 4) ~ - ~ t =
if
(1)
The one-loop amplitude coincides with that computed by Green, Schwarz and Brink [6] for the lower maximal supergravities, namely
o o_,oo 4
e ms)
~---
where Z4(s, t) is the D-dimensional scalar box integral. The one-loop amplitude is finite in D = 11 due to an accidental properties of odd dimensions where one loop divergences are fobidden at oneloop level. Using the technology described previously, the two-loop four-graviton amplitude [3] was obtained in a remarkably simple form
J~4 = (2)6stuMtree(s2~P4 (s,t)-[-S2~P4 (S,U) +s2ZNP(s,t)+s2ZNP(s,u)Tcyclic)
(2)
where 27~ and ZNP are two-loop scalar planar and non planar box integrals. The two-loop amplitude has an ultraviolet divergence in D = 11,
j~D=111 4
u t r e e ( - g ~ , l , 2 , g~')utree(-g~',3,4, g~) i
205
:
Ipole
1 48e (4r) n ×
(a)
l
particles
where the integral is over on-shell g~. We must use this carefully within dimensional regularisation if we wish to determine the LHS correctly to all orders in e. The RHS contains tree amplitudes. Normally we do not regard these as depending upon e, however the momenta gi must match the loop momenta in the LHS. These are in D - 2 e dimensions so that the tree amplitudes should have the momenta g~ in D - 2e and the others in D dimensions. The analysis here is naturally merely indicative and the reader must be referred to elsewhere for the details of how this works and how it may be applied [7,8,10]. 3.
D = 11 S u p e r g r a v i t y A m p l i t u d e s
Eleven dimensional supergravity [2,1] is a fascinating theory whose ultraviolet behavior was suspected but until the last few years has defied definite calculation. We shall review it.
where ~- = (~/2) 6 x s t u M l ree. For four-graviton external states, the linearized counterterms take the form of derivatives acting on
t8tsR 4
=_ t~u2""ust~2"'~8 R.~,2~2
x
R . 3 . , ~ , Ru~.~5~ 6 R,,us~,~8,
(4)
plus the appropriate supersymmetry completion [4]. The D = 11 counterterm is a linear combination of the two tensors
TA
=
tsts • O~,RO~VnROz~pRO~6PR
TB
=
t s t s . O~nRO~nRO~vpRO~PR.
(5)
In each case the indices on the curvatures are contracted with the t8 tensors and the indices on the derivative are contracted with each other. The D = 11 counterterm is 1 --
7r
(2575TA
48e (47r) 11 × 579150-----O\ - i - 2
_~ +
) TB
"
D. Seminara/Nuclear Physics B (Proc. Suppl.) 104 (2002) 204-207
206
\ k 1} . . . .
4. N----1 S u p e r g r a v i t y in D = 10
It is interesting to compare the structures found between types II and type I supergravity (and their lower dimensional descendants.) We have examined the one-loop structures for dimensions 4 _< D < 10 [12]however here we shall restrict presentation to the features of the D = 10 case. The results of calculating the infinities are firstly M N=s'D=I° : 0, as expected. The infinities have two powers more of momentum as compared to D = 8. However we still find that all infinities factorise with one factor of K~. The remaining factor L~ contains the extra two powers of momentum. Specifically we calculate M N = 4 , D = ~ o l ( 2_) 4×= e ×
e
--i 1 X L] (4~r)~ 60480 K~ 1
(47r)a 1440 K~
X Le
where L~ = (e~. e2)(E3, e 4 ) s ( l S u 2 + 41tu + 18t 2) + . . . + 2(el •e2) ( - t z ( l S e 3 . k 4 e 4 . k a + e z . k l e 4 . k 2 ) - u 2 (18~3 • k4e4. k3 q- e3- k2Q. kl) - tu(40e3 • k4e4- ks + ca. kl ¢4" k2 + e3" k2e4. k l ) ) + . ' . + 4 (te~. k3e2. k~ e3. k2 ¢a" k~ +
/
4
(-~E1.k2~2.kzE3,k2E4.k1-u~1-k3~2"k3E 3"
k I E4 • k 2 --I--~E 1 • k3E 2 • k3E 3 • k2E 4 • k I - u~ 1" k2£2" k l E3" k2E4. k2 - U~l, k2£ 2 • k3E3. k l E4-kl - 2Uel"k2E2" k l £3" k l E4' k l - 2UEI" k3E2" k l E3" klE4" k2 -~-~£I" k3E2" kl~3" k2£4" k2 - u~ 1 • k2~ 2 • klE 3 • k2E 4 • k l -~-u£ 1 • k2E 2 • k3E 3 • k l E4' k2 -~"2uE1" k3~2" k l ~3" k2 E4" k l - ~'U~l" k2 ~2" ]gl ~3" k I E4 • k 2 - 2~£ 1 • k2E2 • k 1E3 • k 2 E4 • k I - 2tEl. k2E2- klE3" k2E4. k2 -4- tE 1 • k2E 2 • k3£ 3 • klE 4 • k l -}- 2rE 1 • k2E 2 • k3~ 3 • klE4"k2 - 2 t E l ' k 2 ~ 2 - k 3 £ 3 " k 2 E 4 " k l - t E l " k 2 £ 2 " k l ~ 3 k l E 4 , k 2 - t E I .k2E2.kl~3.k1£4-k I - t E 1 .k3~2-kl£3-kl~ 4. k2-~-tEl.k3E2.k3£3, k1£4-k2--tE1, k3~2.k3~3.k2E4- k l -~tel
m
k3~2" klE3" k2E4" k l - ~£1" k3£2" k1~3" k2£4" k 2 )
The infinities can be cancelled by specific combinations of 0 2 R 4 counterterms. Once more we find local Lorentz invariant counterterms to cancel the infinities. As a working hypothesis we assumed factorisability to make this tractable. The strategy is to calculate arbitrary onshell scalars of the form 0 2 F 4 times the previous t s F 4, to deduce from the resulting three-parameter expression for the amplitude the values of these coefficients and finally to replace the two polynomial solutions in the F ' s by the corresponding invariants in the fourth order in the Riemann tensor. (OnsheU equivalent to the Weyl tensor at this order.) We may choose to express the counterterms in terms of the following set of tensors:
F
5 t E l . k3E 2 • k l e3" k2e4' k2 -I-6tel" k3e2 • k3e3" k1£4"k l -~-
sl = ( OoRp,q,r,sO.P ,q, ,s ) ( Rp,,q,,r,,s, Rp,,q,,r,,s, )
tel. k3e2, kse3. kl e4" k2 - 17tel •k3e2- k3e3. k2~4- k~ + 6tel. k3e2- kae3. k2¢4" k2 - 18te~ • k2e2" kle3- k2~4" k l - 18te~-k2e2.kle3-k2e4,k2 + t e l . k 2 e 2 . k 3 e 3 . k l £ 4 kl-4tel.k2e2.k3e3.kle4.k2-18te~.k2e2.kae3.k2~-4 kl - 23t~1. k2e2' kle3. kle4. k2 - 17te~ • k2e2. kle3 kle4"kl +5tel'k3E2"kl E3" klQ'k2 +6&l'k3E2" klea k~ Q . kl - 18ue~ . kze2 . kl e3 . kl e4 . k~ - 18Uel. k3e2 k~ea. k~ea- k2 q-6Uel, k3e2 . k3e3 . k~ e4 . kl - 23uel k2e2" k l e 3 " k2e4" kl - 17Uel *'k2e2" klE3" k2e4" k2 + 5Uel-k2e2" kae3. kle4" kl + uel. k2e2. k3e3. kl e4" k2 + 5UEI" k2e2" k3E3" k2E4-kl+6Uel-k2E2" k3e3" k2E4" k2 1Tues. k3E2"k3e3" klQ- k2 -~-u£1 •k3£2"k3£3"k2£4" kl + 6ue~. k3£2"k3 £3"k2e4. k2 - 4ue~. k3 £2" k1¢3. k2£4" k~ T U£ 1 • k3£2" k1£3. k2£4" k2 - 18u£~. k~£2. k1£3" klea. k2) and L2 = - (£~. £~) (£3" £4)s (2u 2 + 3tu + 2t 2) + - - . + 2 (£1.
$2 = (O, Rp,q,r, sO~Rp,q,~,t)(Rp,,q,,r,,sRp,,q, #,,t)
£2 ) ( t 2 (2£4"kz£3"k4 +£3"k~ £4"k2)q -u2 (2£34"k4£4"k3 q%
£3 . k2£4 . k ~) + tu( 2e3 . k4 £4 "k3 + e3 "k l e4 "k2 + e3 "k2£4 "
$3 = (OaRp,q,r,sRp,q,r,t)(OaRv,q,,r,,sRV,q,,r,,t) $4 = OaRp,q,r,sO" Rp,q,t,uRe,mv,wRr,s,v,w $5 = OaRp,q,~,~O~ Rp,q,e,uRr,t,~,~Rs,~,v,~ $7 = OaRp,q,r, sOa Rp,t,r,uRt,v,u,wRq,v,s,w Ss : cOaRp,q,r,sOa Rp,t,r,uRt,v,q,wRu,v,s,w $9 = I~,b,~,d R~,~,~,aO,~ R~,f ,~,h O~ R¢,I,~,h
$10 = OpR~,~,t,,~OqR~,~,~,~Rq,d,t,~Rp,d,,~,~. For $1 to $8 the derivatives are contracted with each other and these Si's are related to derivatives acting upon the Ti's of the previous sections. Tensors So and Sm however have the derivatives contracted into the Riemann tensors. Of course many tensors of the form c?2R 4 vanish onshell since they produce amplitudes of the form
D. Seminara/Nuclear Physics B (Proc. Suppl.) 104 (2002) 204-207
~- (s + t + u) x tensor --- 0. In terms of the 5"/'s the infinities are cancelled by the counterterms
4. 5.
c
(4~) ~
'
and __l_(2)4
6.
i_._~cN=4,D=lO (4~) 5
'
where cg=6,D=lO _ I 8~6
1
4.720
($1 \
-
7.
12S2 - 4S3 + 25'4 -t-
+ 16S7 + 8Ss),
cg=4,D=lO _
-1
4.6048
(9S1
\
-
76,-92 + 44S3 + 30S4
8.
-56S5 + 88S6 + 16S7 - 88Ss + 24S9 - 95S10). It is far from obvious that such counterterms lead to infinities which factorise, however we can manipulate them to do so. In fact it is possible to express both tensors in the form
9.
t l O t 8 0 2 R 2 -- tala2aaaaasaeaTaaagal°t 10 8blb2b3bab~ bebvbs Oa D blb2fQ D b3b,. bsbel:~ bTbs (6) 1.L~a~as va4 ~ Lasa6 ~rata8 " ~ a 9 alO "
10.
where we have chosen to contract the derivatives into the tl0 tensor. The specific form of these tensor can be found in Ref. [12]. The counterterms for D = 10, N = 1 supergravities axe given by linear combinations of the vaxious M D = l ° and in fact the infinities are given by K1 × ~ i ciLi since the various infinities factorise in this way. At the level of the counterterms, all type I supergravity divergences contain the factor ts. REFERENCES
1. E. Cremmer, B. Julia and J. Scherk, Phys. Lett. B76 (1978) 409. 2. E. Cremmer and B. Julia, Phys. Lett. B80 (1978) 48; Nucl. Phys. B159 (1979) 141. 3. Z. Bern, L. Dixon, D.C. Dunbar, M. Perelstein and J.S. Rozowsky, Nucl. Phys. B530 (1998) 401, hel>-th/9802162; Class. Quant. Grav. 17 (2000) 979, hep-th/9911194; Z. Bern, L. Dixon, D.C. Dunbar, A.K. Grant,
11.
12.
13.
207
M. Perelstein and J.S. Rozowsky, Nucl. Phys. Proc. Suppl. B88 (2000) 194, hep-th/0002078. S. Deser and D. Seminara, Phys. Rev. Lett. 82 (1999) 2435, hep-th/9812136. M.B. Green, J.H. Schwarz and E. Witten, Superstring Theory, Cambridge University Press, 1987. M.B. Green, J.H. Schwarz and L. Brink, Nucl Phys. B198 (1982) 474. L.D. Landau, Nucl. Phys. B13 (1959) 181; S. Mandelstam, Phys. Rev. 112 (1958) 1344; Phys. Rev. 115 (1959) 1741; R.E. Cutkosky, J. Math. Phys. 1 (1960) 429; Z. Bern, L. Dixon, D.C. Dunbar and D.A. Kosower, Nucl. Phys. B425 (1994) 217; Nucl. Phys. B435 (1995) 59. Z. Bern and D.A. Kosower, Nucl. Phys. B379 (1992) 451; D.C. Dunbar and T. Shimada, Phys. Lett. B312 (1993) 277; D.C. Dunbar and P.S. Norridge, Nucl. Phys. B433 (1995) 181; Z. Bern and D.C. Dunbar, Nucl. Phys. B379 (1992) 562; Z. Bern, Phys. Lett. B296 (1992) 85. I. Robinson, unpublished; L. Bel, Acad. Sci. Paris, Comptes Rend. 247 (1958) 1094 and 248 (1959) 1297. Z. Bern and A.G. Morgan, Nucl. Phys. B467 (1996) 479; Z. Bern, G. Chalmers, L. Dixon and D.A. Kosower, Phys. Rev. Lett. 72 (1994) 2134; Z. Bern and G. Chalmers, Nucl. Phys. B447 (1995) 465. M.B. Green, M. Gutperle and P. Vanhove, Phys. Lett. B409 (1997) 177; J.G. Russo and A.A. Tseytlin, Nucl. Phys. B508 (1997) 245. D.C. Dunbar, B. Julia, D. Seminara and M. Trigiante, J. High Ener. Phys. 0001 (2000) 046. S.A. Fulling, R.C. King, B.G. Wybourne and C.J. Cummins, Class. Quant. Grav. 9 (1992) 1151.