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Non-renormalizable terms and M theory during inflation
12 February 1998
Physics Letters B 419 Ž1998. 57–61
Non-renormalizable terms and M theory during inflation David H. Lyth Department of Physics, UniÕersity of Lancaster, Lancaster LA1 4YB, UK Received 13 October 1997 Editor: M. Dine
Abstract Inflation is well known to be difficult in the context of supergravity, if the potential is dominated by the F term. Non-renormalizable terms generically give < V XX < ; VrM 2 , where V Ž f . is the inflaton potential and M is the scale above which the effective field theory under consideration is supposed to break down. This is equivalent to
1. To achieve slow-roll inflation, the potential V Ž f . must satisfy the flatness conditions e < 1 and
e'
1 2
M Pl2
X
Ž V rV .
2
Ž 1.
h ' M Pl2 V XXrV
Ž 2.
and M Pl s Ž8p G .y1 r2 s 2.4 = 10 18 GeV is the reduced Planck mass. When these are satisfied, the time dependence of the inflaton f is generally given by the slow-roll expression 3 Hf˙ s yV X , where H , 1 y2 3 M Pl V is the Hubble parameter during inflation. On a given scale, the spectrum of the primordial curvature perturbation, thought to be the origin of structure in the Universe, is given by
(
d H2 Ž k . s
1 150p
2
V M Pl4
e
Ž 3.
The right hand side is evaluated when the relevant scale k leaves the horizon. On large scales, the
COBE observation of the cmb anisotropy corresponds to V 1r4re 1r4 s .027M Pl s 6.7 = 10 16 GeV
Ž 4.
The spectral index of the primordial curvature perturbation is given by n y 1 s 2h y 6 e
Ž 5.
A perfectly scale-independent spectrum would correspond to n s 1, and observation already demands < n y 1 < - 0.2. Thus e and h have to be Q 0.1 Žbarring a cancellation. and this constraint will get tighter if future observations move n closer to 1. Many models of inflation predict that this will be the case, some giving a value of n completely indistinguishable from 1. Usually, f is supposed to be charged under at least a Z2 symmetry f ™ yf , which is unbroken during inflation. Then V X s 0 at the origin, and inflation typically takes place near the origin. As a
0370-2693r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 3 7 0 - 2 6 9 3 Ž 9 7 . 0 1 4 9 6 - 2
D.H. Lyth r Physics Letters B 419 (1998) 57–61
58
result e negligible compared with h , and n y 1 s 2h ' 2 M Pl2 V XXrV. We assume that this is the case in what follows. If it is not, the nonrenormalizable terms generically give both
Ž 6.
Supergravity is a non-renormalizable field theory, whose lagrangian presumably contains an infinite number of non-renormalizable terms. Taking the theory to be an effective one, holding up to some scale M, the coefficients of the non-renormalizable terms are expected to be generically of order 1, in units of M. Hopefully, they can be calculated if one understands what goes on at scales above M. The most optimistic view is to take M to be the scale above which field theory itself breaks down, due to gravitational effects such as the quantum fluctuation in the spacetime metric. It is not unreasonable to take this view in the context of inflation, if the inflaton is a gauge singlet, and we adopt it here. 1 With four spacetime dimensions, this means that M s M Pl . If more dimensions open up well below the Planck scale, the answer is less obvious Žsee below.. The non-renormalizable terms respect the gauge symmetries possessed by the renormalizable theory, but they need not respect its global symmetries. On the contrary, it is generally felt that the usual continuous global symmetries Žbuilt out of UŽ1.’s acting on the phases of the complex fields. will not generically be respected, at least if M is the scale at which gravitational effects spoil field theory. This viewpoint derives in part from superstring theory, but that theory also suggests that discrete subgroups of the global symmetries Žbuilt out of ZN ’s. will occur. By imposing suitable discrete symmetries, one can ensure that a global continuous symmetry is approxi-
mately preserved at field values < M w3x. This is important, because several cases are known where an approximate global UŽ1. is phenomenological desirable. The best known case is Peccei-Quinn symmetry, which must be preserved to high accuracy in order to keep the axion sufficiently light. Some time ago w4x, it was emphasized that non-renormalizable terms will contribute to VF , generically giving < VFXX < ; VFrM 2 . 2 Most models of inflation have the F-term dominating Ž F-term inflation. and then the non-renormalizable terms in VF give
Ž 7.
This generic result is too big, since slow-roll inflation per se requires
1
For fields charged under the Standard Model gauge interactions one should use M s MGU T ,10 16 GeV in the low energy theory, and for hiddenrsecluded sector fields below the condensation scale L one should use M s L.
2
The result holds actually for the second derivative in any field direction, as was noted a long time ago w5,6x.
D.H. Lyth r Physics Letters B 419 (1998) 57–61
anyhow sufficiently small values. along with VF Then V , VD ,
j 2g2 2
Re fy1
Ž 8.
where g is the gauge coupling of the relevant UŽ1., and the gauge kinetic function f is a holomorphic function of all of the complex scalar fields fn . If we regard g 2 as fixed, f must be invariant under all internal symmetries. At a given point in field space, one can choose f s 1 corresponding to canonical normalization of the gauge field of the relevant UŽ1.. This point is conveniently taken to be the origin, defined as the fixed point of the usual internal symmetries. Then, along say the f 1 direction, 1rf s 1 q l My2f 12 q PPP
59
strong interaction. It has to be preserved to high accuracy in order to keep the axion sufficiently light, and it appears that discrete symmetries of the model ensure this. 3 Of course, this means that the quadratic term in f is killed. This model has the virtue of making contact with both Peccei-Quinn symmetry and the Standard Model. On the negative side, it so far lacks a definite origin for the Fayet-Illiopoulos term; in contrast with the other models of D-term inflation, a direct origin in the superstring seems impossible since V 1r4 is extremely low. 3. Let us briefly recall the situation for the F term w4x. At least at tree level, 2
VF s e K r M Pl =
Ž 9.
nm Ý Ž Wn q My2 ž Wm q My2 Pl WK n . K Pl WK m /
nm
There is no linear term, unless f 1 is a singlet under all of the internal symmetries that are unbroken during inflation. The quadratic term is allowed if the only symmetry is f 1 ™ yf 1. Then, if inflaton is f s '2 Re f 1 , it gives a contribution h s lŽ M Pl rM . 2 , with l generically of order 1. The offending term is forbidden if there is a ZN symmetry Ž f 1 ™ expŽ i a . f 1 with a s 2prN . with N G 3. It is also forbidden if there is a global UŽ1. symmetry, corresponding to arbitrary a . In all of the models of D-term inflation proposed so far, such a global UŽ1. is present as an R symmetry, with W A f 1. In the simplest case,
Here, W is the superpotential Žholomorphic in the fields. and K is the Kahler potential Ža non-singular ¨ function of the fields and their complex conjugates.. A subscript n denotes ErEfn , and a subscript m denotes ErEfm . Also, K n m is the matrix inverse of K n m . Only the combination G ' K q ln < W < 2 is physically significant. Canonically normalizing the fields at Žsay. the origin corresponds to
W s cf 1 f 2 f 3
ŽAny linear term can be absorbed into W.. Then,
Ž 10 .
where f 2 and f 3 oppositely charged under the relevant gauge UŽ1.. One indeed needs to forbid terms in W of the form f 12fn Ž n / 1. because they would generate a quartic term in the inflaton potential and spoil inflation. But to achieve this it is enough to have the symmetry f 1 ™ yf 1 Žacting on W as an R symmetry.. I am here pointing out that this symmetry is not enough to forbid the disastrous quadratic term in f ; it has to be promoted to Z2 N with N ) 1, or to the full UŽ1.. One hopes that the superstring will ultimately determine which discrete symmetry actually holds, if any. In one model of D-term inflation w15x, the global Ž U 1. is the Peccei-Quinn symmetry, whose non-perturbative breaking ensures the CP invariance of the
< <2 y3My2 Pl W
Ž 11 .
K s Ý < fn < 2 q O Ž fn3 .
Ž 12 .
n
K n m s dn m q My2 Ý l n < fn < 2 q PPP
Ž 13 .
n
In the last expression, the terms not displayed are linear and higher; the quadratic term displayed is a particularly simple one, which cannot be forbidden by any of the usual symmetries. Now make again the assumption that f s Re f 1r '2 ; as we remark in a moment, the opposite assumption that f corresponds instead to the phase
3
so.
In an earlier version of this paper I stated that they will not do
D.H. Lyth r Physics Letters B 419 (1998) 57–61
60
of f 1 would give something dramatically different. For simplicity, also set e K s 1 corresponding to small field values. Then, one can identify some contributions to V XX , V XX s My2 V y < W1 < 2 q My2 Pl
izable terms. A more promising avenue is to suppose that K and W have very special forms w7,21,22x, corresponding roughly to versions of no-scale supergravity. In this case, it may be justified to use the limit of renormalizable global supersymmetry.
Ý l n < Wn < 2 q PPP n
Ž 14 . In the first bracket, V comes from differentiating e K , and the coefficient of y< W1 < 2 is the sum of q2 coming from the first term in the bracket of Eq. Ž11., < <2 and y3 coming from the y3My2 Pl W . The three terms displayed will give a contribution
h s 1 y a q b Ž M Pl rM .
2
Ž 15 .
with generically < b < ; 1. It was pointed out in w4x if W s L2f 1 during inflation, then a s 1. ŽIt is easy to construct models of inflation where this is exactly w4,16x or approximately w17x true.. In that case, one need only require < b < < 1, corresponding to an accidental suppression of the other non-renormalizable contributions. Some authors w17–19x have taken the view that this is an improvement on the generic situation, where one needs in addition the accident < a y 1 < < 1. Notice that Eq. Ž11. contains M Pl , as distinct from the scale M above which field theory is supposed to break down. Taking M Pl to infinity with M fixed converts supergravity into a non-renormalizable globally supersymmetric theory. More usually, one considers the limit where M Pl and M go to infinity together, corresponding to a renormalizable globally supersymmetric theory. It is clear from the form of Eqs. Ž4. and Ž15. that, during inflation, neither of these prescriptions should be used without justification. As we have just seen though, the use of non-renormalizable global supersymmetry can be justified if the superpotential is linear. If b ; 1 Žthe generic case. the only way of evading the result
4. Finally, I note that the scale M might not be as big as M Pl if additional space dimensions open up below M Pl . I have in mind a specific example, where the underlying theory is an M theory w23x. A field theory is valid below some scale Q - M Pl , and it is attractive to choose Q ; 10y2 M Pl ; ))10 16.5 GeV to account for the apparent unification of the gauge couplings. This field theory lives in effectively five spacetime dimensions, until we get down to some still lower scale. If the fifth dimension makes no significant difference, the scale M of the non-renormalizable terms will be M s Q ; 10y2 M Pl . Otherwise M will presumably be lower, though with two mass scales in the underlying theory the concept of a single scale M may not even be useful. As yet nothing has been worked regarding these questions, even in the specific example mentioned. If M is indeed of order )) 10 16.5 GeV, a fieldtheory model of inflation will not make sense w24x if V 1r4 ; 10 16.5 GeV, the maximum allowed by the COBE normalization Eq. Ž4.. A four-dimensional field theory model will not make sense unless V 1r4 is even lower. However, many models have been proposed with low V 1r4 . Of these, the simplest is the D-term model, with the slope of the potential coming from the 1-loop correction w9,10x. In that case COBE normalization requires w11,12x V 1r4
Ž g 2r2.
1r4
s j s 3 = 10 15 GeV
'
Ž 16 .
This might be low enough to justify the use of four-dimensional field theory. Also, if j comes the superstring, it will be a loop suppression factor times M 2 , which might be compatible with the M theory model just mentioned. Again, this has yet to be investigated. The most promising F-term model is probably that of w25x. Starting at the scale f s M with the generic V XX Ž f . , Ž M PlrM . 2 V, renormalization
D.H. Lyth r Physics Letters B 419 (1998) 57–61
group equations are invoked to run this quantity to zero at a scale f < M, where inflation occurs. In this case one can have V 1r4 as low as Mm s where m s s 100 GeV, amply justifying the use of four-dimensional field theory. As given, this model takes M s M Pl , and it is unclear whether the change M ; M Pl r100 would make an important difference.
(
Acknowledgements I thank Toni Riotto for useful comments on a first draft of this paper. This work is partially supported by grants from PPARC and from the European Commission under the Human Capital and Mobility programme, contract No. CHRX-CT94-0423. References w1x For a review of this material concerning inflation, see A.R. Liddle, D.H. Lyth, Phys. Rep. 231 Ž1993. 1. w2x For reviews of supersymmetry and supergravity, see H.P. Nilles, Phys. Rep. 110 Ž1984. 1; H.E. Haber, G.L. Kane, Phys. Rep. 117 Ž1985. 75; D. Bailin, A. Love, Supersymmetric Gauge Field Theory and String Theory, IOP, Bristol, 1994. w3x A useful review of these questions is given by M. Dine, hep-thr9207045. w4x E.J. Copeland, A.R. Liddle, D.H. Lyth, E.D. Stewart, D. Wands, Phys. Rev. D 49 Ž1994. 6410.
61
w5x M. Dine, W. Fischler, D. Nemeschansky, Phys. Lett. B 136 Ž1984. 169. w6x G.D. Coughlan, R. Holman, P. Ramond, G.G. Ross, Phys. Lett. B 140 Ž1984. 44. w7x E.D. Stewart, Phys. Rev. D 51 Ž1995. 6847. w8x J.A. Casas, C. Munoz, Phys. Lett. B 216 Ž1989. 37; J.A. Casas, J.M. Moreno, C. Munoz, M. Quiros, Nucl. Phys. B 328 Ž1989. 272. w9x P. Binetruy, G. Dvali, Phys. Lett. B 388 Ž1996. 241. w10x E. Halyo, Phys. Lett. B 387 Ž1996. 43. w11x R. Jeannerot, hep-phr9706391. w12x G. Dvali, A. Riotto, hep-phr9706408. w13x J.A. Casas, G.B. Gelmini, hep-phr9706439. w14x D.H. Lyth, A. Riotto, hep-phr9707273. w15x M. Bastero-Gil, S.F. King, hep-phr9709502. w16x G. Dvali, Q. Shafi, R. Schaefer, Phys. Rev. Lett. 73 Ž1994. 1886. w17x K.-I. Izawa, T. Yanagida, hep-phr9608359. w18x A.D. Linde, A. Riotto, hep-phr9703209, to be published in Phys. Rev. D. w19x S. Dimopoulos, G. Dvali, R. Rattazzi, hep-phr9705348. w20x K. Freese, J. Frieman, A.V. Olinto Phys. Rev. Lett. 65 Ž1990. 3233; F.C. Adams et al. 1993, Phys. Rev. D 47 Ž1993. 426; L. Knox, A. Olinto, Phys. Rev. D 48 Ž1993. 946; W.H. Kinney, K.T. Mahanthappa, Phys. Rev. D 52 Ž1995. 5529; D 53 Ž1996. 5455; J. Garcia-Bellido, A. Linde, D. Wands, astro-phr9605094. w21x M.K. Gaillard, H. Murayama, K.A. Olive, Phys. Lett. B 355 Ž1995. 71. w22x M.K. Gaillard, H. Murayama, D.H. Lyth, in preparation. w23x E. Witten, Nucl. Phys. B 471 Ž1996. 135. w24x This observation has been made by T. Banks, M. Dine, hep-thr9609046. w25x E.D. Stewart, hep-phr9703232.
Physics Letters B 419 Ž1998. 57–61
Non-renormalizable terms and M theory during inflation David H. Lyth Department of Physics, UniÕersity of Lancaster, Lancaster LA1 4YB, UK Received 13 October 1997 Editor: M. Dine
Abstract Inflation is well known to be difficult in the context of supergravity, if the potential is dominated by the F term. Non-renormalizable terms generically give < V XX < ; VrM 2 , where V Ž f . is the inflaton potential and M is the scale above which the effective field theory under consideration is supposed to break down. This is equivalent to
1. To achieve slow-roll inflation, the potential V Ž f . must satisfy the flatness conditions e < 1 and
e'
1 2
M Pl2
X
Ž V rV .
2
Ž 1.
h ' M Pl2 V XXrV
Ž 2.
and M Pl s Ž8p G .y1 r2 s 2.4 = 10 18 GeV is the reduced Planck mass. When these are satisfied, the time dependence of the inflaton f is generally given by the slow-roll expression 3 Hf˙ s yV X , where H , 1 y2 3 M Pl V is the Hubble parameter during inflation. On a given scale, the spectrum of the primordial curvature perturbation, thought to be the origin of structure in the Universe, is given by
(
d H2 Ž k . s
1 150p
2
V M Pl4
e
Ž 3.
The right hand side is evaluated when the relevant scale k leaves the horizon. On large scales, the
COBE observation of the cmb anisotropy corresponds to V 1r4re 1r4 s .027M Pl s 6.7 = 10 16 GeV
Ž 4.
The spectral index of the primordial curvature perturbation is given by n y 1 s 2h y 6 e
Ž 5.
A perfectly scale-independent spectrum would correspond to n s 1, and observation already demands < n y 1 < - 0.2. Thus e and h have to be Q 0.1 Žbarring a cancellation. and this constraint will get tighter if future observations move n closer to 1. Many models of inflation predict that this will be the case, some giving a value of n completely indistinguishable from 1. Usually, f is supposed to be charged under at least a Z2 symmetry f ™ yf , which is unbroken during inflation. Then V X s 0 at the origin, and inflation typically takes place near the origin. As a
0370-2693r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 3 7 0 - 2 6 9 3 Ž 9 7 . 0 1 4 9 6 - 2
D.H. Lyth r Physics Letters B 419 (1998) 57–61
58
result e negligible compared with h , and n y 1 s 2h ' 2 M Pl2 V XXrV. We assume that this is the case in what follows. If it is not, the nonrenormalizable terms generically give both
Ž 6.
Supergravity is a non-renormalizable field theory, whose lagrangian presumably contains an infinite number of non-renormalizable terms. Taking the theory to be an effective one, holding up to some scale M, the coefficients of the non-renormalizable terms are expected to be generically of order 1, in units of M. Hopefully, they can be calculated if one understands what goes on at scales above M. The most optimistic view is to take M to be the scale above which field theory itself breaks down, due to gravitational effects such as the quantum fluctuation in the spacetime metric. It is not unreasonable to take this view in the context of inflation, if the inflaton is a gauge singlet, and we adopt it here. 1 With four spacetime dimensions, this means that M s M Pl . If more dimensions open up well below the Planck scale, the answer is less obvious Žsee below.. The non-renormalizable terms respect the gauge symmetries possessed by the renormalizable theory, but they need not respect its global symmetries. On the contrary, it is generally felt that the usual continuous global symmetries Žbuilt out of UŽ1.’s acting on the phases of the complex fields. will not generically be respected, at least if M is the scale at which gravitational effects spoil field theory. This viewpoint derives in part from superstring theory, but that theory also suggests that discrete subgroups of the global symmetries Žbuilt out of ZN ’s. will occur. By imposing suitable discrete symmetries, one can ensure that a global continuous symmetry is approxi-
mately preserved at field values < M w3x. This is important, because several cases are known where an approximate global UŽ1. is phenomenological desirable. The best known case is Peccei-Quinn symmetry, which must be preserved to high accuracy in order to keep the axion sufficiently light. Some time ago w4x, it was emphasized that non-renormalizable terms will contribute to VF , generically giving < VFXX < ; VFrM 2 . 2 Most models of inflation have the F-term dominating Ž F-term inflation. and then the non-renormalizable terms in VF give
Ž 7.
This generic result is too big, since slow-roll inflation per se requires
1
For fields charged under the Standard Model gauge interactions one should use M s MGU T ,10 16 GeV in the low energy theory, and for hiddenrsecluded sector fields below the condensation scale L one should use M s L.
2
The result holds actually for the second derivative in any field direction, as was noted a long time ago w5,6x.
D.H. Lyth r Physics Letters B 419 (1998) 57–61
anyhow sufficiently small values. along with VF Then V , VD ,
j 2g2 2
Re fy1
Ž 8.
where g is the gauge coupling of the relevant UŽ1., and the gauge kinetic function f is a holomorphic function of all of the complex scalar fields fn . If we regard g 2 as fixed, f must be invariant under all internal symmetries. At a given point in field space, one can choose f s 1 corresponding to canonical normalization of the gauge field of the relevant UŽ1.. This point is conveniently taken to be the origin, defined as the fixed point of the usual internal symmetries. Then, along say the f 1 direction, 1rf s 1 q l My2f 12 q PPP
59
strong interaction. It has to be preserved to high accuracy in order to keep the axion sufficiently light, and it appears that discrete symmetries of the model ensure this. 3 Of course, this means that the quadratic term in f is killed. This model has the virtue of making contact with both Peccei-Quinn symmetry and the Standard Model. On the negative side, it so far lacks a definite origin for the Fayet-Illiopoulos term; in contrast with the other models of D-term inflation, a direct origin in the superstring seems impossible since V 1r4 is extremely low. 3. Let us briefly recall the situation for the F term w4x. At least at tree level, 2
VF s e K r M Pl =
Ž 9.
nm Ý Ž Wn q My2 ž Wm q My2 Pl WK n . K Pl WK m /
nm
There is no linear term, unless f 1 is a singlet under all of the internal symmetries that are unbroken during inflation. The quadratic term is allowed if the only symmetry is f 1 ™ yf 1. Then, if inflaton is f s '2 Re f 1 , it gives a contribution h s lŽ M Pl rM . 2 , with l generically of order 1. The offending term is forbidden if there is a ZN symmetry Ž f 1 ™ expŽ i a . f 1 with a s 2prN . with N G 3. It is also forbidden if there is a global UŽ1. symmetry, corresponding to arbitrary a . In all of the models of D-term inflation proposed so far, such a global UŽ1. is present as an R symmetry, with W A f 1. In the simplest case,
Here, W is the superpotential Žholomorphic in the fields. and K is the Kahler potential Ža non-singular ¨ function of the fields and their complex conjugates.. A subscript n denotes ErEfn , and a subscript m denotes ErEfm . Also, K n m is the matrix inverse of K n m . Only the combination G ' K q ln < W < 2 is physically significant. Canonically normalizing the fields at Žsay. the origin corresponds to
W s cf 1 f 2 f 3
ŽAny linear term can be absorbed into W.. Then,
Ž 10 .
where f 2 and f 3 oppositely charged under the relevant gauge UŽ1.. One indeed needs to forbid terms in W of the form f 12fn Ž n / 1. because they would generate a quartic term in the inflaton potential and spoil inflation. But to achieve this it is enough to have the symmetry f 1 ™ yf 1 Žacting on W as an R symmetry.. I am here pointing out that this symmetry is not enough to forbid the disastrous quadratic term in f ; it has to be promoted to Z2 N with N ) 1, or to the full UŽ1.. One hopes that the superstring will ultimately determine which discrete symmetry actually holds, if any. In one model of D-term inflation w15x, the global Ž U 1. is the Peccei-Quinn symmetry, whose non-perturbative breaking ensures the CP invariance of the
< <2 y3My2 Pl W
Ž 11 .
K s Ý < fn < 2 q O Ž fn3 .
Ž 12 .
n
K n m s dn m q My2 Ý l n < fn < 2 q PPP
Ž 13 .
n
In the last expression, the terms not displayed are linear and higher; the quadratic term displayed is a particularly simple one, which cannot be forbidden by any of the usual symmetries. Now make again the assumption that f s Re f 1r '2 ; as we remark in a moment, the opposite assumption that f corresponds instead to the phase
3
so.
In an earlier version of this paper I stated that they will not do
D.H. Lyth r Physics Letters B 419 (1998) 57–61
60
of f 1 would give something dramatically different. For simplicity, also set e K s 1 corresponding to small field values. Then, one can identify some contributions to V XX , V XX s My2 V y < W1 < 2 q My2 Pl
izable terms. A more promising avenue is to suppose that K and W have very special forms w7,21,22x, corresponding roughly to versions of no-scale supergravity. In this case, it may be justified to use the limit of renormalizable global supersymmetry.
Ý l n < Wn < 2 q PPP n
Ž 14 . In the first bracket, V comes from differentiating e K , and the coefficient of y< W1 < 2 is the sum of q2 coming from the first term in the bracket of Eq. Ž11., < <2 and y3 coming from the y3My2 Pl W . The three terms displayed will give a contribution
h s 1 y a q b Ž M Pl rM .
2
Ž 15 .
with generically < b < ; 1. It was pointed out in w4x if W s L2f 1 during inflation, then a s 1. ŽIt is easy to construct models of inflation where this is exactly w4,16x or approximately w17x true.. In that case, one need only require < b < < 1, corresponding to an accidental suppression of the other non-renormalizable contributions. Some authors w17–19x have taken the view that this is an improvement on the generic situation, where one needs in addition the accident < a y 1 < < 1. Notice that Eq. Ž11. contains M Pl , as distinct from the scale M above which field theory is supposed to break down. Taking M Pl to infinity with M fixed converts supergravity into a non-renormalizable globally supersymmetric theory. More usually, one considers the limit where M Pl and M go to infinity together, corresponding to a renormalizable globally supersymmetric theory. It is clear from the form of Eqs. Ž4. and Ž15. that, during inflation, neither of these prescriptions should be used without justification. As we have just seen though, the use of non-renormalizable global supersymmetry can be justified if the superpotential is linear. If b ; 1 Žthe generic case. the only way of evading the result
4. Finally, I note that the scale M might not be as big as M Pl if additional space dimensions open up below M Pl . I have in mind a specific example, where the underlying theory is an M theory w23x. A field theory is valid below some scale Q - M Pl , and it is attractive to choose Q ; 10y2 M Pl ; ))10 16.5 GeV to account for the apparent unification of the gauge couplings. This field theory lives in effectively five spacetime dimensions, until we get down to some still lower scale. If the fifth dimension makes no significant difference, the scale M of the non-renormalizable terms will be M s Q ; 10y2 M Pl . Otherwise M will presumably be lower, though with two mass scales in the underlying theory the concept of a single scale M may not even be useful. As yet nothing has been worked regarding these questions, even in the specific example mentioned. If M is indeed of order )) 10 16.5 GeV, a fieldtheory model of inflation will not make sense w24x if V 1r4 ; 10 16.5 GeV, the maximum allowed by the COBE normalization Eq. Ž4.. A four-dimensional field theory model will not make sense unless V 1r4 is even lower. However, many models have been proposed with low V 1r4 . Of these, the simplest is the D-term model, with the slope of the potential coming from the 1-loop correction w9,10x. In that case COBE normalization requires w11,12x V 1r4
Ž g 2r2.
1r4
s j s 3 = 10 15 GeV
'
Ž 16 .
This might be low enough to justify the use of four-dimensional field theory. Also, if j comes the superstring, it will be a loop suppression factor times M 2 , which might be compatible with the M theory model just mentioned. Again, this has yet to be investigated. The most promising F-term model is probably that of w25x. Starting at the scale f s M with the generic V XX Ž f . , Ž M PlrM . 2 V, renormalization
D.H. Lyth r Physics Letters B 419 (1998) 57–61
group equations are invoked to run this quantity to zero at a scale f < M, where inflation occurs. In this case one can have V 1r4 as low as Mm s where m s s 100 GeV, amply justifying the use of four-dimensional field theory. As given, this model takes M s M Pl , and it is unclear whether the change M ; M Pl r100 would make an important difference.
(
Acknowledgements I thank Toni Riotto for useful comments on a first draft of this paper. This work is partially supported by grants from PPARC and from the European Commission under the Human Capital and Mobility programme, contract No. CHRX-CT94-0423. References w1x For a review of this material concerning inflation, see A.R. Liddle, D.H. Lyth, Phys. Rep. 231 Ž1993. 1. w2x For reviews of supersymmetry and supergravity, see H.P. Nilles, Phys. Rep. 110 Ž1984. 1; H.E. Haber, G.L. Kane, Phys. Rep. 117 Ž1985. 75; D. Bailin, A. Love, Supersymmetric Gauge Field Theory and String Theory, IOP, Bristol, 1994. w3x A useful review of these questions is given by M. Dine, hep-thr9207045. w4x E.J. Copeland, A.R. Liddle, D.H. Lyth, E.D. Stewart, D. Wands, Phys. Rev. D 49 Ž1994. 6410.
61
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