Journal ArticleThe present time variation of coupling constants in superstring theories with currently favorable internal backgrounds critically depends on the shape of the potential for the size of the internal space. If the potential is almost flat, as in perturbation theory to all orders, the value of G/G for Newton's gravitational constant is calculable and estimated to be - 1x10- 11 ± yr-1 . If the potential has a minimum with finite curvature due to unknown nonperturbative effects, G/G will become unobservably small. Improvement of the measurement of G/ G would discriminate between  the two situations. Problems with the time variation of other coupling constants are also discussed

Wang, Zi

Wu, Yong-Shi

English

Journal ArticleThe present time variation of coupling constants in superstring theories with currently favorable internal backgrounds critically depends on the shape of the potential for the size of the internal space. If the potential is almost flat, as in perturbation theory to all orders, the value of G/G for Newton\u27s gravitational constant is calculable and estimated to be - 1x10- 11 ± yr-1 . If the potential has a minimum with finite curvature due to unknown nonperturbative effects, G/G will become unobservably small. Improvement of the measurement of G/ G would discriminate between  the two situations. Problems with the time variation of other coupling constants are also discussed

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VOLUME 57, NUMBER 16 PHYSICAL REVIEW LETTERS 20 OCTOBER 1986 Time Variation of Newton's Gravitational Constant in Superstring Theories Yong-Shi Wu Department of Physics, University of Utah, Salt Lake City, Utah 84112 and Zi Wang Department of Physics, Utah State University, Logan, Utah 84322 (Received 29 May 1986) The present time variation of coupling constants in superstring theories with currently favorable internal backgrounds critically depends on the shape of the potential for the size of the internal space. If the potential is almost flat, as in perturbation theory to all orders, the value of G/ G for Newton's gravitational constant is calculable and estimated to be - 1 x 10- 11 ± 1 yr- I . If the poten-tial has a minimum with finite curvature due to unknown nonperturbative effects, G / G will be-come unobservably small. Improvement of the measurement of G/ G would discriminate between the two situations. Problems with the time variation of other coupling constants are also discussed. PACS numbers: 12.1O.Gq, 04.50.+h, 06.20.Jr, 98.80.Dr The time variation of fundamental constants may provide a connection between cosmology and particle physics. This idea can be traced back to Dirac, 1 although his original proposal for variation in Newton's gravitational constant G seems not support-ed by observations.2 Very recently superstring theories3,4 appear to be promising candidates for a consistent quantum theory unifying all known interactions including gravity. They provide a suitable framework for studying the time variation of fundamental constants.5 The con-sistency of superstring theories fixes the space-time dimensionality to be ten, six of which form a very small compact manifold K ( - 10- 32 cm). The metric and other bosonic backgrounds in K are constrained by string-compactification and particle-phenomenology considerations.6,7 The coupling constants in the four-dimensional world are related to those in ten dimen-sions by a factor of the inverse volume of K. The cosmology in the more-dimensional universe governs the evolution of the usual three-space as well as that of K and, through the latter, dynamically determines the time variation of coupling constants in four dimen-sions. Generally in a more-dimensional field-theory approach,s quantum effects in K 9 give rise to an effec-tive potential which may fix the size of the internal space R6 in vacuum and influences its cosmological evolution. But in superstring theories, Witten's non-renormalization theorem lO tells us that such a potential for R6 is flat up to all orders in perturbation theory. So far, the study of nonperturbative supersymmetry-breaking effects, 11,12 including world-sheet instantons, also has failed to produce a potential with a minimum at finite R6, whose existence is expected by the con-ventional wisdom. In this Letter we will show that the time variation of coupling constants critically depends on the shape of this potential. If the potential is flat the present value of G/ G is calculable; for example, for an open universe 0) where Ho is the Hubble constant, to the age of the universe, qo the deceleration parameter, and flo == 811' x Gopol3HJ the density parameter. Here Po is the density in ordinary three-space and the subscript 0 denotes the present value of the quantity. We esti-mate ( G / G)o to be in the range (2) which overlaps the present observational upper bound13 (3) However, if the potential really has a minimum at fin-ite R 6, (G / G)o will be suppressed and become unob-servably small. So an improvement on the measure-ments of G/ G will give us important information about the shape of the potential. Here we concentrate on G/ G, since theoretically it is independent of the di-lation field and experimentally extracting it from data is simple and direct. Some remarks about time varia-tion of other coupling constants in superstring theories are given at the end of the Letter. We start with the following equations of motion in 1978 © 1986 The American Physical Society VOLUME 57, NUMBER 16 PHYSICAL REVIEW LETTERS 20 OCTOBER 1986 ten dimensions: RAB - tgABR = t-K1orf>-3/2(HAMNHBMN- t;gAB Hl1NP) +9K1o\1M(rf>-3/2HAPQRMBPQ) + trf>-2[a Arf>a Brf>- tgAB(aMrf»2] + 3~ KIorf>-3/4(TrFAMFBM- t-gAB TrFl1N) + tKtorf> -3/4[ tgAB( Rl1NPQ -- 4Rl1N + R2) - 2RRAB +4RAMRBM +4RAMBNRMN - 2RAMNPRBMNP] +KIo TAB, (4) \1M(rf>-3/2HMNP) =0, (5) DM(rf>-3/4FMPQ) +9KIo(rf>-3/2FMNQHMNP) =0, (6) 6\1 M(rf> -2aMrf» + 6rf> -3(aMrf»2 + 6K1orf> -5/2 Hl1NP + KIorf> -7/4[ 3~ Tr Fl1N - (Rl1NPQ - 4Rl1N + R2)] = 0, (7) where A,B,M,N=O,I, ... ,9; gAB, rf>, FMNQ, and HMNP are the metric, dilation, Yang-Mills, and Kalb-Ramond strengths. TAB is the thermal energy-momentum tensor; we have neglected the effects of matter on other bosonic backgrounds. KIo is the gravitational constant in ten dimensions. These equations can be derived from the follow-ing action in the field-theory limit of superstring theoriesl4 : S = dlo x -v - g -2-R - -KIorf> 3/2 Hl1NP - - -2- (rf> la Mrf»2 f ~{I 3 - 91_ 2KlO 4 16 KIO -! rf> -3/4[ -fo Tr Fl1N - (Rl1NPQ - 4Rl1N + R2) [ + L f}' (8) In addition, the following Bianchi identity has to be satisfiedl5 : dH = tr R /I. R - 3~ Tr F /I. F. (9) We assume that the cosmological metric is of the form gMN = diag[ - I ,Rf ( t}gij (x ),Rg (t}gmn (y) 1, 00) where i,j=1,2,3; m,n=4, ... ,9; R3(t} and R6(r) are the scale factors. gij (x) is assumed to be maximal-ly symmetric in three-space. For gmn (Y) and other bo-sonic backgrounds we will adopt the following Ansatz: gmn (Y) is Calabi-Yau, HMNP=O (M,N,P=O, ... ,9), FMNOI.{3 = Rmn 0I.{3 if (M,N) = (m,n), = ° otherwise, rf> = const, (Ila) o I b) Olc) (] Id) where ex, f3 are internal-space vielbein indices. It is the same as the static vacuum configuration of Ref. 6 ex-cept for the metric 00). One can alter Eq. (11 b) such that both the internal HMNP 16, 17 and an appropriate gluino condensate become nonvanishing and their contributions to the cosmological constant cancelY Alternatively, (11 a) can be relaxed: gmn (y) is Ricci flat. These changes would not affect the following dis-cussion of R 3(r), R6 (t), and GIG, but supersym-metry in four dimensions would be broken. The key observation5,18 is that the Ansatze (Ila)-(I Id) make Eqs. (5), (6), and (9) satisfied in the time-dependent case 00), as in the static case.6 We also assume (I2) In the matter-dominant era, p = p' = 0. 19 The conser-vation of TAB gives P (t) Rl ( t) Rt ( t) = const. (13) [We have normalized gmn(Y) such that f d 6y x (detg) 1/2 = 1.] With the above Ansiitze and assumptions and ne-glecting terms of order Ktol rex. (tpl t)2 (where tp - 10- 43 sec is the Planck time), the dilation Eq. (7) is satisfied and the Einstein equations (4) become R3 +2 R6 = _ ~ KtoPo (R 3(tO»)3 (4) R3 R6 24 Rt R3(r) , 2k + R3 + 2Rl +6 R3R6 = ~ KIoPo ( R3( to»)3 Rf R3 Rf R3 R6 8 Rt R3( t) , (5) R6 Rt R3R6 1 KIoPo ( R3( to) )3 -+5-+3--=---R6 Rg R3 R 6 8 Rt R3( r) , (6) where we have used R = t-KIoP; k is the Robertson-Walker parameter. For an open universe (k = - 1), in the large- t limit we have the following asymptotic solution,5 which is stable under perturbations: R3(r)=t+c, R6{t)=R60 =const. (7) Now we assume that Po can be treated as a small quan-1979 VOLUME 57, NUMBER 16 PHYSICAL REVIEW LETTERS 20 OCTOBER 1986 tity. Define'3 ( t) and '6 ( t) by R 3 ( I) = ( I + C )[ 1 + '3 ( t) 1. R 6 ( t} = R 60 [ 1 + '6( t} 1. (8) and treat them as small quantities. Up to first order, Eqs. (4)-06) are reduced to linear differential equa-tions for '3(t} and '6(t} with Ka(r)=817'G(r)=KToi Rt ( t) replaced by KH to) = 817' Go. The solution is 5 2 3 In (ht) b '3( () = - -HoHo 10 - -. (l9a) 8 I ~ 09b) where hand b are integration constants and we have made the change I+C- I. From qo= -(R3RJiR 3)0 one can determine b. The final result for (GI G)o = - 6'6( (0) is given by Eq. (1). The parameter 10 may differ from the age of the universe by a factor of order 1 which we will neglect. Astronomical observations have produced quite diverse values for the cosmological parameters flo, Ho, qo, and 10' (For details, see Rowan-Robinson. 20 ) Using the most "satisfactory" set of parameters rec-ommended by Ref. 20, i.e., (flo, qo, Ho) = (0.05, 0.025,67 km sec- 1 Mpc- 1) and to= 1.6x 1010 yr, we obtain «(;IG)0=-3.6 x lO- 12 yr- l . (20) The extreme sets are given by (flo, qo, Ho) = (0.05, -0.925, 100 km seC I Mpc- I) and 0, 0.5, 40 km sec-I Mpc- I). Correspondingly, «(;IG)o= -7.1xlO- 11 yr- 1 or -1.2x 10- 11 yr- 1• (20 Thus we estimate the range for (GIG)o as given by Eq. (2). Rigorously speaking, if flo is close to 1, the above perturbation calculation breaks down. One needs to use a computer for solving Eqs. (4)-(16), but this would not change the estimation (2). The same is ex-pected to be true for k = 0 or k = + I cases. The key point here is that Eq. (16) with Po~ 0 does not allow R6=0. Thus GIG=-6RJR6~0. Since to is the only relevant cosmological time scale, «(;1 G)o must be proportional to 11 to with a coefficient of order unity or, probably, one to two orders lower. Now assume that there is an effective potential term for R6 due to unknown nonperturbative quantum ef-fects in the action (8). If the potential is flat near R60 , the result is the same as given above. If the potential has a minimum for finite R6. R60 must be located there. To first order, it adds a term p.,2'6( t) to the left-hand side of Eq. (16), where the mass p., is deter-mined by the curvature of the potential at R60 . If we assume (p.,t)2» 1, then '6( r) = - t-floHJtJ p.,1r3 + 1-3/2A COS(p.,I+O). 09c) The second term is oscillatory and vanishes after being averaged over the period 217'1p.,. The first term, com-pared to that in Eq. (19b), is suppressed by the factor (p., tO)-2=[(lO.-32 eV)/p.,F. So a very tiny mass IL would make (GIG)o in this case unobservably small. The conventional wisdom favors a not very small p." since in four dimensions '6 represents a Brans-Dicke-type21 scalar field which would compete with gravitons and would have been observed if it is mass-less. However, the coupling of this field to matter might be anomalously weak; if so, a flat potential for '6 is not in conflict with observations. As for the time variation of particle-physics con-stants, such as CI' and strong or weak coupling con-stants, including the masses of the electron and pro-ton, etc., the following remarks are in order. First, the time dependence of R6 will lead also to a variation in the grand unification (GU) coupling constant Cl'Gu(R 6- 1 )=g6u (R6- 1 )/417'=q}/4R6- 6/417', (22) which, in turn, gives rise to a variation in almost every coupling constant and mass measured at low energies. This is an important feature of unified string theories, in constrast to the usual assumption made in previous analyses of experimental data that only the quantity considered is varying alone. Second, there is a renormalization-group (RG) running of coupling con-stants,22 which relates those measured at low energy p., ( « R6- 1 ) to Cl'GU (R 6- 1 ) calculated from Eq. (22) as follows: (23) where i= 1,2, and 3 correspond to U(l)em, SU(2) ... , and SU(3)c; the sum is over j=leptons, quarks, gluons, W ±, etc. The C/j are well-known numbers depending on the spin and group representation of the jth particle. If one neglects the variation of the second term, then a/(p.,) CI'/(p.,) aGU(R6 1 ) CI'/(p.,) G CI'/(p.,) = Cl'Gu(R 6- 1 ) 0Gu(R6- 1 ) 0Gu(R6- 1 ) G' (24) 1980 VOLUME 57, NUMBER 16 PHYSICAL REVIEW LETTERS 20 OCTOBER 1986 For a this is two orders of magnitude lower than GI G. Thus, though both G and atomic clocks vary in the time defined by the metric (10), the latter probably varies more slowly. Similarly, there is also RG run-ning of particle masses which, however, depends very much on the presence or absence of heavy families. 23 Also, as shown in Eq. (22), the time dependence of the background cP, which might arise upon appropriate modification of our Ansatz, would lead to an extra con-tribution to aGul aGU' It might be important to in-clude this in considering the variation of, e.g., a over a long period such as 5 x 109 yr, as in some previous determinations of al a .24 In conclusion, further improvement in measuring GI G can discriminate between different shapes of the potential in superstring theories for the size of internal space. If the potential is almost flat or has no minimum for finite R 6, probably we are on the edge of observing GIG. We encourage that old data be reanalyzed and new experiments be done. Especially, new clever ideas for precise short-time laboratory ex-periments would be most welcome. We thank J. Ball, C. DeTar, M. Dine, R. Rohm, S. H. H. Tye, and J. Schwarz for useful discussions. One of us (Y.S.W.) was supported in part by National Sci-ence Foundation Grant No. PHY-8405648. Ip. A. M. 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Time variation of Newton\u27s gravitational constant in superstring theories

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Time variation of Newton's gravitational constant in superstring theories

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