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Density-dependent couplings and astrophysical bounds on light scalar particles
Volume 228, number 2
PHYSICS LETTERS B
14 September 1989
D E N S I T Y - D E P E N D E N T C O U P L I N G S AND ASTROPHYSICAL BOUNDS O N L I G H T SCALAR PARTICLES John ELLIS a, S. K A L A R A b, K.A. O L I V E b and C. W E T T E R I C H c CERN, CH-1211 Geneva 23, Switzerland b School of Physics andAstronomy, University of Minnesota, Minneapolis, MN 55455, USA DESY, D-2000 Hamburg, FRG Received 16 June 1989
The couplings of a light scalar to ordinary matter induce non-vanishing fields 0 inside and around stars. In turn, the masses and couplings or ordinary particles depend on 0 in general. Thus, fundamental coupling constants can be altered by dense matter nearby. We estimate the resulting effect on the cooling of neutron stars, the neutrino burst from supernova 1987A and the period of the pulsar recently discovered in its remnant, and obtain bounds on the mass and couplings of scalar particles. We also c o m m e n t on cosmological and other constraints on such scalar particles, and discuss in particular bounds on the mass of the dilaton in string theories.
Weakly-coupled scalar particles with a very small mass/2 appear in a variety of theoretical contexts, notably spontaneously broken anomalous dilatation symmetry [ 1 ] and in string theories [ 2 ]. If the scalar field ¢ is a singlet of the low-energy gauge group SU (3) × SU (2) × U ( 1 ), simple arguments based on renormalizability and chirality indicate that its couplings to ordinary particles (gauge bosons, quarks .... ) must be suppressed by the inverse of some mass scale f f r o m beyond the standard model. If f is sufficiently large, such scalars cannot be produced in conventional particle physics experiments. On the other hand, forces radiated by the exchange of light scalar particles may compete with gravity if# is sufficiently small. It has been proposed [ l ] that weakly-coupled scalars with/2 ~ 10- 9 eV ( -~ (200 m ) - ~) could be responsible for the deviations from newtonian gravity reported in some "fifth force" experiments [3], although these experimental reports still require clarification. For a range 2 =/2- l of the scalar interaction between a few km and the extension of our solar system, a large body of astronomical and satellite observations impose stringent bounds [4] on the scalar coupling strength which depend on/2. Similar, albeit somewhat weaker, bounds are derived from Cavendish-type experiments in the centimetre range. 264
These bounds follow directly from the observed validity of the l / r dependence of the gravitational potential, since a scalar, unlike a pseudoscalar, mediates an attractive interaction that leads to a correction to the newtonian potential of the form
Mm
AV= --GN - -
r
a exp( --/2r).
(1)
Here a measures the strength of the scalar coupling to ordinary atoms, which we assume to he proportional to mass in leading order. Another possible aspect of a scalar interaction is an induced density-dependence of fundamental couplings. Scalar couplings to nucleons and photons of the form
L~ = CN ~ mN ~VN+ Cv ~ F~F ~
(2)
generate not only a modification to the newtonian potential of the form ( 1 ) with a given by CZN m2p a= ---4re f 2 ,
(3)
but also imply that in a non-zero background field the nucleon mass and the electromagnetic coupling
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PHYSICS LETTERSB
[ 5 ] differ from their vacuum values when # # 0:
8mN
CN~
mN-
'
8e = 2Cv ~
e
f"
the mass dimensions ~2 of the corresponding quantities, e.g. (4)
Similar ~-dependences can arise for quark and electron masses and the strong gauge coupling. A 0-dependence of the Higgs mass a n d / o r its self-interaction can induce a ~-dependence of the Fermi constant, via a 0-dependence of the vacuum expectation value (VEV) of the standard model Higgs doublet ~"
8GF ~ _ CG ¢ 6F f'
14 September 1989
(5)
CF--0, CN
Cc; =-2:a. S G F = - 2 ( 8 m N ' ) CN GF \ mN / '
and a material dependence of the scalar interaction would arise if the C~/CN deviated from these values. When it is large compared with the scale of inhomogeneity in the material medium, the derivative term in (6)must be taken into account. For g}at a distance R from the centre of a sphere with uniform density p and radius/~ one obtains
CNp e x p ( - # R )
0= ~ The local value of the scalar field ~ in turn depends on the surrounding material according to the field equation
etc.(9)
R>k,
[p3~ cosh (/d~) - s i n h ( / d ~ ) ] ,
(10a)
CNp -- fll3 R [/~R- (/~k+ 1 ) e x p ( - U k ) sinh(/LR) ],
(6) R
p [ g c m - 3 ] . (7)
\ mN f0 The variations in other parameters are scaled by the ratios of coefficients C~/CN, e.g.
8GF -- CG ( SmN ) Gv
(8)
CN \ mN ,/"
Upper limits on the material dependence of the scalar interaction constrain these ratios. The scalar would couple exactly to mass if the ratios C~/CN reflected ~' Wedo not commitourselvesin this paper to any specificmodel for generating the electroweak Higgs VEV. In the minimal standard model with V(H)=-v21HI2+2(IHI2) 2, 8Gv/
GF=f2/2--28U/V.
(10b)
and hence corresponding expressions for (SmN/mN ) etc. On the Earth, where p ~ 5 g c m -3, the densitydependences of masses and couplings are too small to be observed now, if one takes values of a ( _ 10-2) and 2 ( = 300 m) suggested by "fifth force" experiments [ 3 ]. In this letter, we investigate whether the large densities in the core of supernova 1987A and neutron stars could lead to observable changes in fundamental "constants". We keep o~ and 2 as free parameters and derive bounds on their allowed values from observations [ 8 ] of the neutrino burst from supernova 1987A, from the period of its recently-discovered pulsar remnant [ 9 ], and from the cooling of neutron stars [ 10 ]. In the case of the supernova, a substantial change in GF would have modified the energy and duration of the neutrino burst, whilst changes in GF or mN would change neutrino emissivities and hence the cooling rates of neutron stars. Changes in mN would alter the central density of the SN 1987A pulsar remnant, and hence its period. As an example, we apply our bounds on a and 2 to the case of a light dilaton with couplings suggested by string theories. ~2 The naive mass dimensions of operators should be corrected for anomalous dimensions, inducing small corrections in relations like (9). 265
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We also comment on cosmological bounds on weaklycoupled light scalars. We now consider neutrino emission from a supernova. We first discuss the effects due to a change in the Fermi constant Gv, and then more indirect effects due to variations in other fundamental quantities. The neutrinos detected [ 8 ] are emitted from a neutrinosphere located approximately at a radius R which is equal to the neutrino mean free path 2 ~, i.e. when 2v-~R--- ( 1 5 - 2 0 ) km. The core profile is essentially independent of Gv [ 11 ]. However, the neutrino mean free path is given by 2 ~-~ 1 / an, where a~: G 2 T 2 is the low-energy neutrino-nucleon elastic scattering crosssection and nocT 3 is the nucleon density, so that 2y ~oc G ~ T 5. The neutrino energy emission rate Q~ is given by the black-body formula at R---2~, so that Q,, oc R z T 4oQG ~-4 T 6 The observed duration [ 8 ] of the supernova 1987A neutrino pulse agree with calculations [ 11 ] based on the standard model, and estimates [12] of the total energy E~=fdtQ~ emitted in the form o f neutrinos are in the range E~= ( 2 4) × 1053 erg and consistent with the calculated [ 11 ] total binding energy of a newly-born neutron star. Variations in Q~ by more than a factor of 2 would have led to significant observable changes in the neutrino pulse. We therefore constrain a possible variation in Gv by requiring that --
1 Qv R2 T4 2 < Qv~ - R~ To4 < 2 ,
(11)
where quantities with (without) suffices are those calculated in the standard model (with a modified value of GF). Since the star density varies very rapidly as a function of the radius in the range of interest, we expect that
R/Ro -~ 1,
(12a)
and hence 2/2o -~ 1, i.e.
G~oT5 G~To
1.
(12b)
Substituting the approximations (12) into ( 11 ) we obtain the bounds 2_5/8< Gv
~o <
25/8
(13) •
Using the parametrization (5) of variations in Gv, 266
14 September 1989
eq. (13) can be used to bound C / f and hence a and p. Since we are interested in GF at T=2v, which is where the matter density p ~ 10 ~ g cm -3, we need to compute the field ¢ outside the core of the supernova where the matter has nuclear density. We approximate the core by a uniform sphere of radius Rc = 10 km and density Pc = 10 ~5 g c m - 3. If the range 2 = / z of the scalar interaction is larger than a few km we should use eq. (10a) w i t h / ~ = R c and P=Pc, obtaining the bound 3
-0.14<
f Ro "~ Ca k2~-k-~m) ce ~ ( / t R o ) 3 e x p ( - # R o )
× [#Rc cosh (/ZRc) -sinh(~tRc) ] <0.093.
(14)
I f 2 = ~ t ~<< 1 km, one can neglect the derivative term in (6) and obtain an expression for 6GF/Gv analogous to eq. (7) for fimN/rnN. Taking in this case p = I0 ~l g cm -3, one obtains -6.3×10
< 4 . 1 × 1 0 6.
(15)
Bounds for 10 k m > ) . > 1 km can also be obtained, but they depend more sensitively on the details of the neutron star model. The above bounds on 5Gv/Gv arise under the assumption that all couplings and masses except Gv are held fixed. However, the neutrino energy emission also depends implicitly on other parameters. For example, the radius of the core would be sensitive to a variation in the strong interaction scale, which would change the nucleon mass, nuclear density, and hence the binding energy of the neutron star. One can bound a possible variation in the nucleon mass by requiring the total binding energy to be within a factor 2 of the estimated energy of the neutrino pulse. The binding energy E~ GNM2/Rc~ GNNZm~/Rc, where N is the total n u m b e r of nucleons in the core. From the balancing o f gravitational and Fermi energies we have N - 2 / 3 ~ GNm~, and the radius Rc of the core is related to N and the baryon number density n by Re~ (N/n) ~/3. The number density n is close to the nuclear matter density, so that n~A~cD ~ m 3, where the constant o f proportionality depends on details of the equation of state. We allow an uncertainty
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PHYSICS LETTERS B
of a factor of 10 in this constant, and hence o f 101/3 in the proportionality constants in
G~ I/2 1 R~~ ~ , E ~ i~,.3/2~" m~q
'~N
(16)
It~N
Comparing the expected binding energy with that observed in neutrinos, we can constrain the density variation of mN, assuming GN to be fixed: 0.48~2_t/210_1/6
< mN < 2 1 / 2 1 0 J / 6 mNa
2.1.
(17)
For an estimate of O/f inside the core we use eq. (10b) with p ~ 1015g cm -3 to obtain 2 2 --1.3X103~< (30---0~m) O~
where cn is the specific heat o f normal neutrons: cn -~2X 102°T9 erg c m - 3 K - j .
Ru,c, ~- 1.7× 1021T89erg c m - 3 s - i.
dT dt
- -
=
1 Ru .... c,
(21)
whose solution is [ 13,10] (with t in s)
(17N)
It is worth noting that time, energy and temperature scales also become density-dependent inside the neutron star. The conversion to the vacuum time and temperature scales relevant at low densities on and beyond the surface introduces density-dependent factors. In the special case of a scalar field coupling exactly to mass the situation simplifies considerably. There is a c o m m o n 0-dependent physics scale, since all masses are fixed in units of the neutron mass, and E, t - t and T are all proportional to raN. Since only the ratio m~/rnp is observable, where me is the Planck mass, we can use an appropriate Weyl rescaling of the metric to make a reformulation where mp is 0-dependent whilst all other physics masses and couplings are 0-independent. Then we have to examine the effects of a density-dependence o f Newton's constant whereas mN is kept fixed. Next we consider the cooling o f mature neutron stars. These are generally believed to have no internal source of heat. Starting with a high initial temperature T~~ 10 ~ K provided by the collapse of a supernova, they cool at a rate that is expected to be dominated by neutrino emission until the temperature T falls to ~ 108 K, with [ 13,10 ] --
(20)
Density-induced changes in couplings would modify Rurca (20) a n d / o r c,, and thus modify the cooling rate ( 18 ) and the temperature of the neutron star observed now. Defining Q - ( Rur~a/ Cn) / ( Rurcao/ Cno), we can rewrite eq. ( 18 ) as
log,o T9 = ~ (7.3 - l o g l o t - log,o Q). <6.1×102.
(19)
For p = 1015 g c m - 3, and 7"9= 7"/10 9 K, the heat emission rate for the modified Urca process n + n - , n + p + e + % ~3is [13]
dT9 = 0 . 9 × 10-ST7Q s-1, dt
ltR~+lexp(-lzRc)sinh(gR)#R
X(1
14 September 1989
(22)
The ratio Rur~a/C, depends on several fundamental parameters: Rurca
cn
~
2
ozG~mNn
I/3 T7
--
(23)
m~'
As a guide to estimating the sensitivity to density dependence, we assume that all particle physics mass parameters scale in the same way. Since (Rur~a/Cn)/ T 7 has the dimension of (mass) -5, we can write Q=Q.(mNo/mN) 5 and derive from eq. (22) the relation In ( m ~ ) = ½ 1 n
(~o)+
6 In ( ~ o ) '
(24)
where Q/(~o allows for a possible uncertainty in the calculation of RurcJCn even if the fundamental physics is specified, and T/To represents a deviation of the observed temperature (T) from that expected
(To). Observations of X-ray emission from the Crab pulsar[ 14] and R C W l 0 3 [ 15] are consistent with surface temperatures Ts= (1.5-2.5) X 106 K. The conversion from the surface temperature to the internal temperature TI depends on the equation of state assumed. For example, Ts--- 1.5 X 106 K corresponds to T~ ~ 5 X 107 K for a magnetized star with a soft equa-
(18)
~3 We take the effective neutron mass m* =0.8 m°. 267
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tion of state, and to T~ ~- 2 × 108 K for an unmagnetized star with a stiffequation of state [ 16 ]. A recent analysis [ 17 ] of the behaviour of the Crab pulsar following glitches in 1969 and 1975 indicates that T 1 = ( 4 + 2 ) × 1 0 8 K. Eq. (22) would give T~o---3× l0 s K if Q = 1, so we conclude that (T~/T~ o) = 2 °-+1. The prediction for R C W 103 is comparably accurate. Using (T~/TIo) = 2 o + ~ and allowing ((~/ Qo) = 10 °-+j, we obtain 0 . 2 7 < mh <3.6,
14 September 1989
[18]. In this case, one estimates that P-~ 2 n ( G N p ) - l / 2 , where p is the core density [18], and the observed period of SN 1987A is compatible with a central density p ~ 10 ~5 g c m - 3 as expected. Assuming that all particle masses scale in the same way so that p~:m 4, we can write P-2=dGNm4 where the dimensionless coefficient d may be known to within a factor of 10: d/do= 10 °-+ i reflecting uncertainties in the equation of state, etc. Comparing the observed period (P) with that expected (Po), we therefore have
(25a)
mNo
In (m~--~NNo)=--}In (ff-~o) --¼1n ( ~ o ) ,
and hence - 2 . 6 < CN ~ <0.73.
(25b)
To obtain a constraint on the scalar mass and coupling strength, we use the previous solution (10b) for ¢ inside the neutron star, which we take to have uniform density and radius/~ = 10 km. Averaging (10b) over the neutron star's interior, we find
(~/~) 3 exp(-- #~)
2
-1.5<
× [ / ~ cosh (p.R) - sinh (/~/~) ] ) (27)
The quantity in large brackets varies between 1 (g---,oo) and 0 (#--,0). The recent observation [ 9 ] of a pulsar with short period P ~ 5 × l 0 - 4 S in the remnant of supernova 1987A allows us to set a slightly more stringent constraint on mN/mNo and hence on 8 C / f and/~. This infant neutron star is believed to spin close to breakup, with a surface velocity close to the speed of light 1¢4 Here we introduce C~ defined by 8m~/m~= -C~¢/f
268
<0.6,
f
1.7 ×
(25be)
103 2
2
3 (/d~+ 1) exp(-/d~) (U~) 3
\ × [/u~ cosh (p3~) - sinh (~R) ] ) < 700,
exp(-~/~)
<850.
(25ap)
and hence
(26)
--3.0X 10 -3
(~)3
<2.5,
and finally
Imposing the upper limit (25b) on this expression, we obtain the constraint ~4
2
MN mNo
-
X [/t/~ cosh (/t/~) - sinh (/t~) ] ) .
P/Po= 20+- i we finally obtain
and taking 0.4<
(24p)
(27p)
Our bounds are displayed in fig. 1, together with various previous bounds [ 4,19 ]. A few comments are in order: ( 1 ) The solid lines in the figure are calculated assuming a material-independent scalar interaction: CG = -- 2C., etc. Bounds for other model couplings are easy to derive, and we will give an example at the end o f this paper. (2) As compared to bounds from Cavendish-type experiments, astronomical and satellite observations etc., our bounds are competitive when 2 < 10- 3 m. (3) The changes in masses and couplings corresponding to our bound are not small. Therefore, the linear approximation to the scalar-matter couplings assumed above may not be valid. However, we ex-
Volume 228, number 2
PHYSICS LETTERS B
i 16 I
t 4 September 1989
V;
•
%
12
/
~',"
/2"
"?771" OsdUations I'fflllrl 1 , " "rt.FOfllD.lll.."
~
N.SZ.~
44.,
•~ x ~
II[?tlrflllll'l'l?f-"Ii'1 "
-4
-8
I -6
-/+
~ -
i
I
I
I
I
I
I
0
2
4
6
8
10
12
14
log (X/m)
Fig. 1. Compilation of constraints on the dilaton-matter coupling strength oz as functions of the range 2= 1/,u. The solid line is taken from ref. [4 ], and is a compilation of bounds from Cavendish-type experiments, astronomical and satellite observations, etc. The hatched lines are the astrophysical and cosmological bounds discussed in this paper. The line marked SN is derived from the agreement [ 12] of neutrino observations [ 8] of SN 1987A with theoretical predictions [ 11 ]. The line marked N.S. is derived from the agreement between neutron star cooling calculations [ 13,10 ] and observations [ 14,15 ], and from the observed [9 ] period of the SN 1987A pulsar. The dotted line is the porous cosmological constraint [19] on coherent dilaton field oscillations. These bounds are shown for materialindependent couplings. In addition, we display (dashed line) bounds from EOtv6s-typeexperiments for a material-dependent interaction, assuming a scalar coupling proportional to baryon number. pect that realistic non-linear couplings would yield qualitatively similar bounds. ( 4 ) The d o t t e d lines in the figure are cosmological b o u n d s [19]. These d e p e n d on some specific ass u m p t i o n s that we now discuss. To treat cosmology, we replace the right-hand side o f eq. ( 6 ) by a term (Cv/f)pu which parametrizes the coupling o f ~ to the d o m i n a n t c o m p o n e n t o f the cosmic energy density. We expect Cu to be negligible during the r a d i a t i o n - d o m i n a t e d epoch, since we can always rescale the fields so that kinetic terms are ind e p e n d e n t o f ~ from which it is clear that the scalar does not couple to an ideal gas in thermal equilibrium. A non-vanishing value o f Co could only arise from m o d i f i c a t i o n s o f the equation o f state due to Od e p e n d e n t interactions. In contrast, couplings o f ~ to visible or d a r k non-relativistic m a t t e r could induce a substantial value o f Cu during the m a t t e r - d o m i n a t e d
epoch. The earlier arguments strongly constrain couplings to matter, but couplings to dark m a t t e r could be much larger. F o r an equation o f state Pn= (n/3-1)pn with n = 4 ( 3 ) for the radiation ( m a t t e r ) d o m i n a t e d epoch, a n d assuming the Hubble constant H obeys the condition H2//t2<< 1/12 ( 2 / 9 ) , the generic homogeneous and isotropic solution o f the k - - 0 cosmological equations [ 6 ] ~'+ 3 H ~ + / t 2 ~ = t i p ,
p+nHp+ t i p S = 0 , 6MZH 2 = p + V+ ~ 2 (where
(28)
M2=--m2/16n,fl- ½x/-~Cu/CN)is a d a m p e d 269
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oscillation of ~ around a time-dependent minimum given by 0o(t) = ~_MP(t).
(29)
The oscillating field behaves like non-relativistic matter and should not dominate the energy density of the Universe too early. This gives bounds [ 20 ] on the initial amplitude ~---~-~o. Whether these bounds are satisfied or not depends on the detailed dynamics of the QCD phase transition or, in the case of small ~t2, the dynamics of the onset of ~ oscillations once H 2 decreases below/ft. The dotted line in the figure corresponds to maximal oscillations I ~l = a - I / 2 M (_~fwhen CN--~I) at T = m i n ( 1 0 0 MeV, (pM) ~/2), as would have been natural for the similar case of axion oscillations. For a < 1 we take I~1 = M as a maximal amplitude. When /z2<
O=co + a M l n - - , to
H=rlt -~, p=potot -2,
(30)
where 2
24fl
~l= n + 1 2 f l 2 / ( 6 _ n ) , a= n ( 6 _ n ) + 1 2 f l 2, pot~ = ( O / 2 - ½a2)M 2.
(31)
Nucleosynthesis and the microwave background radiation place limits on q [22 ] and hence on fl during the radiation-dominated epoch: fl<0.28 (0.20)
14 September 1989
y
0"+3H~+ ~ - ~ ( O - ~ ) p = 0 ,
fl=~,
(33)
we obtain standard cosmology with ~ approaching q~o as a negative power of t:
~-~=At-'", N/(3q 3q-1 m= ~ - -
1 )2
--6yr/2 ,
2
q= -
(34)
n
When a small mass term is included, ~ approaches a time-dependent effective minimum at
0o(t)=0o (1 +
yp(t)]
-' "
(35)
The evolution of ~ before the onset of oscillations (/z 2 << H 2 ) is relevant for the magnitude of the initial amplitudes ~, since we expect fl and 7 to be non-negligible during the QCD phase transition (but also the dynamics to be more complex than in our simple discussion). This illustrates the difficulty of obtaining cosmological bounds in our case. In particular, a scalar with range 2_~300 m and o¢-~ 10 - 2 cannot be completely excluded, given our present ignorance of the cosmological evolution of such a scalar field. Finally, we specialize the above analysis to one example of matter-dependent scalar couplings, namely a light "dilaton" from string theory. In this case, the dominant effect comes from the q~-dependence of the strong coupling due to the term [ 2 ] ~ = - ¼exp ( -
(36)
~¢M)FI,~F"",
(32) which implies
for a HeliUm abundance Yp~<0.26 (0.25 ). However, as discussed above, we expect fl to be very small during this epoch. When # - ~ is bigger than our present horizon size, fl is bounded by the age of the Universe qHff ~. It should be noted, though, that a linear approximation to the 0-matter coupling is not well-motivated when ~t2 << H 2. If we postulate instead a scalar evolution equation ~5 This invalidates the bound given in ref. [ 21 ], since during nucleosynthesis ~'+ 3H~ >>/~-'~ for the value o f # quoted.
270
8gs 1 gs - 2 M "
(37)
This leads [ 6 ] to a variation in the strong coupling scale: 8A
[
1
A - ~ 2b3g~(Mx) where
d
Mx M'
(38)
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1
A~-Mx e x p ( 2 b 3 g f ( M x ) )
(39)
with Mx the scale at which S U ( 3 ) is unified with other interactions in some higher symmetry and b3 is the coefficient of g3 in the renormalization group equation for gs. The first term in eq. (38) dominates, since g~(Mx) must be small. The nucleon mass is proportional to A at zeroth order in the small u, d and s quark masses, so that 5mN/mN~-SA/A and one obtains O{'~
")
1 4
b~gs (Mx)
(40)
"
Numerical values of ol (40) in the minimal SU(5) G U T where b3=(33-2Nq)/48z~ 2 and a s ( M x ) = g2 (Mx)/4~r_~ 1/42, and the minimal supersymmetric SU(5) G U T where b3=(27-3Nq)/48n 2 and c~s(Mx) ~ 1/24, for N , = 6 are a ~ 250 -~ 190
without superrsymmetry,
(41 )
with supersymmetry.
To first order in the light u, d and s quark masses there is an additional contribution to the possible variation in the nucleon mass:
(6mN) = \ m N ./q
~
q=u.cl.s
(6m+~ \---'~-u./ mq(N[ElqlN) mN
(42)
If we assume, for definiteness, that 8mq/mq is universal for the u, d and s quarks, and use the estimate [23] Zq=u.d.smq(NI4qlN) =]mN, we find that
( 6mN ~ 1 5mq mh ./ u 3 mq
(43)
14 September 1989
~-dependence of v a n d / o r 2 would be more complicated if symmetry breaking is induced by radiative corrections. We assume conservatively that
~mq =O(1 ) ~ M mq
(45)
in which case the dominant source of dilaton couplings to matter is ~A/A, resulting in a relatively large material-independent coupling a (40), (41) and smaller material-dependent couplings suppressed by O(1/a). A scalar with such a large coupling strength (40), (41 ) and material-dependent couplings 2 or 3 orders of magnitude smaller would have been seen in E6tv6stype experiments or satellite observations, unless its range were very small. We therefore conclude that the dilaton mass must be bounded from below by / 1 > 2 × 10 -~ eV.
(46)
The dilaton mass must therefore be generated by some physics beyond the QCD scale, since otherwise a typical order of magnitude would be # ~A 2/M as for an axion. We should note, however, the possibility that the spectrum of light string scalars is richer. A contribution of these with a small dilaton component would have a correspondingly reduced value of oL, and could have a mass in the range detectable by experiment. One of the authors (C.W.) would like to thank the members o f l T P for the kind hospitality in Santa Barbara, where part of this work was performed and supported by the National Science Foundation under Grant No. PHY82-17853, supplemented by funds from the National Aeronautics and Space Administration.
Since quark masses mq=2qV,where the 2q are Yukawa couplings and v is a Higgs VEV, we have 8 m ~ _ ~2q + ~5___v. m~ ).q V
References
(44)
In the type of string models discussed here 82q/ 2q= 8g/g as a first approximation and is hence given by eq. (39). The value of 5v/v depends on the detailed mechanism of spontaneous electroweak symmetry breaking. There are contributions to 6v/v from the ~-dependence of the Higgs scalar mass u and the quartic coupling 2 mentioned above. One would expect that 2ocg 2 and hence that 62/2-~ 28g/g, but the
[ 1] R.D. Peccei, J. Sola and C. Wenerich, Phys. Len. B 195 (1987) 18. [ 2 ] M.B. Green, J.H. Schwarz and E. Witten, Superstring theory (Cambridge U.P., Cambridge, 1986 ), see especially Ch. 16. [3] S.C. Holding, F.D. Stacey and G.J. Tuck, Phys. Rev. D 33 (1986) 348; P. Thieberger, Phys. Rev. Lett. 58 (1987) 1066; C.W. Stubbs et al., Phys. Rev. Len. 58 (1987) 1070; T.M. Niebauer et al., Phys. Rev. Lett. 59 (1987) 609; E.G. Adelberger et al., Phys. Rev. Len. 59 (1987) 849; P.E. Boynton et al., Phys. Rev. Lett. 59 (1987) 1385;
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V.L. Fitch et al., Phys. Rev. Lett. 60 ( 1988 ) 1801; D.H. Eckhardt et al., Phys. Rev. Lett. 60 (1988) 2567. [4] A. de Rtijula, Phys. Lett. B 180 (1986) 213. [5] C.T. Hill, Phys. Lett. B 135 (1984) 47; Q. Shaft and C. Wetterich, Phys. Rev. Lett. 52 (1984) 875. [6] C. Wetterich, Nucl. Phys. B 302 (1988) 645, 668. [ 7 ] C.T. Hill and G.G. Ross, Phys. Lett. B 203 ( 1988 ) 125; Nucl. Phys. B311 (1988)253. [8] Kamiokande Collab, K. Hirata et al., Phys. Rev. Lett. 58 (1987) 1490; IMB Collab., R.M. Bionta et al., Phys. Rev. Lett. 58 ( 1987 ) 1494. [9] J. Kristian et al., Nature 338 (1989) 234. [ 10] J. Ellis and K.A. Olive, Nucl. Phys. B 223 (1983) 252. [ 11 ] For a review and references, see D.N. Schramm, Comm. Nucl. Part. Phys. 17 (1987) 239. [ 12] J. Ellis and K.A. Olive, Phys. Lett. B 193 (1982) 525; D.N. Schramm, Proc. Rencontre de Moriond, ed. J. Tran Thanh Van (1987). [13] B.L. Friman and O.V. Maxwell, Astrophys. J. 232 (1979) 541. [ 14] F.R. Harnden et al., Bull. Amer. Astron. Soc. 11 (1979) 789,424.
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[ 15] I. Tuohy and G. Garmire, Astrophys. J. 239 (1980) L107. [ 16 ] G. Glen and P. Sutherland, Astrophys. J. 239 (1980) 671. [ 17 ] M.A. Alpar, R. Nandkumar and D. Pines, Astrophys. J. 288 (1985) 191. [ 18] S.L. Shapiro and S.A. Teukolsky, Black holes, white dwarfs and neutron stars (Wiley, New York, 1983 ). [19]J. Ellis, N. Tsamis and M.B. Voloshin, Phys. Lett. B 194 (1987) 291. [20] J. Preskill, M.B. Wise and F. Wilczek, Phys. Lett. B 120 (1983) 127; L. Abbott and P. Sikivie, Phys. Lett. B 120 (1983) 133; M. Dine and W. Fischler, Phys. Lett. B 120 (1983) 137. [21] S.M. Barr and P.K. Mohapatra, Phys. Rev. D 38 (1988) 3011; C.A. Vayonakis, Phys. Left, B 213 (1988) 419. [22] J. Yang, M.S. Turner, G. Steigman, D.N. Schramm and K.A. Olive, Astrophys. J. 281 (1984) 493. [23] T.P. Cheng, Phys. Rev. D 13 (1976) 2161; for a recent review, see J. Gasser, H. Leutwyler, M.P. Locher and M.E. Sainio, Bern/SIN preprint BUTP-88/13-BERN, SIN-PR-88-13 (1988).
PHYSICS LETTERS B
14 September 1989
D E N S I T Y - D E P E N D E N T C O U P L I N G S AND ASTROPHYSICAL BOUNDS O N L I G H T SCALAR PARTICLES John ELLIS a, S. K A L A R A b, K.A. O L I V E b and C. W E T T E R I C H c CERN, CH-1211 Geneva 23, Switzerland b School of Physics andAstronomy, University of Minnesota, Minneapolis, MN 55455, USA DESY, D-2000 Hamburg, FRG Received 16 June 1989
The couplings of a light scalar to ordinary matter induce non-vanishing fields 0 inside and around stars. In turn, the masses and couplings or ordinary particles depend on 0 in general. Thus, fundamental coupling constants can be altered by dense matter nearby. We estimate the resulting effect on the cooling of neutron stars, the neutrino burst from supernova 1987A and the period of the pulsar recently discovered in its remnant, and obtain bounds on the mass and couplings of scalar particles. We also c o m m e n t on cosmological and other constraints on such scalar particles, and discuss in particular bounds on the mass of the dilaton in string theories.
Weakly-coupled scalar particles with a very small mass/2 appear in a variety of theoretical contexts, notably spontaneously broken anomalous dilatation symmetry [ 1 ] and in string theories [ 2 ]. If the scalar field ¢ is a singlet of the low-energy gauge group SU (3) × SU (2) × U ( 1 ), simple arguments based on renormalizability and chirality indicate that its couplings to ordinary particles (gauge bosons, quarks .... ) must be suppressed by the inverse of some mass scale f f r o m beyond the standard model. If f is sufficiently large, such scalars cannot be produced in conventional particle physics experiments. On the other hand, forces radiated by the exchange of light scalar particles may compete with gravity if# is sufficiently small. It has been proposed [ l ] that weakly-coupled scalars with/2 ~ 10- 9 eV ( -~ (200 m ) - ~) could be responsible for the deviations from newtonian gravity reported in some "fifth force" experiments [3], although these experimental reports still require clarification. For a range 2 =/2- l of the scalar interaction between a few km and the extension of our solar system, a large body of astronomical and satellite observations impose stringent bounds [4] on the scalar coupling strength which depend on/2. Similar, albeit somewhat weaker, bounds are derived from Cavendish-type experiments in the centimetre range. 264
These bounds follow directly from the observed validity of the l / r dependence of the gravitational potential, since a scalar, unlike a pseudoscalar, mediates an attractive interaction that leads to a correction to the newtonian potential of the form
Mm
AV= --GN - -
r
a exp( --/2r).
(1)
Here a measures the strength of the scalar coupling to ordinary atoms, which we assume to he proportional to mass in leading order. Another possible aspect of a scalar interaction is an induced density-dependence of fundamental couplings. Scalar couplings to nucleons and photons of the form
L~ = CN ~ mN ~VN+ Cv ~ F~F ~
(2)
generate not only a modification to the newtonian potential of the form ( 1 ) with a given by CZN m2p a= ---4re f 2 ,
(3)
but also imply that in a non-zero background field the nucleon mass and the electromagnetic coupling
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[ 5 ] differ from their vacuum values when # # 0:
8mN
CN~
mN-
'
8e = 2Cv ~
e
f"
the mass dimensions ~2 of the corresponding quantities, e.g. (4)
Similar ~-dependences can arise for quark and electron masses and the strong gauge coupling. A 0-dependence of the Higgs mass a n d / o r its self-interaction can induce a ~-dependence of the Fermi constant, via a 0-dependence of the vacuum expectation value (VEV) of the standard model Higgs doublet ~"
8GF ~ _ CG ¢ 6F f'
14 September 1989
(5)
CF--0, CN
Cc; =-2:a. S G F = - 2 ( 8 m N ' ) CN GF \ mN / '
and a material dependence of the scalar interaction would arise if the C~/CN deviated from these values. When it is large compared with the scale of inhomogeneity in the material medium, the derivative term in (6)must be taken into account. For g}at a distance R from the centre of a sphere with uniform density p and radius/~ one obtains
CNp e x p ( - # R )
0= ~ The local value of the scalar field ~ in turn depends on the surrounding material according to the field equation
etc.(9)
R>k,
[p3~ cosh (/d~) - s i n h ( / d ~ ) ] ,
(10a)
CNp -- fll3 R [/~R- (/~k+ 1 ) e x p ( - U k ) sinh(/LR) ],
(6) R
p [ g c m - 3 ] . (7)
\ mN f0 The variations in other parameters are scaled by the ratios of coefficients C~/CN, e.g.
8GF -- CG ( SmN ) Gv
(8)
CN \ mN ,/"
Upper limits on the material dependence of the scalar interaction constrain these ratios. The scalar would couple exactly to mass if the ratios C~/CN reflected ~' Wedo not commitourselvesin this paper to any specificmodel for generating the electroweak Higgs VEV. In the minimal standard model with V(H)=-v21HI2+2(IHI2) 2, 8Gv/
GF=f2/2--28U/V.
(10b)
and hence corresponding expressions for (SmN/mN ) etc. On the Earth, where p ~ 5 g c m -3, the densitydependences of masses and couplings are too small to be observed now, if one takes values of a ( _ 10-2) and 2 ( = 300 m) suggested by "fifth force" experiments [ 3 ]. In this letter, we investigate whether the large densities in the core of supernova 1987A and neutron stars could lead to observable changes in fundamental "constants". We keep o~ and 2 as free parameters and derive bounds on their allowed values from observations [ 8 ] of the neutrino burst from supernova 1987A, from the period of its recently-discovered pulsar remnant [ 9 ], and from the cooling of neutron stars [ 10 ]. In the case of the supernova, a substantial change in GF would have modified the energy and duration of the neutrino burst, whilst changes in GF or mN would change neutrino emissivities and hence the cooling rates of neutron stars. Changes in mN would alter the central density of the SN 1987A pulsar remnant, and hence its period. As an example, we apply our bounds on a and 2 to the case of a light dilaton with couplings suggested by string theories. ~2 The naive mass dimensions of operators should be corrected for anomalous dimensions, inducing small corrections in relations like (9). 265
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We also comment on cosmological bounds on weaklycoupled light scalars. We now consider neutrino emission from a supernova. We first discuss the effects due to a change in the Fermi constant Gv, and then more indirect effects due to variations in other fundamental quantities. The neutrinos detected [ 8 ] are emitted from a neutrinosphere located approximately at a radius R which is equal to the neutrino mean free path 2 ~, i.e. when 2v-~R--- ( 1 5 - 2 0 ) km. The core profile is essentially independent of Gv [ 11 ]. However, the neutrino mean free path is given by 2 ~-~ 1 / an, where a~: G 2 T 2 is the low-energy neutrino-nucleon elastic scattering crosssection and nocT 3 is the nucleon density, so that 2y ~oc G ~ T 5. The neutrino energy emission rate Q~ is given by the black-body formula at R---2~, so that Q,, oc R z T 4oQG ~-4 T 6 The observed duration [ 8 ] of the supernova 1987A neutrino pulse agree with calculations [ 11 ] based on the standard model, and estimates [12] of the total energy E~=fdtQ~ emitted in the form o f neutrinos are in the range E~= ( 2 4) × 1053 erg and consistent with the calculated [ 11 ] total binding energy of a newly-born neutron star. Variations in Q~ by more than a factor of 2 would have led to significant observable changes in the neutrino pulse. We therefore constrain a possible variation in Gv by requiring that --
1 Qv R2 T4 2 < Qv~ - R~ To4 < 2 ,
(11)
where quantities with (without) suffices are those calculated in the standard model (with a modified value of GF). Since the star density varies very rapidly as a function of the radius in the range of interest, we expect that
R/Ro -~ 1,
(12a)
and hence 2/2o -~ 1, i.e.
G~oT5 G~To
1.
(12b)
Substituting the approximations (12) into ( 11 ) we obtain the bounds 2_5/8< Gv
~o <
25/8
(13) •
Using the parametrization (5) of variations in Gv, 266
14 September 1989
eq. (13) can be used to bound C / f and hence a and p. Since we are interested in GF at T=2v, which is where the matter density p ~ 10 ~ g cm -3, we need to compute the field ¢ outside the core of the supernova where the matter has nuclear density. We approximate the core by a uniform sphere of radius Rc = 10 km and density Pc = 10 ~5 g c m - 3. If the range 2 = / z of the scalar interaction is larger than a few km we should use eq. (10a) w i t h / ~ = R c and P=Pc, obtaining the bound 3
-0.14<
f Ro "~ Ca k2~-k-~m) ce ~ ( / t R o ) 3 e x p ( - # R o )
× [#Rc cosh (/ZRc) -sinh(~tRc) ] <0.093.
(14)
I f 2 = ~ t ~<< 1 km, one can neglect the derivative term in (6) and obtain an expression for 6GF/Gv analogous to eq. (7) for fimN/rnN. Taking in this case p = I0 ~l g cm -3, one obtains -6.3×10
< 4 . 1 × 1 0 6.
(15)
Bounds for 10 k m > ) . > 1 km can also be obtained, but they depend more sensitively on the details of the neutron star model. The above bounds on 5Gv/Gv arise under the assumption that all couplings and masses except Gv are held fixed. However, the neutrino energy emission also depends implicitly on other parameters. For example, the radius of the core would be sensitive to a variation in the strong interaction scale, which would change the nucleon mass, nuclear density, and hence the binding energy of the neutron star. One can bound a possible variation in the nucleon mass by requiring the total binding energy to be within a factor 2 of the estimated energy of the neutrino pulse. The binding energy E~ GNM2/Rc~ GNNZm~/Rc, where N is the total n u m b e r of nucleons in the core. From the balancing o f gravitational and Fermi energies we have N - 2 / 3 ~ GNm~, and the radius Rc of the core is related to N and the baryon number density n by Re~ (N/n) ~/3. The number density n is close to the nuclear matter density, so that n~A~cD ~ m 3, where the constant o f proportionality depends on details of the equation of state. We allow an uncertainty
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of a factor of 10 in this constant, and hence o f 101/3 in the proportionality constants in
G~ I/2 1 R~~ ~ , E ~ i~,.3/2~" m~q
'~N
(16)
It~N
Comparing the expected binding energy with that observed in neutrinos, we can constrain the density variation of mN, assuming GN to be fixed: 0.48~2_t/210_1/6
< mN < 2 1 / 2 1 0 J / 6 mNa
2.1.
(17)
For an estimate of O/f inside the core we use eq. (10b) with p ~ 1015g cm -3 to obtain 2 2 --1.3X103~< (30---0~m) O~
where cn is the specific heat o f normal neutrons: cn -~2X 102°T9 erg c m - 3 K - j .
Ru,c, ~- 1.7× 1021T89erg c m - 3 s - i.
dT dt
- -
=
1 Ru .... c,
(21)
whose solution is [ 13,10] (with t in s)
(17N)
It is worth noting that time, energy and temperature scales also become density-dependent inside the neutron star. The conversion to the vacuum time and temperature scales relevant at low densities on and beyond the surface introduces density-dependent factors. In the special case of a scalar field coupling exactly to mass the situation simplifies considerably. There is a c o m m o n 0-dependent physics scale, since all masses are fixed in units of the neutron mass, and E, t - t and T are all proportional to raN. Since only the ratio m~/rnp is observable, where me is the Planck mass, we can use an appropriate Weyl rescaling of the metric to make a reformulation where mp is 0-dependent whilst all other physics masses and couplings are 0-independent. Then we have to examine the effects of a density-dependence o f Newton's constant whereas mN is kept fixed. Next we consider the cooling o f mature neutron stars. These are generally believed to have no internal source of heat. Starting with a high initial temperature T~~ 10 ~ K provided by the collapse of a supernova, they cool at a rate that is expected to be dominated by neutrino emission until the temperature T falls to ~ 108 K, with [ 13,10 ] --
(20)
Density-induced changes in couplings would modify Rurca (20) a n d / o r c,, and thus modify the cooling rate ( 18 ) and the temperature of the neutron star observed now. Defining Q - ( Rur~a/ Cn) / ( Rurcao/ Cno), we can rewrite eq. ( 18 ) as
log,o T9 = ~ (7.3 - l o g l o t - log,o Q). <6.1×102.
(19)
For p = 1015 g c m - 3, and 7"9= 7"/10 9 K, the heat emission rate for the modified Urca process n + n - , n + p + e + % ~3is [13]
dT9 = 0 . 9 × 10-ST7Q s-1, dt
ltR~+lexp(-lzRc)sinh(gR)#R
X(1
14 September 1989
(22)
The ratio Rur~a/C, depends on several fundamental parameters: Rurca
cn
~
2
ozG~mNn
I/3 T7
--
(23)
m~'
As a guide to estimating the sensitivity to density dependence, we assume that all particle physics mass parameters scale in the same way. Since (Rur~a/Cn)/ T 7 has the dimension of (mass) -5, we can write Q=Q.(mNo/mN) 5 and derive from eq. (22) the relation In ( m ~ ) = ½ 1 n
(~o)+
6 In ( ~ o ) '
(24)
where Q/(~o allows for a possible uncertainty in the calculation of RurcJCn even if the fundamental physics is specified, and T/To represents a deviation of the observed temperature (T) from that expected
(To). Observations of X-ray emission from the Crab pulsar[ 14] and R C W l 0 3 [ 15] are consistent with surface temperatures Ts= (1.5-2.5) X 106 K. The conversion from the surface temperature to the internal temperature TI depends on the equation of state assumed. For example, Ts--- 1.5 X 106 K corresponds to T~ ~ 5 X 107 K for a magnetized star with a soft equa-
(18)
~3 We take the effective neutron mass m* =0.8 m°. 267
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PHYSICS LETTERS B
tion of state, and to T~ ~- 2 × 108 K for an unmagnetized star with a stiffequation of state [ 16 ]. A recent analysis [ 17 ] of the behaviour of the Crab pulsar following glitches in 1969 and 1975 indicates that T 1 = ( 4 + 2 ) × 1 0 8 K. Eq. (22) would give T~o---3× l0 s K if Q = 1, so we conclude that (T~/T~ o) = 2 °-+1. The prediction for R C W 103 is comparably accurate. Using (T~/TIo) = 2 o + ~ and allowing ((~/ Qo) = 10 °-+j, we obtain 0 . 2 7 < mh <3.6,
14 September 1989
[18]. In this case, one estimates that P-~ 2 n ( G N p ) - l / 2 , where p is the core density [18], and the observed period of SN 1987A is compatible with a central density p ~ 10 ~5 g c m - 3 as expected. Assuming that all particle masses scale in the same way so that p~:m 4, we can write P-2=dGNm4 where the dimensionless coefficient d may be known to within a factor of 10: d/do= 10 °-+ i reflecting uncertainties in the equation of state, etc. Comparing the observed period (P) with that expected (Po), we therefore have
(25a)
mNo
In (m~--~NNo)=--}In (ff-~o) --¼1n ( ~ o ) ,
and hence - 2 . 6 < CN ~ <0.73.
(25b)
To obtain a constraint on the scalar mass and coupling strength, we use the previous solution (10b) for ¢ inside the neutron star, which we take to have uniform density and radius/~ = 10 km. Averaging (10b) over the neutron star's interior, we find
(~/~) 3 exp(-- #~)
2
-1.5<
× [ / ~ cosh (p.R) - sinh (/~/~) ] ) (27)
The quantity in large brackets varies between 1 (g---,oo) and 0 (#--,0). The recent observation [ 9 ] of a pulsar with short period P ~ 5 × l 0 - 4 S in the remnant of supernova 1987A allows us to set a slightly more stringent constraint on mN/mNo and hence on 8 C / f and/~. This infant neutron star is believed to spin close to breakup, with a surface velocity close to the speed of light 1¢4 Here we introduce C~ defined by 8m~/m~= -C~¢/f
268
<0.6,
f
1.7 ×
(25be)
103 2
2
3 (/d~+ 1) exp(-/d~) (U~) 3
\ × [/u~ cosh (p3~) - sinh (~R) ] ) < 700,
exp(-~/~)
<850.
(25ap)
and hence
(26)
--3.0X 10 -3
(~)3
<2.5,
and finally
Imposing the upper limit (25b) on this expression, we obtain the constraint ~4
2
MN mNo
-
X [/t/~ cosh (/t/~) - sinh (/t~) ] ) .
P/Po= 20+- i we finally obtain
and taking 0.4<
(24p)
(27p)
Our bounds are displayed in fig. 1, together with various previous bounds [ 4,19 ]. A few comments are in order: ( 1 ) The solid lines in the figure are calculated assuming a material-independent scalar interaction: CG = -- 2C., etc. Bounds for other model couplings are easy to derive, and we will give an example at the end o f this paper. (2) As compared to bounds from Cavendish-type experiments, astronomical and satellite observations etc., our bounds are competitive when 2 < 10- 3 m. (3) The changes in masses and couplings corresponding to our bound are not small. Therefore, the linear approximation to the scalar-matter couplings assumed above may not be valid. However, we ex-
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i 16 I
t 4 September 1989
V;
•
%
12
/
~',"
/2"
"?771" OsdUations I'fflllrl 1 , " "rt.FOfllD.lll.."
~
N.SZ.~
44.,
•~ x ~
II[?tlrflllll'l'l?f-"Ii'1 "
-4
-8
I -6
-/+
~ -
i
I
I
I
I
I
I
0
2
4
6
8
10
12
14
log (X/m)
Fig. 1. Compilation of constraints on the dilaton-matter coupling strength oz as functions of the range 2= 1/,u. The solid line is taken from ref. [4 ], and is a compilation of bounds from Cavendish-type experiments, astronomical and satellite observations, etc. The hatched lines are the astrophysical and cosmological bounds discussed in this paper. The line marked SN is derived from the agreement [ 12] of neutrino observations [ 8] of SN 1987A with theoretical predictions [ 11 ]. The line marked N.S. is derived from the agreement between neutron star cooling calculations [ 13,10 ] and observations [ 14,15 ], and from the observed [9 ] period of the SN 1987A pulsar. The dotted line is the porous cosmological constraint [19] on coherent dilaton field oscillations. These bounds are shown for materialindependent couplings. In addition, we display (dashed line) bounds from EOtv6s-typeexperiments for a material-dependent interaction, assuming a scalar coupling proportional to baryon number. pect that realistic non-linear couplings would yield qualitatively similar bounds. ( 4 ) The d o t t e d lines in the figure are cosmological b o u n d s [19]. These d e p e n d on some specific ass u m p t i o n s that we now discuss. To treat cosmology, we replace the right-hand side o f eq. ( 6 ) by a term (Cv/f)pu which parametrizes the coupling o f ~ to the d o m i n a n t c o m p o n e n t o f the cosmic energy density. We expect Cu to be negligible during the r a d i a t i o n - d o m i n a t e d epoch, since we can always rescale the fields so that kinetic terms are ind e p e n d e n t o f ~ from which it is clear that the scalar does not couple to an ideal gas in thermal equilibrium. A non-vanishing value o f Co could only arise from m o d i f i c a t i o n s o f the equation o f state due to Od e p e n d e n t interactions. In contrast, couplings o f ~ to visible or d a r k non-relativistic m a t t e r could induce a substantial value o f Cu during the m a t t e r - d o m i n a t e d
epoch. The earlier arguments strongly constrain couplings to matter, but couplings to dark m a t t e r could be much larger. F o r an equation o f state Pn= (n/3-1)pn with n = 4 ( 3 ) for the radiation ( m a t t e r ) d o m i n a t e d epoch, a n d assuming the Hubble constant H obeys the condition H2//t2<< 1/12 ( 2 / 9 ) , the generic homogeneous and isotropic solution o f the k - - 0 cosmological equations [ 6 ] ~'+ 3 H ~ + / t 2 ~ = t i p ,
p+nHp+ t i p S = 0 , 6MZH 2 = p + V+ ~ 2 (where
(28)
M2=--m2/16n,fl- ½x/-~Cu/CN)is a d a m p e d 269
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oscillation of ~ around a time-dependent minimum given by 0o(t) = ~_MP(t).
(29)
The oscillating field behaves like non-relativistic matter and should not dominate the energy density of the Universe too early. This gives bounds [ 20 ] on the initial amplitude ~---~-~o. Whether these bounds are satisfied or not depends on the detailed dynamics of the QCD phase transition or, in the case of small ~t2, the dynamics of the onset of ~ oscillations once H 2 decreases below/ft. The dotted line in the figure corresponds to maximal oscillations I ~l = a - I / 2 M (_~fwhen CN--~I) at T = m i n ( 1 0 0 MeV, (pM) ~/2), as would have been natural for the similar case of axion oscillations. For a < 1 we take I~1 = M as a maximal amplitude. When /z2<
O=co + a M l n - - , to
H=rlt -~, p=potot -2,
(30)
where 2
24fl
~l= n + 1 2 f l 2 / ( 6 _ n ) , a= n ( 6 _ n ) + 1 2 f l 2, pot~ = ( O / 2 - ½a2)M 2.
(31)
Nucleosynthesis and the microwave background radiation place limits on q [22 ] and hence on fl during the radiation-dominated epoch: fl<0.28 (0.20)
14 September 1989
y
0"+3H~+ ~ - ~ ( O - ~ ) p = 0 ,
fl=~,
(33)
we obtain standard cosmology with ~ approaching q~o as a negative power of t:
~-~=At-'", N/(3q 3q-1 m= ~ - -
1 )2
--6yr/2 ,
2
q= -
(34)
n
When a small mass term is included, ~ approaches a time-dependent effective minimum at
0o(t)=0o (1 +
yp(t)]
-' "
(35)
The evolution of ~ before the onset of oscillations (/z 2 << H 2 ) is relevant for the magnitude of the initial amplitudes ~, since we expect fl and 7 to be non-negligible during the QCD phase transition (but also the dynamics to be more complex than in our simple discussion). This illustrates the difficulty of obtaining cosmological bounds in our case. In particular, a scalar with range 2_~300 m and o¢-~ 10 - 2 cannot be completely excluded, given our present ignorance of the cosmological evolution of such a scalar field. Finally, we specialize the above analysis to one example of matter-dependent scalar couplings, namely a light "dilaton" from string theory. In this case, the dominant effect comes from the q~-dependence of the strong coupling due to the term [ 2 ] ~ = - ¼exp ( -
(36)
~¢M)FI,~F"",
(32) which implies
for a HeliUm abundance Yp~<0.26 (0.25 ). However, as discussed above, we expect fl to be very small during this epoch. When # - ~ is bigger than our present horizon size, fl is bounded by the age of the Universe qHff ~. It should be noted, though, that a linear approximation to the 0-matter coupling is not well-motivated when ~t2 << H 2. If we postulate instead a scalar evolution equation ~5 This invalidates the bound given in ref. [ 21 ], since during nucleosynthesis ~'+ 3H~ >>/~-'~ for the value o f # quoted.
270
8gs 1 gs - 2 M "
(37)
This leads [ 6 ] to a variation in the strong coupling scale: 8A
[
1
A - ~ 2b3g~(Mx) where
d
Mx M'
(38)
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1
A~-Mx e x p ( 2 b 3 g f ( M x ) )
(39)
with Mx the scale at which S U ( 3 ) is unified with other interactions in some higher symmetry and b3 is the coefficient of g3 in the renormalization group equation for gs. The first term in eq. (38) dominates, since g~(Mx) must be small. The nucleon mass is proportional to A at zeroth order in the small u, d and s quark masses, so that 5mN/mN~-SA/A and one obtains O{'~
")
1 4
b~gs (Mx)
(40)
"
Numerical values of ol (40) in the minimal SU(5) G U T where b3=(33-2Nq)/48z~ 2 and a s ( M x ) = g2 (Mx)/4~r_~ 1/42, and the minimal supersymmetric SU(5) G U T where b3=(27-3Nq)/48n 2 and c~s(Mx) ~ 1/24, for N , = 6 are a ~ 250 -~ 190
without superrsymmetry,
(41 )
with supersymmetry.
To first order in the light u, d and s quark masses there is an additional contribution to the possible variation in the nucleon mass:
(6mN) = \ m N ./q
~
q=u.cl.s
(6m+~ \---'~-u./ mq(N[ElqlN) mN
(42)
If we assume, for definiteness, that 8mq/mq is universal for the u, d and s quarks, and use the estimate [23] Zq=u.d.smq(NI4qlN) =]mN, we find that
( 6mN ~ 1 5mq mh ./ u 3 mq
(43)
14 September 1989
~-dependence of v a n d / o r 2 would be more complicated if symmetry breaking is induced by radiative corrections. We assume conservatively that
~mq =O(1 ) ~ M mq
(45)
in which case the dominant source of dilaton couplings to matter is ~A/A, resulting in a relatively large material-independent coupling a (40), (41) and smaller material-dependent couplings suppressed by O(1/a). A scalar with such a large coupling strength (40), (41 ) and material-dependent couplings 2 or 3 orders of magnitude smaller would have been seen in E6tv6stype experiments or satellite observations, unless its range were very small. We therefore conclude that the dilaton mass must be bounded from below by / 1 > 2 × 10 -~ eV.
(46)
The dilaton mass must therefore be generated by some physics beyond the QCD scale, since otherwise a typical order of magnitude would be # ~A 2/M as for an axion. We should note, however, the possibility that the spectrum of light string scalars is richer. A contribution of these with a small dilaton component would have a correspondingly reduced value of oL, and could have a mass in the range detectable by experiment. One of the authors (C.W.) would like to thank the members o f l T P for the kind hospitality in Santa Barbara, where part of this work was performed and supported by the National Science Foundation under Grant No. PHY82-17853, supplemented by funds from the National Aeronautics and Space Administration.
Since quark masses mq=2qV,where the 2q are Yukawa couplings and v is a Higgs VEV, we have 8 m ~ _ ~2q + ~5___v. m~ ).q V
References
(44)
In the type of string models discussed here 82q/ 2q= 8g/g as a first approximation and is hence given by eq. (39). The value of 5v/v depends on the detailed mechanism of spontaneous electroweak symmetry breaking. There are contributions to 6v/v from the ~-dependence of the Higgs scalar mass u and the quartic coupling 2 mentioned above. One would expect that 2ocg 2 and hence that 62/2-~ 28g/g, but the
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