Download Nucleosynthesis and the time dependence of

c. .H7
‘.’
-Q 4-
23 February
1995
PHYSICS
physics
El St-Mm
Letters
B 345 ( 1995)
LETTERS
B
429-434
Nucleosynthesis and the time dependence of fundamental
couplings
Bruce A. Campbell a, Keith A. Olive b
a Department
of Physics,
Universiry
of Alberta. Eihnonton.
Alto., Canada T6G 2JI
h School of Physics and Astronomy.
University
a/ Minnesota,
Minneapolis.
MN SSd55B USA
Received 22 November
Editor: M. Dine
1994
Abstract
We consider the effects of the time dependenceof coupling: due to their dependence on a dilaton field, as occurs in
superstringtheory, as well as in gravity theories of the Jordan-Brans-Dicketype. Becausethe scaleparametersof couplings
set by dimensional transmutation depend exponentially on ihe dilaton vev. we may obtain stringent limits on the shift of
the dilaton from the requirement that the inducd shift in the couplings not vitiate the successfulcalculationsof element
abundancesfor big-bang nucleosynthesis.These limits can be substantially stronger than those obtained directly from the
dilaton-induced change in the gravitational coupling.
The successful predictions of the light element abundances in the standard model of big bang nucleosynthesis
(SBBN) [ I] provides a basis to test extensions to the standard model of particle interactions. While deviations
to SBBN typically induce changes in all of the light element abundance predictions (D, 3He, ‘He, and ‘Li).
particle physics models are most!y constrained by the 4He mass fraction, 5,. In the SI3BN, the abundances
are primarily sensitive to only a single parameter, the baryon-to-photon ratio, 7. Consistency between the
predictions of SBBN and the observational determinations of the light element abundances restricts 77 to a
narrow range between 2.8 x IO-” and 4 x IO-lo. In this range the calculated 4He mass fraction lies in
the range YP = 0.239-0.246 [2]. This is slightly high when compared with the observationally inferred best
primordial value, Y, = 0.232 f 0.003 f 0.005 [ 31, where the errors are 1 u statistical and systematic errors
respectively. Indeed, consistency relies on these errors and even so, allows for very little breathing room for
any enhancement in primordial 4He. This is the reason that one can obtain very tight limits on the number of
neutrino liavors [4,3,5].
The 4He abundance is primarily determined by the neutron-to-proton ratio just prior to nucleosynthzs:c which
before the freeze-out of the weak interaction rates at a temperature TJ N 1 MeV, is given approximately by the
equilibrium condition
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B.A. Campbell,
K.A. Olive/Physics
Le~fer.~ B 345 (1995)
4~~34
where AInN = 1.29 MeV is the neutron-proton
mass difference.
(The ratio is slightly altered by free neutron
decays between T, and the onset of nuclcosynthcsis
at about T N 0. I MeV.) Furthermore,
freeze-out
is
determined by the competition
between the weak interaction
rates and the expansion rate of the Universe
(2)
where N counts the total number of relativistic particle species. The presence OS additional neutrino flavors (or
any other relativistic species) at the time of nucleosynthesis
increases the overall energy density of the Universe
a’nd hence the expansion rate leading to a larger value of TJ, (n/p),
and ultimately
YP. Because of the form
of Eq. (2) it is cl=
that -just as one can place limits on N, any changes in the weak or gravitational
coupling
constants can be simiiarly canstrained
[ 6]-[ 1 I 1.
Constraints on C:. aqrl C;C have often beer obtained under the assumption that these quantities have varied
in time as a power-law,
G o( 1.‘. Colrstramts on SC/G yield an acceptable range for x [ 6,111. Limits on these
couplings as well as the fine structure constant and neutron-proton
mass difference were considered in [7]. In
general, 5, is most sensitive to changes in ANIN [ 71. It was pointed out in [9] however, that as the Fermi
constant can be written directly as the vev of the Higgs boson in the standard model, G~/fi
= 1/2u*, changes
in GF will naturally induce changes in fermion masses and hence AmN. In this con:ext, temporal as well as
spatial changes in GF were considered in [ IO].
In string theory, the vev of the dilaton field, acts as the string loop counting parameter [ 121. At the (string)
tree level, changes in the vacuum value of the dilaton corresponds
directly to changes in the gravitational
coupling GN. This can be seen by writing down the action in the string frame [ 131,
where 4 is the dilaton field, y is an arbitrary scalar lield and rc/ is an arbitrary fermion. D, is the gaugecovariant derivative corresponding
to gauge fields with field strength F,,,. K’ = 8rrG~. However, by performing
a confcrmal
transformation,
g,,, --) c - b&g g@,, we can rcwri+e (3) in the Einstein frame as
Now it is apparent that changes in the dilaton vev will i.lduce changes in gauge coupling5 and fermion masses
(set details helow). Thus, we will he able to limit ch;,lges ir, ;he d&ton vev from the rime of nucleosynthesis
til: today from the observed ‘Hc ahundancc. As we will see, we will be able to obtain particularly
stringent
limits bccausc the dcpcndence on d, of gauge arid Yukawa couplings induce changes in quantities such ijs’ <ne
Higgs vev, i!, and hQco which are exponentially
dependent on the dilaton vev through the renormalization
g:oup
equations and dil.lcn?ional
transmutation.
In what follows. we will first derive the general (though approximate)
relations between Y,, clld the various couplings and bmN md the ccrrcsponding
limits on these: quantities. WC
will then derive the induced changes in these quantities from changes in the dilaton vev and hence derive limits
on the changes of the dilaton vev in the context of string theory. We will also consider these same effects in
the co,rtext of Jordan-Brans-Dicke
gravity.
As is well known, the 4He abundance is predominanily
detennined
by the neutron-to-proton
ratio and is
easily estimated assuming that all neutrons are incorporated
into ‘He,
B.A. Campbell,
K.A. Olive/Physics
Leners
B 345 (1995)
4.29434
431
(5)
so that
.yx
Y
l
1 +
A(nfp)
(n/p)
(6)
(n/p)
From Eqs. (1) and (2) it is clear that changes in any of the quantities CF. GNPor N, will lead to a change in
T/ and hence q,. If we keep track of the changes in T.f and AmN separately, we can write,
(7)
where b*m,v is the change in Am,v. Combining these equations we obtain
AY
--x
Y
AT/
--( Tf
A*!nN
A% >
(8)
where the factor hN/Tf(
! + (n/p))
M 1. From the consistency of the light elements, we will take the limit
-0.08 < AyIY< 0.01 assuming a SBBN value of 5, = 0.240 and the observed range to be 0.221 < YP< 0.243.
As noted above, changes in T, are induced by changes in the weak and gravitational couplings and can be
readily determined from (2). Changes in Am,v can come from a number of sources. One can write the nucleon
mass difference as
where u and b are dimensionless consiants giving the relative contributions from the clectlomagnetic and weak
interactions. In (9). u is the standard model Higgs expectation value. A discussion on the contributions to Am,v
can be found in [ 141. We will take n N -.8MeV/ ~,,,,,&JQ, where &Do is the present (low energy value)
of AQcn, a;;;, = 137 and b N 2.1 MeV/uo where 00 is the standard value of the Higgs expectation value,
00 N 247 C-C% Our results will not be particularly sensitive to the precise values of a and b.
In what follows we will consider the effects of changes in gauge and Yukawa coupling constants. From
Eq. (9) we see that a change in the electromagnetic coupling constant will directly induce a change in Arn,v.
Changes in the strong coupling constant however can bc seen to have dramatic consequencrs from the running
of the renormalization group equstions ’ . Indeed the QCD scale A is determined by dimensional transmutation
(10)
or for Sj- = 3
-48~”
A2= M2pCXF( 27gf(M’,) )
Clearly, changes in go will induce (exponentially) large changes in AQCD and therefore in Amw and 4,.
Similarly, changes in Yukawa couplings can induce large changes in AmN. In models in which the ele:troweak
symmetry is broken radiatively, the weak scale is also determined by dimensional transmutation 1151. This
mechanism is based on the solution for the renormalization scale at which the Higgs mass* goes negative, being
’ This
observafirm
was made
ty Dixit
rind Sher
[ 91 in their criticism
of the dependence
of Y, on (r? in [ 8 J.
B.A. Campbell,
432
K.A. Olive /Physics
Lefters
B 345 (I 995) 429434
driven by a Yrukawe ~upling, presumably h,. The v.*eakscale and the Higgs expectation value then corresponds
to the renormalization point and is given qualitatively by
lJ 0’ MfJ exp( -&%/a,)
(12)
.vhere c is a constant of otdcr I, and Q, = hT/47r. Thus small changes in h, will induce large changes in o and
hence in Tf and AmN.
Let us now look at the implications of a rolling dilaton in string theory. We will work in the Einstein frame
and therefore, we will not consider changes in the gravitational coupling GN. From the form of the action in (4)
one can see that although scalars and fermions have canonical kinetic terms after the conformal transformation,
there remains a dilaton dependence in their masses as well as in the coefficient of the gauge field strength.
From (4), WC,dcfine the gauge coupling constant
a’e-JZK6
I
~2,>
=
2K2
a’&
(13)
= - 2K
S is the (chiral) multiplet in which the real part of the sca!ar is associated with the dilaton and is used here for
convenience;. a’ is the string tension. From (I I), we see that Awu is in fact doubly exponentially dependent
on the dilaron2 4
(14)
Therefore we can write the induced change in Amv as
(15)
Similarly, we see from (4) that Yukawa couplings are also expected to be dilaton dependent. If we assume
that fermion masses are generated by the Higgs mechanism, then the corresponding Yukawa term in the
Lagrangian would be
and the JI rntis is given by mII = he*~fJZ(H) = h t?6/fi~/&.
The effective Yukawa coupling is therefore
given by hcK+lJ2. Now, as a conscqucnce of Eq. (12). the Higgs vev is determined from
(17)
In this case, the induced change in Am&
is
A*mN
bttM,v d
-AITIN = -imN.j$-;
(18)
Note that in (9) it is really the quark mass difference which contributes to Al&v so that the dependcncc is hu
rather than c.imply u. In addition, changes in u will also induct changes in the frcczc-out. tcmpcrlture Tf,
4
-=--=
Tf
4Au
3
11
.--32rr*cuA&
311;
-* The dependence
of hw:D
the expansion
of the Universe
on the dilaton
in 1 17 1
(19)
was utilized
in n diccussion
regarding
the dilaton
coupling
IO matter
[ 161 and t I our notion
of
B.A. Campbell.
K.A. Olive
/Physics
Lerters
B 345 (I 995) 429-434
433
Clearly e\cn small changes in the dilaton expectation value will have dramatic consequences on the jHe
abundance.
The contributions in Eqs. (15.1% and 19) can all be summed to give a net change in l$. For c N h, r~ K& N 1
and 2 N 0.1, we have from (8)
(20)
which in order to be. consistent with SBBN gives
-4
X ‘o-4
2
K&
2
5 X
lo-”
(21)
The corresponding limits on other quantitiescan be easily obtained from (21) using relations such as AGF/Gr: N
150~A&, A+,,,/Q,, = -A&/&,
Ah/h = -A&/2&.
This is >ur main result.
Before co.lcluding, it is interesting to consider these same limits in the context of Jordan-Brans-Dicke gravity.
If we write down the analogous action to Eq. (3)
and perform the analogous conformal transformation, g,, -+
(o+ ;,
2.+
&R--
-+Y%$
-
’
( 2K2f$)3/2
--+yPy
rn.&,b
-
LF
Q2 P’
~K~c&,,
we
have
- -
Fp”
Notice first, that the coefficient of Fp,Fp’ ’ is independent of 4 and therefore we do not expect that gauge
couplings will vary. This is a result of the conformal invariance of the gauge kinetic term. Furthermore, notice
also that the fermion and boson kinetic terms are no longer canonical (as well as that of 4). If we rescale
fermions and bosons by (9, y) -+ m
($, y) and we assume that masses are generated by the expectation
value of a scalar (through interaction: of the form H2y2 and H$# and H is similarly resealed) then we see
that the 4 dependence of the masses drop out. Therefope. we dn not expect effects based on transoimensional
mu:ation as neither gauge nor Yukawa couplings will dcpccd OP $ to induce changes in Awn and II. The
Higgs expectation value will probably still depend on r$ if its value is determined from a Higgs potential of the
form‘i’=AH4m2H2. If in the standard model u2 N m2/A, then in tl-.- conformally transformed JBD theory,
Thus GF cx ve2 x 4 and AGF/GF = A ;b/+. this could be put in the form of a constraint
V2 w m2 j.1(2K2g).
on the JBD parameter w, but will yield constraints which have bren discussed recently in the literature [ 181
and will not be rcpcated here.
In summary, we have derived limits on any possible time variation (from the time of nucleosynthesis to the
present) in the dilaton expectation value (in the context of string gravity) due to its effect on standard model
parameters such as gauge and Yukawa couplings as well as &o
and the Higgs expectation value from big
ban8 nucleosynthcsis. The induced variation in the latter two quantities (noting that their scales are generated
through dimensional transmmation) provides us with stringent limits on A& which can be translated into limits
on other couplings. In the JBD theory of gravity, effects as large were not found as the JBD scalar does not
automatically alter gauge and Yukawa couplings.
We would like to tha.>L R. Madden for comment:, on the manuscript. This work was supported in part by
the Natural Scrences and Engineering Research Councir cf Canada and by DOE grant DE-FCl02-94ER40823.
434
B.A. Campbell,
K.A. Olive/
Physics
Lerfers
B 345 {1995)
429434
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