This document summarizes a talk on the variation of fundamental constants over time. It discusses several methods for measuring potential variations, including analyses of the cosmic microwave background, quasar absorption spectra, radioactive decay rates from the natural nuclear reactor at Oklo, and comparisons of atomic clock rates. Measurements from big bang nucleosynthesis and quasar data suggest the fine structure constant may have been smaller in the early universe, varying on the order of 10^-15 per year. However, results are not conclusive and depend on theoretical models. Ongoing work using improved atomic clocks aims to more precisely measure any drift of fundamental constants like the fine structure constant and quark-mass ratios over time.

1. Variation of Fundamental
Constants
Pinghan Chu
Department of Physics
University of Illinois at Urbana-Champaign
Sack Lunch Talk, April 5, 2010
Variation of Fundamental Constants – p. 1/2

2. Dimensionless Constants
In 1937(Nature 139, 323), Dirac pointed out the interest of time-varying fundamental
constants, like fine structure constant and the mass ratio of the proton to the electron.
Many theories suggest that time variation in fundamental constants can be correlated
to each others (e.g. Langacker, et. al. Phys. Lett. B, 528, 121(2002))
If these constants were different at early universe than they are today, their relative
shifts can only be determined by the underlying theory which causes the changes. For
example, in the grand unifies theories, all couplings are correlated to the GUT scale
parameter, which varies with time.
Since variation of dimensional constants cannot be distinguished from variation of
units, it only makes sense to consider variation of dimensionless constants.
e2 1
Fine structure constant α = c
= 137.036
.
me,q me
Electron or quark mass/QCD strong interaction scale, ΛQCD
,or mp
since
mp ∼ 3ΛQCD where me,q are proportional to Higgs vacuum (weak scale).
Variation of Fundamental Constants – p. 2/2

3. Variation of Coupling Constants
The variations of EM, weak and strong couplings should be related at high energy.
The inverse coupling constants have the following dependence on the scale µ and
normalization point µ0 (∼ 3 × 10−16 GeV):
α−1 (µ) = α−1 (µ0 ) + bi ln(µ/µ0 )
i i (1)
Variation of Fundamental Constants – p. 3/2

4. Variation of Fundamental Constants
GUT predicts the variation of QCD scale ΛQCD in terms of variation of α(fine structure
constant):
αs (r) ∼ 1/ ln(rΛQCD / c) → δΛQCD /ΛQCD ≈ +34δα/α. (2)
The variation of quark mass and electron masses are given by δm/m ∼ +70δα/α, so
that
δ(m/ΛQCD ) δm δΛQCD δα
∼ − ∼ 35 . (3)
(m/ΛQCD ) m ΛQCD α
The big coefficient implies the running strong-coupling constant and Higgs constants
run faster than α.
If these models are correct, the variation of mq /ΛQCD can be searched for
experimentally.
Variation of Fundamental Constants – p. 4/2

5. Search for Variation of Fundamental Constants
Deuteron binding energy.
Quasar Absorption Spectra.
Oklo natural nuclear reactor.
Atomic clocks (based on atomic and molecular calculations).
Variation of Fundamental Constants – p. 5/2

6. Variation of Deuteron Binding Energy
Deuteron binding, Qnow , is well measured to be 2.22 MeV(25.82 × 109 K) at the
present time.
Deuteron binding energy at the time of big bang necleosynthesis can be deduced from
the isotope abundances of D, 4 He, Li, and the baryon to photon ratio η.
η has been accurately measured by WMAP, and Yp =4 He/H, YD = D/H and
YLi = Li/H have also been well measured.
Need model for big bang nucleosynthesis to relate QBBN to η and Yp , YD and YLi .
Variation of Fundamental Constants – p. 6/2

7. Variation of Deuteron Binding Energy
Dmitriev et. al. (Phys. Rev. D 69, 063506 (2004)) used big bang nucleosynthesis
calculations and light element abundance data to constrain the relative variation of the
deuteron binding energy = Q, so that ∆Q = Q(BBN ) − Q(now) .
The vertical line shows the present
value of Qnow = 25.82 × 109 K.
The shaded regions illustrate the 1σ
range in the data of the element
abundance, Yi .
Calculate the light elements abundances
as a function of the deuteron binding
energy QBBN using ηwmap = 6.14 ×
10−10 .
Variation of Fundamental Constants – p. 7/2

8. Variation of Deuteron Binding Energy(cont’)
8
1σ contour ellipse shows the QBBN
and η by fitting YD , Yp and YLi . 7
The lighter shaded region shows
CMB-WMAP data for ηwmap .
Η 1010
6
The darker shaded region is the
1σ-range for η from BBN calculations
using the present-day value of the
deuturon binding energy, Qnow . 5
Using ηwmap , ∆Q/Q = −0.019±0.005.
4
24 24.5 25 25.5 26
Q 109 K
Variation of Fundamental Constants – p. 8/2

9. Variation of Deuteron Binding Energy (cont’)
From previous slide, ∆Q/Q = −0.019 ± 0.005. It can be interpreted as
X ≡ ms /ΛQCD of the strange quark mass and strong scale
δX/X = (1.1 ± 0.3) × 10−3 .
The reason that the deuteron binding energy is sensitive to the strange quark mass
are (Flambaum and Shuryak, Phys. Rev. D 67, 083507 (2003) ):
there is strong cancellation between σ-meson and ω-meson contributions into
nucleon-nucleon interaction (Walecka model), therefore, a minor variation of
σ-meson mass leads to a significant change in the strong potential.
the σ-meson contains valence s¯ quarks which give large contribution to its mass.
s
Variation of Fundamental Constants – p. 9/2

10. Optical Spectra
Compare cosmic and laboratory optical spectra to measure the time variation of α.
The relative value of any relativistic corrections to atomic transition frequencies is
proportional to α2 .
The relativistic corrections vary very strongly from atom to atom and can have opposite
signs in different transitions.
Relativistic many-body calculations are used to reveal the dependence of atomic
frequencies on α for a range of atomic species (like Mg, Mg II, Fe II, Cr II, Ni II, Al II, Al
III, Si II, Zn II, etc.).
The transition frequencies can be presented as
ω = ω0 + qx, (4)
where x = (α/α0 )2 − 1 and ω0 is a laboratory frequency of a particular transition.
Three classes:
positive shifters, q > 1000 cm−1 .
negative shifters, q < −1000 cm−1 .
anchor lines with small values of q.
It gives us an excellent control of systematic errors.
Variation of Fundamental Constants – p. 10/2

11. Observations of the Variation of α
Spectroscopic observations of gas clouds seen in absorption against background
quasars.
The alkali doublet method. The fine structure splitting is proportional to α2 .
E(p3/2 ) − E(p1/2 ) = A(Zα)2 . (5)
Simple but inefficient. It compares transitions with respect to the same ground state.
Si IV alkali doublet
P3/2
6
1393.8Å
P1/2
6
1402.8Å
S1/2
Varshalovich et. al.(Astron. Lett., 22, 6(1996)) showed δα/α = (0.2 ± 0.7) × 10−4 and
|α/α| < 1.6 × 10−14 yr −1 .
˙
Variation of Fundamental Constants – p. 11/2

12. Observations of the Variation of α
α1 α2
P3/2 P3/2
6 6
P1/2 P1/2
6 6
S1/2
S1/2
The many-multiplet method. It allows the simultaneous use of any combination of
transitions from many multiplets, comparing transitions relative to different
ground-states. Relativistic correction to electron energy En :
En 1
∆n = (Zα)2 [ − C(Z, j, l)], (6)
ν j + 1/2
where C ≈ 0.6 is the contribution of the many-body effect and ν is effective principal
quantum number.
Compare heavy (Z ≈ 30) and light (Z < 10) atoms, or, compare s → p and d → p
transitions in heavy atoms. Shifts can be of opposite sign.
Variation of Fundamental Constants – p. 12/2

13. Observations of the Variation of α (cont’)
Webb et al. (Phys.Rev.Lett.82, 884, 1999 and 87, 091301, 2001) report observations
of quasar absorption lines at redshift of z = 1 − 2 that suggest the variation of the fine
structure constant is at the level of 10−5 .
The fit of the data (over 12 billion years) (Mon. Not. R. Astron. Soc. 345, 609, 2003) is
δα α
˙
= (−0.543 ± 0.116) × 10−5 and = (6.40 ± 1.35) × 10−16 year −1 . (7)
α α
Variation of Fundamental Constants – p. 13/2

14. Oklo Natural Nuclear Reactor
The discovery of the Oklo natural nuclear reactor in Gabon (West Africa) was in 1972.
About 1.8 billion years ago within a rich vein of uranium ore, the natural reactor went
critical, consumed a portion of its fuel and then shut down.
Measure the number of atoms per unit volume NA for 149 Sm, 147 Sm, 235 U, etc.
The 149 Sm concentrations is due to the yields from the fissions and the effect of
neutron captures. Solve the neutron capture cross section of 149 Sm,
n +149 Sm →150 Sm + γ by using the observed abundances ratio 149 Sm/235 U, etc.
The neutron capture cross section can be described by the Breit-Wigner formula,
Γn (E)Γγ
σ(E) ∝ g 1
. (8)
(E − Er )2 + 4 Γ2
A. I. Shlyakhter (Nature, 264, 5584, 340 (1976)) and Damourb, Dyson (Nuclear
Physics B, 480, 37 (1996)) showed that the resonance Er for the neutron capture
cross section is sensitive to the value of the fine structure constant
(Oklo) (now)
∆ ≡ Er − Er = −|αdEr /dα|(αOklo − αnow )/α so that the final result is
α
˙
−6.7 × 10−17 yr −1 < < 5.0 × 10−17 yr −1 . (9)
α
Variation of Fundamental Constants – p. 14/2

15. Oklo Natural Nuclear Reactor (cont’)
Fujii et. al. (Nuclear Physics B, 573, 377 (2000)) showed
(−0.2 ± 0.8) × 10−17 yr −1 < α/α < (4.9 ± 0.4) × 10−17 yr −1 .
˙
Assuming the reactor temperature (200 − 400◦ C), use the measured cross section
from the abundance of 149 Sm to estimate ∆Er upper and lower bounds.
200 600
20°C Calculated resonance shift for n + 149Sm
n+ 149
Sm
Capture Cross Section σ (kb)
100°C
200°C 500
150 300°C
^
500°C
Temperature T (°C)
1000°C 400
100 300
200
50
100
0 0
− 200 − 150 − 100 − 50 0 50 100 150 200 − 20 − 10 0 10 20
Resonance Position Change ∆Er (meV) Resonance Position Change ∆Er (meV)
Variation of Fundamental Constants – p. 15/2

16. Atomic clocks
Compare rates of different clocks over long period of time. Optical transitions are
related to α and microwave transitions are related to α and mq /ΛQCD .
Very narrow lines, high accuracy of measurements.
Flexibility to choose lines with larger sensitivity to variation of fundamental constants.
Simple interpretation (local time variation).
For example, Blatt et. al. (Phys. Rev.
Lett. 100, 140801 (2008)) measured
(a) [15] (c)
1 S −3 P clock transition frequency in 75
0 0
87 Sr related to the Cs standard. 2.1 Hz
neutral [12]
[13] [14]
They showed the variations are 72
5/06 11/06 5/07 11/07
[18]
85 (b)
[15]
α
˙ νSr − ν0 (Hz)
[13] [14]
= (−3.3 ± 3.0) × 10−16 /yr, 75
α [22] [23]
[12]
65 [17]
Tokyo
(10) Paris
55
Boulder
45
1/05 7/05 1/06 7/06 1/07 7/07 1/08
Need detailed atomic, nuclear and QCD Date
calculations. Refer to Dinh, et. al.’s work
(Phys. Rev. A 79, 054102 (2009)).
Variation of Fundamental Constants – p. 16/2

17. Summary
Big Bang Nucleosynthesis: may be interpreted as a variation of mq /ΛQCD .
Quasar data : MM method provided sensitivity increase 100 times. Anchors, positive
and negative shifters–control of systematics.
Oklo : no positive conclusion.
Atomic clocks : present time variation of α, mq /ΛQCD .
Variation of Fundamental Constants – p. 17/2

18. Conclusion
The experiments and observations mentioned here are related to different time
intervals and there is no reliable method for their model-independent comparison.
The lab results on optical measurements in terms of the effective rate ∂α/∂t are the
most reliable since they are related to optical clocks.
A possible variation of the fine structure constant is at the level of 10−15 per year?
Variation of Fundamental Constants – p. 18/2

19. Parameters of the Standard Model(Variations?)
There are 19 parameters in SM, determined by experiments.
Mass of three leptons, me , mµ , mτ .
Mass of six quarks, mu , md , ms , mc , mt , mb .
CKM three angle, θ12 , θ23 , θ13 .
CKM phase, δ.
Three coupling constants, g1 , g2 , g3 .
QCD vacuum angle, θQCD .
Higgs quadratic coupling µ and Higgs self-coupling strength λ.
We only discuss the variation of the coupling constants (fine structure constant) and
mass ratio. Is it possible other parameters are also varied?
Variation of Fundamental Constants – p. 19/2

20. References
E. W. Kolb and M. S. Turner, The Early Universe, Westview Press (1990).
J. P. Kneller and G. C. McLaughlin, Phys.Rev. D68 (2003) 103508.
Jean-Philippe Uzan, Rev. Mod. Phys. 75, 403 (2003).
S. G. Karshenboim, A. Yu. Nevsky, E. J. Angstmann, V. A. Dzuba and V. V. Flambaum,
J.Phys.B.At.Mol.Opt.Phys.39:1937-1944,2006.
V. V. Flambaum, AIP Conf.Proc.869:29-36,2006.
V. V. Flambaum and V. A. Dzuba, Can. J. Phys. 87, 25-33 (2009).
B. Fields, Lecture Notes for Astronomy 596 NPA, Fall 2009.
V. V. Flambaum, Variation of Fundamental Constants, Workshop at University of
Adelaide, February 2010.
Variation of Fundamental Constants – p. 20/2