Dependence of nuclear magnetic moments on quark masses and limits on temporal variation of fundamental constants from atomic clock experiments

Abstract

We calculate the dependence of the nuclear magnetic moments on the quark masses including the spin-spin interaction effects and obtain limits on the variation of the fine structure constant $\alpha$ and $(m_q/\Lambda_{QCD})$ using recent atomic clock experiments examining hyperfine transitions in H, Rb, Cs, Yb$^+$ and Hg$^+$ and the optical transition in H, Hg$^+$ and Yb$^+$ .

Full text

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arXiv:nucl-th/0601050v3 8 Mar 2006
Dependence of nuclear magnetic moments on quark masses and limits on temporal
variation of fundamental constants from atomic clock experiments
V.V. Flambaum, A.F. Tedesco
School of Physics, The University of New South Wales, Sydney NSW 2052, Australia
(Dated: February 9, 2008)
We calculate the dependence of the nuclear magnetic moments on the quark masses including the
spin-spin interaction effects and obtain limits on the variation of the fine structure constant α and
(mq/ΛQCD) using recent atomic clock experiments examining hyperfine transitions in H, Rb, Cs,
Yb+and Hg+and the optical transition in H, Hg+and Yb+.
PACS numbers: 06.20.Jr, 06.30.Ft, 21.10.Ky
I. INTRODUCTION
Theories unifying gravity with other interactions sug-
gest a possibility of temporal and spatial variation of
the fundamental constants of nature (see e.g.
[1] where the theoretical models and results of measure-
ments are presented). There are hints for the variation
of the fundamental constants in Big Bang nucleosynthe-
sis, quasar absorption spectra and Oklo natural nuclear
reactor data. However, a majority of publications report
only limits on possible variations. For example, compari-
son of different atomic clocks gives limits on present time
variation of the fundamental constants.
A large fraction of the publications discuss the vari-
ation of the fine structure constant α = e2/?c.
hypothetical unification of all interactions implies that
a variation in α should be accompanied by a variation
of the strong interaction strength and the fundamental
masses. For example, the grand unification model dis-
cussed in Ref. [2] predicts the quantum chromodynamics
(QCD) scale ΛQCD(defined as the position of the Lan-
dau pole in the logarithm for the running strong coupling
constant) is modified as δΛQCD/ΛQCD≈ 34 δα/α. The
variation of quark and electron masses in this model is
given by δm/m ∼ 70 δα/α, giving an estimate of the
variation for the dimensionless ratio
review
The
δ(mq/ΛQCD)
(mq/ΛQCD)
∼ 35δα
α
(1)
The coefficient here is model dependent but large val-
ues are generic for grand unification models in which
modifications come from high energy scales; they appear
because the running strong-coupling constant and Higgs
constants (related to mass) run faster than α. If these
models are correct, the variation in quark masses and the
strong interaction scale may be easier to detect than a
variation in α.
One can only measure the variation of dimensionless
quantities. We want to extract from the measurements
the variation of the dimensionless ratio mq/ΛQCDwhere
mqis the quark mass (with the dependence on the renor-
malization point removed). A number of limits on the
variation of mq/ΛQCDhave been obtained recently from
consideration of Big Band nucleosynthesis, quasar ab-
sorption spectra and the Oklo natural nuclear reactor,
which was active about 1.8 billion years ago [3, 4, 5].
Karshenboim [6] has pointed out that measurements of
ratios of hyperfine structure intervals in different atoms
are sensitive to variations in nuclear magnetic moments.
However, the magnetic moments are not the fundamental
parameters and can not be directly compared with any
theory of the variations. Atomic and nuclear calculations
are needed for the interpretation of the measurements.
Below, we calculate the dependence of nuclear magnetic
moments on mq/ΛQCD by building on recent work and
incorporating the effect of the spin-spin interaction be-
tween nucleons. We obtain limits on the variation of
mq/ΛQCD from recent experiments that have measured
the time dependence of the ratios of the hyperfine struc-
ture intervals of133Cs and87Rb [8],133Cs and1H [9],
171Yb+and133Cs [10],199Hg+and1H [11], the ratio
of the optical frequency in199Hg+to the hyperfine fre-
quency of133Cs [12], the ratio of the optical frequency
in1H to the hyperfine frequency of133Cs [13], and the
ratio of the optical frequency in171Yb+to the hyperfine
frequency of133Cs [14]. It has been suggested in Ref.[15]
that the effects of the fundamental constants variation
may be enhanced 2-3 orders of magnitude in diatomic
molecules like LaS, LaO, LuS, LuO. Therefore, we also
present the results for139La.
During the calculations, we shall assume (for no-
tational convenience) that the strong interaction scale
ΛQCD does not vary and so we shall speak about the
variation of masses (this means that we measure masses
in units of ΛQCD). We shall restore the explicit appear-
ance of ΛQCDin the final answers.
The hyperfine structure constant can be presented as
A = const × (mee4
?2)[α2Frel(Zα)](µme
mp) (2)
The factor in the first set of brackets is an atomic unit
of energy. The second “electromagnetic” set of brackets
determines the dependence on α and includes the rela-
tivistic correction factor (Casimir factor) Frel. The last
set of brackets contains the dimensionless nuclear mag-
netic moment µ (that is, the nuclear magnetic moment
M = µ[e?/2mpc] ) and the electron and proton masses
meand mp. We might also have included a small correc-

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tion due to the finite nuclear size but its contribution is
insignificant.
The ratio of two hyperfine structure constants for dif-
ferent atoms will cancel out some factors such as atomic
unit of energy and me/mp, and any time dependence
falls on two values: the ratio of the factors Frel (which
depends on α) and the ratio of the nuclear magnetic mo-
ments (which depends on mq/ΛQCD).
For the Frel component, variation in α leads to the
following variation of Frel[11]
δFrel
Frel
= Krelδα
α
(3)
and one can use the s−wave electron approximation for
Frelto get
Krel=(Zα)2(12γ2− 1)
γ2(4γ2− 1)
(4)
where γ =
body calculations [16] give more accurate results, with
a slightly higher value of Krel than that given by this
formula. A comparison is shown in Table I.
The other component contributing to the ratio of two
hyperfine structure constants is the nuclear magnetic mo-
ment µ. Theoretical values for µ in the valence shell
model are based on the unpaired valence nucleon and are
given by Schmidt values
?
1 − (Zα)2. However, numerical many-
µ =
?1
2[gs+ (2j − 1)gl]for j = l +1
2
j
2(j+1)[−gs+ (2j + 3)gl] for j = l −1
2
(5)
The orbital gyromagnetic factors are gl= 1 for a va-
lence proton and gl= 0 for a valence neutron. The spin
gyromagnetic factors are gs(= gp) = 5.586 for protons
and gs(= gn) = −3.826 for neutrons. These g-factors
depend on mq/ΛQCD and previous work exploring this
dependence [17, 18] is summarized below. We then use
these results to consider the more realistic situation of µ
having both a valence nucleon contribution and a non-
valence nucleon contribution due to the spin-spin inter-
action.
II.VARIATION OF MAGNETIC MOMENT
WITH VARIATION IN QUARK MASS
A.Variation in µ using valence model magnetic
moment
As a preliminary to our results and as a comparison for
evaluating the effects of our calculations, we include the
results of work previously done in this area [17, 18]. This
work was essential to our results as the authors calcu-
lated the variation in the neutron and proton magnetic
moments (µn and µp) with the variation in mq/ΛQCD
using chiral perturbation theory.
As mentionedabove, theg-factorsdepend on
mq/ΛQCD. The light quark mass mq= (mu+ md)/2 ≈
5MeV and in the chiral limit mu = md = 0, the nu-
cleon magnetic moment remains finite. Thus one might
assume that corrections to the spin g-factors gpand gn
are small. However, the quark mass contribution is en-
hanced by π-meson loop corrections to the nuclear mag-
netic moments, which are proportional to π-meson mass
mπ∼?mqΛQCD. Since mπ = 140 MeV, the contribu-
Full details of these calculations are given in Ref
[17, 18]. They give the following results, which relate
variations in µn and µp with variations in light and
strange quark masses (mqand ms):
tion can be significant.
δµp
µp
δµp
µp
δµn
µn
δµn
µn
= −0.087δmq
mq
(6)
= −0.013δms
ms
(7)
= −0.118δmq
mq
(8)
= +0.0013δms
ms
(9)
Using these relations and the valence model approxi-
mations for µ, we can obtain expressions of the form
δµ
µ
= κqδmq
mq
+ κsδms
ms
(10)
Hence for nuclei with even Z and a valence neutron
δµ
µ
=δgn
gn
= −0.118δmq
mq
+ 0.0013δms
ms
.
For valence protons, the orbital gyromagnetic factor gl
also has an effect. Thus for133Cs with its valence proton
and j = l −1
2,
δµ
µ
= 0.110δmq
mq
+ 0.016δms
ms
while for87Rb with its valence proton but j = l +1
2,
δµ
µ
= −0.64δmq
mq
− 0.010δms
ms
These results can be presented using the ratio of the
hyperfine constant A to the atomic unit of energy E =
mee4/?2by defining the parameter V through the rela-
tion
δV
V
=δ(A/E)
A/E
. (11)
The values for κq and κs in the results for δµ/µ can
then be combined with the corresponding values of Krel
in Table I to give results of the form:
V (M) = α2+Krel
?
mq
ΛQCD
?κq?
ms
ΛQCD
?κsme
mp
(12)

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TABLE I: Variational factor Kreland κ values for various atoms obtained using simple valence shell model (method A) as used
in equation (12). The first row is given by equation (4). The second row presents the results of the more accurate many-body
calculations (see ref. [16]). The numerical results marked by∗are obtained by an extrapolation from other atoms.
Atom
1H
2H
3He
87Rb
111Cd
129Xe
133Cs
171Yb
199Hg
Krel (analytical)000 0.290.530.71 0.741.422.18
Krel (numerical)000 0.340.6∗
0.8∗
0.831.5∗
2.28
κq
-0.087-0.020 -0.118-0.064 -0.118 -0.1180.110-0.118 -0.118
κs
-0.013-0.044 0.0013 -0.0100.00130.0013 0.0160.0013 0.0013
The calculated values of Kreland κ to be used in the
expression for V(M) for various atoms are summarized in
Table I. The factor me/mpwill cancel out when a ratio
of hyperfine transitions is used. It will, however, survive
in a comparison with optical and molecular transitions.
B. Variation in µ incorporating the effect of
non-valence nucleons by using the experimental
magnetic moment
The results of the previous section were used to cal-
culate the variation of µ with mq/ΛQCD based on the
single particle approximation for µ (one valence nucleon)
within the shell model. That is, it was assumed that
the dimensionless nuclear magnetic moment is given by
µ = gp?sz?o+?lz?0for a valence proton, and µ = gn?sz?o
for a valence neutron. Here, gpand gnare the spin gy-
romagnetic factors for free protons and neutrons respec-
tively, ?lz?0= jz−?sz?o, and ?sz?ois the spin expectation
value of the single valence nucleon in shell model:
?sz?o=
?1
2
−
for j = l +1
2
j
2(j+1)for j = l −1
2
(13)
However, it is well known that this theoretical value
is only an estimate of µ and the magnetic moment of
the valence nucleon tends to be offset by a contribution
from the core nucleons. An empirical rule is that the spin
contribution of a valence nucleon should be reduced by
40% to obtain a reasonable value for the nuclear mag-
netic moment. This reduction may be explained by the
contribution of core nucleons, which should be negative
since proton and neutron magnetic moments are large
and have opposite signs.
For example, a valence proton polarizes, by the spin-
spin interaction, core neutrons and these core neutrons
give a negative contribution to the nuclear magnetic mo-
ment (polarization of the core protons by the valence
proton is not important). We can estimate this offset by
considering contributions to µ from the valence and core
nucleons. This means we have both neutron and proton
spin contributions to µ:
µ = gn?szn? + gp?szp? + ?lzp? (14)
We neglected here a small contribution of the exchange
currents into the magnetic moment.
We want to evaluate the corrections to the valence
model results using the very accurate experimental val-
ues of nuclear magnetic moments. Since there are three
unknown parameters (?szn?,?szp?,?lzp?) and only one ex-
perimental value (the total magnetic moment) available,
to perform an estimate we need to make approximations.
However, as we will show below, the result is not sen-
sitive to particular approximations if we can reproduce
the experimental magnetic moment exactly. Indeed, us-
ing the equations above, we can present the variation of
the magnetic moment in the following form
δµ−δµo= [0.45(?szn?−?szn?o)−0.56(?szp?−?szp?o)]δmq
mq
(15)
Here δµois the valence model value. For brevity we here
assume that
mq
=
ms
(the coefficient before
small anyway)).
Let us start with the case of a valence proton. The sim-
plest assumption is that the spin-spin interaction trans-
fers part of the proton spin to the core neutron spin, i.e.
(?szn? − ?szn?o) = −(?szp? − ?szp?o) and ?lzp? = ?lzp?o.
Then we can solve the equation for the magnetic moment
and obtain the deviation from the valence model value
δmq
δms
δms
ms
is
δµ − δµo= −0.11(µ − µo)δmq
mq
(16)
To test the stability of the result, we can try different
“extreme” assumptions. For example, if the angular mo-
mentum exchange occurs exclusively between the pro-
ton spin and proton orbital angular momentum, then
?lzp? − ?lzp?o= −(?szp? − ?szp?o), ?szn? = ?szn?o. In
this case
δµ − δµo= −0.12(µ − µo)δmq
mq
(17)
Finally, we can try an unreasonable assumption that the
exchange happens between the proton spin and neutron
orbital angular momentum: ?lzn? − ?lzn?o= −(?szp? −
?szp?o), ?szn? = ?szn?o, ?lzp? = ?lzp?o. Then
δµ − δµo= −0.10(µ − µo)δmq
mq
(18)

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We see that the results are very stable, the difference
in the correction to the valence model is about 10 %
only. The results for a valence neutron are similar. The
coefficients are -0.11 for (?szn? − ?szn?o) = −(?szp? −
?szp?o), -0.12 for ?lzn? − ?lzn?o= −(?szn? − ?szn?o), and
-0.09 for ?lzp? − ?lzp?o= −(?szn? − ?szn?o).
To present the final results, we will use a physical ap-
proximation which gives results somewhere in between
the “extreme” assumptions. The nuclear magnetic mo-
ments are reproduced with a reasonable accuracy by the
RPA calculations. In the RPA approximation there are
two separate conservation laws for the total proton jp
and total neutron jn angular momenta (see e.g. [19]).
We also assume that total orbital angular momentum
?lzn? + ?lzp? and total spin ?szn? + ?szp? are conserved
(this assumption corresponds to neglect of the spin-orbit
interaction). We repeat again that we only need these
approximations to obtain specific numbers which are in
between “extreme” model values. Then we can write
?sz?o= ?szp? + ?szn?
?jzp? = ?lzp? + ?szp?
(19)
(20)
where ?jzp? = I for a valence proton and ?jzp? = 0 for a
valence neutron. Using equations (20) and (19) to elimi-
nate ?lzp? and ?szp? in equation (14) we get
µ = gn?szn? + (gp− 1)?szp? + ?jzp? (21)
?szn? =µ − ?jzp? − (gp− 1)?sz?o
gn+ 1 − gp
(22)
?szp? = ?sz?o− ?szn? (23)
We thus have taken into account both the proton and
neutron contributions to the nuclear magnetic moment
and can more accurately estimate how a variation in
quark mass relates to a variation in µ. From equation
(14) we see immediately that
δµ = δgn?szn? + δgp?szp? (24)
and thus
δµ
µ
=δgn
gn
Kn+δgp
gp
Kp
(25)
where
Kn=
?gn?szn?
µ
?
andKp=
?gp?szp?
µ
?
From the definition of the g-factor for free protons and
free neutrons, we know δgn/gn= δµn/µnand δgp/gp=
δµp/µp. We can now use equations (6 - 9) to explicitly
relate the variation in µ to the variation in quark masses.
Thus
δµ
µ
= κqδmq
mq
+ κsδms
ms
(26)
where clearly
κq= −0.118Kn− 0.087Kp
κs= 0.0013Kn− 0.013Kp
We are now in a position to evaluate the coefficients
in specific cases. For133Cs, Iπ= 7/2+and µ = 2.5820
and it has a valence proton. Thus ?jzp? =
and ?sz?o= −7
18. Therefore equations (22) and (23)
immediately give us
7
2= l −1
2
?szn? = −0.103
?szp? = −0.286
and thus
δµ
µ
= 0.152δgn
gn
− 0.619δgp
gp
giving
δµ
µ
= 0.0358δmq
mq
+ 0.00824δms
ms
.
The dependence on the strange quark mass is relatively
weak and it is convenient to assume that the relative vari-
ation of the strange quark mass is the same as the relative
variation in the light quark masses (this assumption is
motivated by the Higgs mechanism of mass generation).
We restore the explicit notation for the strong-coupling
constant and conclude
δµ
µ
= 0.0441δ(mq/ΛQCD)
(mq/ΛQCD)
for
133Cs. (27)
Values for87Rb,199Hg,171Yb,111Cd and129Xe can
be similarly calculated and are presented in Table II
For2H and3He, the magnetic moments are pretty close
to naive values, therefore we have not tried to improve
the results.
These results can be summarized in Table II. A com-
parison with the earlier results (using the valence model
method) for the total variational relation (κq+ κs) is
shown later in Table III. We see that δµ/µ in the nuclei
with a valence proton is very sensitive to the core polar-
ization effects. However, there is no such sensitivity in
the nuclei with a valence neutron. To explain this conclu-
sion one should note that a neutron does not give an or-
bital contribution to the nuclear magnetic moment. The
orbital contribution of core protons is relatively small.
As a result, the core polarization effect changes δµ and µ
in a similar way, i.e. it practically does not change their
ratio. In the nuclei with a valence neutron
δµ
µ
= −0.117?szn? − 1.25?szp?
?szn? − 1.20?szp?
δmq
mq
(28)

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TABLE II: Variation in µ incorporating the effect of non-valence nucleons in various atoms. ?sz?ois the spin expectation value,
?szn? and ?szp? are (either, depending on nucleus) the valence and non-valance nucleon contributions to this spin, Kn and Kp
are defined in equation 25, and κq and κs are defined in equation 26.
Atom
1H
2H
3He
87Rb
111Cd
129Xe
133Cs
171Yb
199Hg
?sz?o
01 0.5 0.50.5 0.5-0.389 -0.167-0.167
?szn?0 0.50.5 0.124 0.343 0.365-0.103 -0.150-0.151
?szp? 0.50.50 0.376 0.157 0.135 -0.286-0.017 -0.016
Kn
––– -0.1722.21 1.800.1521.16 1.14
Kp
––– 0.764-1.47 -0.969 -0.619 -0.194-0.173
κq
-0.087 -0.020 -0.118 -0.046-0.133 -0.1280.036 -0.120-0.120
κs
-0.013 -0.044 0.0013-0.0100.022 0.0150.0080.0040.004
κq+ κs
-0.100 -0.064-0.117 -0.056 -0.111 -0.1130.044 -0.116-0.116
C. Effect of variation of the the spin-spin
interaction
In previous subsection we have not taken into account
that ?szp? and ?szn? may depend on the quark mass.
However, this dependence appears since the spin-spin in-
teraction depends on the quark masses. Below we want
to perform a rough estimate of this effect using the one-
boson-exchange model of the strong interaction.
Consider, for example, a nucleus with a valence proton.
As above we assume that the total spin of nucleons is
conserved and so
?sz?o= ?szp? + ?szn? (29)
where ?sz?ois again the valence model value. It is con-
venient to use the following notations:
?szn? = b?sz?o
?szp? = (1 − b)?sz?o
(30)
(31)
where b is a coefficient determined by the spin-spin in-
teraction. We need to determine the dependence of b
on mq/ΛQCD to complete our calculations. It can be
estimated using perturbation theory as
b ∼
??o|Vss|k?
Eo− Ek
?2
(32)
where
Vss= V (|r1− r2|)S1· S2
and Eo− Ekis the spin-orbit splitting (see e.g.[19]). As
mq→ 0, Eo−Ekremains finite and so it can only have a
weak dependence on mq/ΛQCD. The major dependence
comes from the π-meson mass which vanishes in the chi-
ral limit mq→ 0. According to review [20] the π-meson
exchange gives about 1/3 of the spin-spin interaction.
The other most significant contribution is given by the
ρ-meson exchange. We neglect other meson contributions
and assume that the remaining 2/3 of the spin-spin in-
teraction is given by the ρ-meson. The result is not very
sensitive to this assumption since the ρ−meson and other
vector mesons have approximately the same and rather
weak sensitivity to a variation in mq. According to Ref.
[21]
δmρ
mρ
= 0.021δmq
mq
whereas for the π−meson
mπ∼?mqΛQCD
so we have
δmπ
mπ
=1
2
δmq
mq
.
The dominating contribution is therefore given by π-
meson exchange.The exchange contribution of π±is
small due to the small overlap between ψp(r) and ψn(r)
and is not important. The main contribution is when a
neutron is excited through π0exchange into a spin-orbit
doublet, j = l+1
2. The strongest dependence
thus originates from the π0pion mass.
The momentum space representation of the nucleon-
nucleon interaction due to a π-meson is
2to j = l−1
Vπ(q) = g2
π(? τ1· ? τ2)(? σ1· q)(? σ2· q)
1
m2
π+ q2
where q is the momentum transfer q = p1− p2, ? τ is the
isotopic spin and ? σ is the Pauli spin matrix.
We separate this into tensor and scalar parts
(? σ1· q)(? σ2· q) =?(? σ1· q)(? σ2· q) −1
3? σ1· ? σ2q2?
+?1
3? σ1· ? σ2q2?
(33)