Gyromagnetic Factors and Atomic Clock Constraints on the Variation of Fundamental Constants
aa r X i v : . [ h e p - ph ] D ec G YROMAGNETIC F AC TORS AND A TOMIC C LOC K C ONSTRAINTS ON THE V AR IATION OF F UNDAMENTAL C ONSTANTS F ENG L UO ∗ , K EITH
A. O
LIVE , , † J EAN -P HILIPPE U ZAN , , ‡ School of Physics and Astronomy, University of Minnesota, Minneapolis, MN 55455, USA William I. Fine Theoretical Physics Institute, University of Minnesota, Minneapolis, MN 55455, USA Institut d’Astrophysique de Paris, UMR-7095 du CNRS, Universit´e Paris VI Pierre et Marie Curie98 bis bd Arago, 75014 Paris, France Astrophysics, Cosmology and Gravitation CentreDepartment of Mathematics and Applied Mathematics, University of Cape TownRondebosch 7701, South Africa National Institute for Theoretical Physics (NITheP), Stellenbosch 7600, South Africa
November 5, 2018
UMN–TH–3006/11, FTPI–MINN–11/16
Abstract
We consider the effect of the coupled variations of fundamental constants on the nucleon mag-netic moment. The nucleon g -factor enters into the interpretation of the measurements of variationsin the fine-structure constant, α , in both the laboratory (through atomic clock measurements) and inastrophysical systems (e.g. through measurements of the 21 cm transitions). A null result can betranslated into a limit on the variation of a set of fundamental constants, that is usually reduced to α .However, in specific models, particularly unification models, changes in α are always accompaniedby corresponding changes in other fundamental quantities such as the QCD scale, Λ QCD . This worktracks the changes in the nucleon g -factors induced from changes in Λ QCD and the light quark masses.In principle, these coupled variations can improve the bounds on the variation of α by an order ofmagnitude from existing atomic clock and astrophysical measurements. Unfortunately, the calcula-tion of the dependence of g -factors on fundamental parameters is notoriously model-dependent. Any definitive measurement of a temporal or spatial variation in a fundamental constant, such as thefine-structure constant α , would signal physics beyond the standard model, and in particular a violationof the equivalence principle which is one of the foundations of general relativity. In many cases, such anobservation would indicate the existence of a new light (usually scalar) degree of freedom [1]. Indeed,there has been considerable excitement during the last decade over the possible time variations in α fromobservations of quasar absorption systems [2, 3, 4, 5, 6, 7].In effectively all unification models of non-gravitational interactions, and certainly in models inwhich one imposes gauge coupling unification at some high energy scale, a variation in α is invari-ably accompanied by variations in other gauge couplings [8, 9]. In particular, variations in the strong ∗ E–mail: [email protected] † E–mail: [email protected] ‡ E–mail: [email protected] α s , will induce variations in the QCD scale, Λ QCD , as can be seen from the low energyexpression for Λ QCD when mass thresholds are included Λ QCD = µ (cid:18) m c m b m t µ (cid:19) exp (cid:20) − π α s ( µ ) (cid:21) , (1)for a renormalization scale µ > m t up to the unification scale [8, 9, 10], where m c , b , t are the masses ofthe charm, bottom, and top quarks. Because fermion masses are proportional to hv where h is a Yukawacoupling and v is the Higgs vacuum expectation value (vev), variations in Yukawa couplings will alsoaffect variations in Λ QCD so that ∆Λ QCD Λ QCD = R ∆ αα + 227 (cid:18) vv + ∆ h c h c + ∆ h b h b + ∆ h t h t (cid:19) . (2)Typical values for R are of order 30 in many grand unified theories, but there is considerable model-dependence in this coefficient [11].Furthermore, in theories in which the electroweak scale is derived by dimensional transmutation,changes in the Yukawa couplings (particularly the top Yukawa) lead to exponentially large changes inthe Higgs vev. In such theories, the Higgs expectation value is related to the Planck mass, M P , by [9] v ∼ M P exp (cid:18) − πcα t (cid:19) , (3)where c is a constant of order 1, and α t = h / π . For c ∼ h t ∼ , ∆ vv ∼ S ∆ h t h t , (4)with S ∼ , though there is considerable model-dependence in this value as well. For example, insupersymmetric models, S can be related to the sensitivity of the Z gauge boson mass to the top Yukawa,and may take values anywhere from about 80 to 500 [12]. This dependence gets translated into a variationin all low energy particle masses [13].In addition, in many string theories, all gauge and Yukawa couplings are determined by the expecta-tion value of a dilaton and we might expect [9] ∆ hh = 12 ∆ αα , (5)assuming that all Yukawa couplings vary similarly, so that they all reduce to h . Therefore, once we allow α to vary, virtually all masses and couplings are expected to vary as well, typically much more stronglythan the variation induced by the Coulomb interaction alone.Irrespective of the purported observations of a time variation in α , many experiments and analyseshave led to limits on possible variations [14, 15]. Furthermore, the use of coupled variations has led tosignificantly improved constraints in a wide range of environments ranging from big bang nucleosyn-thesis [9, 16, 17, 18, 19, 20, 21], the Oklo reactor [22, 24], meteoritic data [22, 23, 24], the microwavebackground [25, 20] and stellar evolution [26].This article explores the possibility that the strongest existing limits on the fine-structure constant,namely those derived from atomic clock measurements, can also be enhanced by considering such cou-pled variations. We expect the effect of induced variations in Λ QCD and the light quark masses to enterthrough the nucleon magnetic moment. Existing experimental limits on α from atomic clock experi-ments assume constant µ p , n . Indeed, limits on the variations of quark masses in units of the QCD scale,i.e. m q / Λ QCD , from atomic clock measurements have been derived [27, 28]. Given a (model-dependent)calculation of the nucleon magnetic moment (or equivalently its g -factor), we can derive sharper bounds ATOMICCLOCKCONSTRAINTS 3on the variation of α from existing data. Unfortunately, because of the model-dependence, we find thatwhile the limits are generally improved (by as much as an order of magnitude), there is considerableuncertainty in the precise numerical limit. As a corollary, we apply our results to astrophysical measure-ments such as those which rely on the 21 cm line which also depends on µ p , n .The article is organized as follows: In section 2, we outline the procedure of obtaining limits on α from atomic clock experiments. In particular, we examine the detailed dependence on the nuclear g -factors which will be subject to variation. In section 3, we derive the dependence of the nucleon mag-netic moment on Λ QCD and the light quark masses. Because there is no unique (or rigorous) method forcalculating baryon magnetic moments, we consider several different approaches. The most straightfor-ward employs the constituent quark model. Surprisingly, this model is quite effective in matching theobserved baryon magnetic moments. Even within this broad approach, our result will depend on thecalculation of the nucleon mass, as well as the calculation of the constituent quark mass; each carrying asignificant degree of uncertainty. We also consider an approach based on chiral perturbation theory, anda method based partially on lattice results. In section 4, we apply these results to atomic clock measure-ments and derive “improved” limits on the variation of α . Finally, in section 5, we extend these resultsto measurements involving the 21 cm line and summarize our results. The comparison of atomic clocks provides a constraint on the relative shift of the frequencies of the twoclocks as a function of time, on time scales of the order of a couple of years. This observation (or lackthereof) can be translated into a constraint on the time variation of a fundamental constant. Using QED,the frequency of the atomic transitions can be expressed (see e.g. [29]) in terms of the fine structureconstant α , the electron-to-proton mass ratio, µ ≡ m e /m p and the gyromagnetic factor g i = 2 µ i /µ N ,where µ i is the nuclear magnetic moment, and µ N = e m p is the nuclear magneton.The hyperfine frequency in a given electronic state of an alkali-like atom is given by ν hfs ≃ R ∞ c × A hfs × g i × α × µ × F hfs ( α ) , (6)where R ∞ the Rydberg constant, A hfs is a numerical factor depending on the atomic species and F hfs ( α ) is a factor taking into account relativistic corrections (including the Casimir contribution) which dependson the atom. We omitted the effect of the finite nuclear radius on hyperfine frequency in Eq. (6), sincethe effect of varying the nuclear radius is shown to be smaller [30, 31] than the effects of varying otherparameters which we consider in this work. Similarly, the frequency of an electronic transition is well-approximated by ν elec ≃ R ∞ c × A elec × F elec ( Z, α ) , (7)where, as above, A elec is a numerical factor depending on each particular atom and F elec is the functionaccounting for relativistic effects, spin-orbit couplings and many-body effects. Even though an electronictransition should also include a contribution from the hyperfine interaction, it is generally only a smallfraction of the transition energy and thus should not carry any significant sensitivity to a variation of thefundamental constants.Relativistic corrections are important [32] and are computed by means of relativistic N -body cal-culations [33, 34, 35, 36]. These can be characterized by introducing the sensitivity of the relativisticfactors to a variation of α defined by κ α = δ ln Fδ ln α . (8)The values of these coefficients for the transitions that we shall consider below are summarized in Table 1. ATOMICCLOCKCONSTRAINTS 4Table 1: Sensitivity of various transitions on a variation of the fine structure constant. From Refs. [33,34, 35, 36]. Atom Transition Sensitivity κ α H s − s Rb hf 0.34 Cs S / ( F = 2) − ( F = 3) Yb + 2 S / − D / Hg + 2 S / − D / –3.2 Sr S − P Al + 1 S − P Over the past several years, many comparisons of atomic clocks have been performed. We consider onlythe latest result of each type of comparison for our analysis. • Rubidium : The comparison of the hyperfine frequencies of rubidium and caesium in their elec-tronic ground state between 1998 and 2004 [29] yields dd t ln (cid:18) ν Cs ν Rb (cid:19) = (0 . ± . × − yr − . (9)From Eq. (6), and using the values of the sensitivities κ α , we deduce that this comparison con-strains ν Cs ν Rb ∝ g Cs g Rb α . . (10) • Atomic hydrogen : The s − s transition in atomic hydrogen was compared to the ground statehyperfine splitting of caesium [37] in 1999 and 2003, setting an upper limit on the variation of ν H of ( − ± Hz within 44 months. This can be translated in a relative drift dd t ln (cid:18) ν Cs ν H (cid:19) = (32 ± × − yr − . (11)Since the relativistic correction for the atomic hydrogen transition nearly vanishes, we have ν H ∼ R ∞ so that ν Cs ν H ∝ g Cs µ α . . (12) • Mercury : The Hg + 2 S / − D / optical transition has a high sensitivity to α (see Table 1) sothat it is well suited to test its variation. The frequency of the Hg + electric quadrupole transitionat 282 nm was thus compared to the ground state hyperfine transition of caesium first during a twoyear period [38] and then over a 6 year period [39] to get dd t ln (cid:18) ν Cs ν Hg (cid:19) = ( − . ± . × − yr − . (13)While ν Cs is still given by Eq. (6), ν Hg is given by Eq. (7). Using the sensitivities of Table 1, weconclude that this comparison test the stability of ν Cs ν Hg ∝ g Cs µ α . . (14) ATOMICCLOCKCONSTRAINTS 5 • Ytterbium : The S / − D / electric quadrupole transition at 688 THz of Yb + was comparedto the ground state hyperfine transition of caesium. The constraint of [40] was updated, after acomparison over a six year period, which leads to [41] dd t ln (cid:18) ν Cs ν Yb (cid:19) = (0 . ± . × − yr − . (15)This tests the stability of ν Cs ν Yb ∝ g Cs µ α . . (16) • Strontium : The comparison of the S − P transition in neutral Sr with a caesium clock wasperformed in three independent laboratories. The combination of these three experiments [42]leads to the constraint dd t ln (cid:18) ν Cs ν Sr (cid:19) = (1 . ± . × − yr − . (17)Similarly, this tests the stability of ν Cs ν Sr ∝ g Cs µ α . . (18) • Atomic dyprosium : The electric dipole (E1) transition between two nearly degenerate opposite-parity states in atomic dyprosium should be highly sensitive to the variation of α [34, 35, 43, 44].The frequencies of two isotopes of dyprosium were monitored over a 8 months period [45] showingthat the frequency variation of the 3.1-MHz transition in Dy and the 235-MHz transition in
Dy are 9.0 ± ± ˙ αα = ( − . ± . × − yr − , (19)at 1 σ level, without any assumptions on the constancy of other fundamental constants. • Aluminium and mercury single-ion optical clocks : The comparison of the S − P transition in Al + and S / − D / in Hg + over a year allowed one to set the constraint [46] dd t ln (cid:18) ν Al ν Hg (cid:19) = ( − . ± . × − yr − . (20)Proceeding as previously, this tests the stability of ν Al ν Hg ∝ α . , (21)which, using Eq. (21) directly sets the constraint ˙ αα = ( − . ± . × − yr − , (22)since it depends only on α .Experiments with diatomic molecules, as first pointed out by Thomson [47] provide a test of thevariation of µ . The energy difference between two adjacent rotational levels in a diatomic molecule isinversely proportional to M r − , r being the bond length and M the reduced mass, and the vibrationaltransition of the same molecule has, in first approximation, a √ M dependence. For molecular hydrogen M = m p / so that the comparison of an observed vibro-rotational spectrum with a laboratory spectrum ATOMICCLOCKCONSTRAINTS 6gives an information on the variation of m p and m n . Comparing pure rotational transitions with elec-tronic transitions gives a measurement of µ . It follows that the frequency of vibro-rotation transitions is,in the Born-Oppenheimer approximation, of the form ν ≃ E I ( c elec + c vib √ µ + c rot µ ) , (23)where c elec , c vib and c rot are some numerical coefficients.The comparison of the vibro-rotational transition in the molecule SF6 was compared to a caesiumclock over a two-year period, leading to the constraint [48] dd t ln (cid:18) ν Cs ν SF6 (cid:19) = ( − . ± . ± . × − yr − , (24)where the second error takes into account uncontrolled systematics. Now, using Table 1 again and Eq. (6)for Cs, we deduce that for a vibrational transition, ν Cs ν SF6 ∝ g Cs √ µ α . . (25) g -factors All the constraints involve only 4 quantities, µ , α and the two gyromagnetic factors g Cs and g Rb . Itfollows that we need to relate the nuclear g -factors that appeared in the constraints of the previous sub-section, with the proton and neutron g -factors that will be calculated in Section 3.An approximate calculation of the nuclear magnetic moment is possible in the shell model and isrelatively simple for even-odd (or odd-even) nuclei where the nuclear magnetic moment is determinedby the unpaired nucleon. For a single nucleon, in a particular ( l, j ) state within the nucleus, we can write g = (cid:26) lg l + g sjj +1 [2( l + 1) g l − g s ] for (cid:26) j = l + j = l − (26)where g l = 1(0) and g s = g p ( g n ) for a valence proton (neutron).From the previous discussion, the only g -factors that are needed are those for Rb and
Cs. Forboth isotopes, we have an unpaired valence proton. For Rb, the ground state is in a p / state so that l = 1 and j = , while for Cs, the ground state is in a g / state corresponding to l = 4 and j = .Using Eq. (26), the nuclear g -factor can easily be expressed in terms of g p alone. Using g p = 5 . , wefind g = 7 . for Rb and g = 3 . for Cs, while the experimental values are g = 5 . for Rb and g = 5 . for Cs.The differences between the shell model predicted g -factors and the experimental values can beattributed to the effects of the polarization of the non-valence nucleons and spin-spin interaction [27, 31].Taking these effects into account, the refined formula relevant for our discussion of Rb and
Cs is g = 2 [ g n b h s z i o + ( g p − − b ) h s z i o + j ] , (27)where g n = − . , h s z i o is the spin expectation value of the single valence proton in the shell modeland it is one half of the coefficient of g s in Eq. (26), and b is determined by the spin-spin interaction andit appears in the expressions for the spin expectation value of the valence proton h s z p i = (1 − b ) h s z i o and non-valence neutrons h s z n i = b h s z i o . Following the preferred method in [27, 31], it is found h s z n i = g − j − ( g p − h s z i o g n + 1 − g p , (28)and h s z p i = h s z i o − h s z n i . (29) ATOMICCLOCKCONSTRAINTS 7Therefore, the variation of the g -factor can be written as δgg = δg p g p g p h s z p i g + δg n g n g n h s z n i g + δbb g n − g p + 1) h s z n i g . (30)From Eq. (28), (29) and (30), we find, by using the experimental g -factors, δg Rb g Rb = 0 . δg p g p − . δg n g n − . δbb , (31) δg Cs g Cs = − . δg p g p + 0 . δg n g n + 0 . δbb . (32) Given the discussion in the two previous subsections, and in particular Eqs. (31) and (32), the atomicclock experiments give constraints on the set { g p , g n , b, µ, α } and thus variations in the relative frequencyshift ν AB = ν A /ν B are given by δν AB ν AB = λ g p δg p g p + λ g n δg n g n + λ b δbb + λ µ δµµ + λ α δαα , (33)or equivalently ˙ ν AB ν AB = λ g p ˙ g p g p + λ g n ˙ g n g n + λ b ˙ bb + λ µ ˙ µµ + λ α ˙ αα , (34)with the coefficients { λ g p , λ g n , λ b , λ µ , λ α } summarized in Table 2.For the sake of comparison, the shell model gives δg Rb g Rb ≃ . δg p g p (35)and δg Cs g Cs ≃ − . δg p g p . (36)The main difference arises from the dependence in g n and b but the order of magnitude is similar.Table 2: Summary of the constraints of the atomic clock experiments and values of the coefficients { λ g p , λ g n , λ b , λ µ , λ α } entering the decomposition (34).Clocks ν AB λ g p λ g n λ b λ µ λ α ˙ ν AB /ν AB (yr − )Cs - Rb g Cs g Rb α . − .
383 0 .
325 0 . .
49 (0 . ± . × − Cs - H g Cs µ α . − .
619 0 .
152 0 . .
83 (32 ± × − Cs - Hg + g Cs µ α . − .
619 0 .
152 0 . .
03 ( − . ± . × − Cs - Yb + g Cs µ α . − .
619 0 .
152 0 . .
93 (0 . ± . × − Cs - Sr g Cs µ α . − .
619 0 .
152 0 .
335 1 2 .
77 (1 . ± . × − Cs - SF g Cs √ µα . − .
619 0 .
152 0 . ( − . ± . ± . × − Dy α ( − . ± . × − Hg + - Al + α − . − .
208 (5 . ± . × − NUCLEONMAGNETICMOMENTS,CURRENTQUARKMASSESAND Λ QCD Λ QCD
In this section, we will review several approaches in the literature in calculating the nucleon magnetic mo-ments, including the non-relativistic constituent quark model (NQM), chiral perturbation theory ( χ PT),and a method combining the results of χ PT and lattice QCD. We will try to extract the dependence of thenucleon magnetic moments on the current quark masses and Λ QCD from the expressions given by eachof these approaches.
The NQM, which approximates hadrons as bound states of their constituent quarks gives a good approx-imation to the measured baryon magnetic moments [49]. In this model, the baryon magnetic momentsare expressed in terms of the Dirac magnetic moments of their constituent quarks, with the coefficientsgiven by the baryon spin/flavor wave functions. For the proton and neutron, the magnetic moments are µ p = 43 µ u − µ d and µ n = 43 µ d − µ u , (37)where µ u = e M u and µ d = − e M d . Here, M u and M d are the constituent u and d quark masses,respectively, with their values around a third of the nucleon mass, to be compared with the much smaller u and d current quark masses, m u and m d , which are several MeV. For the three light flavors ( u , d and s ), the main part of their constituent quark masses have a strong interaction origin, with the dynamics ofthe virtual gluons and quark-antiquark sea being responsible for the large masses [50], while the currentquark masses which contribute only a small portion of their corresponding constituent quark masses areof pure electroweak origin.From Eq. (37), the nucleon magnetic moment in units of the nuclear magneton µ N = e m p , that is,the g -factor of the nucleon, can be written as g NQM = 2 (cid:18) c u m p M u + c d m p M d (cid:19) , (38)where c u = 8 / and c d = 1 / for the proton, and c u = − / and c d = − / for the neutron. Inthe study of hadron properties, the constituent quark masses are usually taken as fitting parameters,with M u = M d often assumed [49], since isospin is a good approximate symmetry. We will assumethis relation in the following calculations to simplify the algebra, but we emphasize that δM u may notnecessarily be equal to δM d . By differentiating Eq. (38), we obtain a general expression for the variationof the g -factor δg NQM g NQM = δm p m p − (cid:18) c u c u + c d δM u M u + c d c u + c d δM d M d (cid:19) . (39)The proton mass, m p , and M u , d are functions of the fundamental constants, and they can be formallywritten as m p = m p ( v , v , · · · , v n ) and M u , d = M u , d ( v , v , · · · , v n ) , where the v i ’s are fundamentalconstants including m u , m d , m s , Λ QCD , etc.. Therefore, Eq. (39) becomes δg NQM g NQM = n X i =1 δv i v i (cid:20) v i m p ∂m p ∂v i − (cid:18) c u c u + c d v i M u ∂M u ∂v i + c d c u + c d v i M d ∂M d ∂v i (cid:19)(cid:21) ≡ n X i =1 δv i v i κ i . (40)This is our key equation in studying the dependence of the g -factors on fundamental constants in theNQM approach, and the problem amounts to finding the expressions for m p ( v , · · · , v n ) and M u , d ( v , · · · , v n ) . NUCLEONMAGNETICMOMENTS,CURRENTQUARKMASSESAND Λ QCD Λ QCD dependence of m p To get the coefficients of δm p m p , that is, the first term in the square bracket of Eq. (40), we follow theprocedure of [51, 52, 53], by defining B q ( q = u, d, s ) and the π -nucleon sigma term, Σ π N , in terms ofproton matrix elements, m q B q ≡ h p | m q ¯ qq | p i = m q ∂m p ∂m q , (41) Σ π N ≡ h p | ˆ m (¯ uu + ¯ dd ) | p i = ˆ m ∂m p ∂ ˆ m , (42)where ˆ m ≡ ( m u + m d ) . The latter equalities of the above two equations come from the Hellmann-Feynman theorem [54] as noted by Gasser [55].By using the strangeness fraction of the proton, y ≡ B s B d + B u = 1 − σ Σ π N , (43)where σ is the shift in the nucleon mass due to nonzero quark masses, and a relation from the energy-momentum tensor trace anomaly [56] for the baryon-octet members [57, 51, 58], z ≡ B u − B s B d − B s = m Ξ + m Ξ − − m p − m n m Σ + + m Σ − − m p − m n ≈ . , (44)we can derive from Eqs. (41) and (42) the current quark masses dependence of m p , denoted as f T q ’s, as f T u ≡ m u B u m p = 2Σ π N m p (cid:16) m d m u (cid:17) (cid:16) B d B u (cid:17) ,f T d ≡ m d B d m p = 2Σ π N m p (cid:16) m u m d (cid:17) (cid:16) B u B d (cid:17) , (45) f T s ≡ m s B s m p = (cid:16) m s m d (cid:17) Σ π N ym p (cid:16) m u m d (cid:17) , where B d B u = 2 + y ( z − z − y ( z − . (46)Motivated by the trace anomaly expression for m p , m p = m u B u + m d B d + m s B s + gluon term , (47)we will write the remaining fundamental constants dependence of m p as f T g ≡ Λ QCD m p ∂m p ∂ Λ QCD = 1 − X q = u,d,s f T q , (48)which is the coefficient of δ Λ QCD / Λ QCD in δm p /m p . The argument behind Eq. (48) is the following:the gluon term has its origin in the strong interaction, and Λ QCD , which is approximately the scale atwhich the strong interaction running coupling constant diverges, is the only mass parameter of the stronginteraction in the chiral limit m u = m d = m s = 0 , and therefore in this limit all of the other finitemass scales of the strong interaction phenomena, including pion decay constant, the spontaneous chiral NUCLEONMAGNETICMOMENTS,CURRENTQUARKMASSESAND Λ QCD Λ QCD by some pure number of order one [59]. Note that theheavy quark ( c, b, t ) masses do not explicitly appear in Eq. (47) as discussed in [57]. Then as the onlyother variable besides the light current quark masses, we get Eq. (48) for the Λ QCD dependence in m p .We note that the f T q ’s and the f T g are also needed in the next section when we vary the electron-to-protonmass ratio, µ ≡ m e /m p .In calculating the f T q ’s and the f T g , we take the central values given in [60] for the current quarkmass ratios, m u m d = 0 . and m s m d = 18 . , the central value of the π -nucleon sigma term suggestedin [53], Σ π N = 64 MeV, and we take σ = 36 MeV [61, 62] and m p = 938 . MeV. The results are f T u = 0 . , f T d = 0 . , f T s = 0 . , f T g = 0 . . (49)In the isospin-symmetric limit such that m u = m d = ˆ m , which will be needed in subsections 3.2and 3.3, Eqs. (45) and (48) take simpler forms, f T ˆ m = Σ π N m p , f T s = m s ˆ m Σ π N y m p , f T g = 1 − f T ˆ m − f T s . (50)In calculating the values for this isospin-symmetric limit case, we take m s ˆ m = 25 [63], and the results are f T ˆ m = 0 . , f T s = 0 . , f T g = 0 . . (51) M u , d without an explicit quark sea To get the coefficients of δM u M u and δM d M d , we need to model the constituent quark masses. Intuitively, M u , d can be written as M q = m q + a q , int Λ QCD ( q = u, d ) , ( A ) (52)where a q , int ’s are pure dimensionless numbers. The argument behind this form is the following: if thestrong interaction were switched off, the constituent quark mass would be identical to its correspondingvalence current quark mass which is obtained from the electroweak symmetry breaking. On the otherhand, in the chiral limit, m u = m d = m s = 0 , the strong interaction is responsible for the entire con-stituent quark mass. The above intuitive expression for the constituent quark masses does not explicitly take into account the sea quark contribution, which if included will depend on the current quark masses,similar to the terms m q B q in the proton mass trace anomaly formula Eq. (47) [51]. However, one couldargue that the sea quark contribution is already included implicitly in the second term of Eq. (52) to-gether with the virtual gluons contribution, since the dynamics of the quark sea and virtual gluons aredetermined by strong interaction, which is characterized in the second term.From Eq. (52), we obtain the coefficients of δM u M u and δM d M d as m u M u ∂M u ∂m u = m u M u , m d M u ∂M u ∂m d = m s M u ∂M u ∂m s = 0 , Λ QCD M u ∂M u ∂ Λ QCD = 1 − m u M u ,m d M d ∂M d ∂m d = m d M d , m u M d ∂M d ∂m u = m s M d ∂M d ∂m s = 0 , Λ QCD M d ∂M d ∂ Λ QCD = 1 − m d M d . (53)In calculating the above coefficients, we will use m u m d = 0 . , the central value of m d = 9 . MeV in themodified minimal subtraction (MS) scheme at a renormalization scale of GeV [60], and we will choose M u = M d = 335 MeV. M u , d with an explicit quark sea – linear form A method explicitly taking into account the sea quark contribution can be traced back to the internalstructure of the constituent quarks [50]. Then, for m p (Λ QCD , m u , m d , m s ) a linear realization of this NUCLEONMAGNETICMOMENTS,CURRENTQUARKMASSESAND Λ QCD M u = a lin Λ QCD + b u , lin m u + b d , lin m d + b s , lin m s ,M d = a lin Λ QCD + b d , lin m u + b u , lin m d + b s , lin m s , ( B ) (54)where we have related the coefficients in M d with those in M u following [50]. The coefficients a lin and b q , lin ’s are pure numbers. Each of the four terms of M u , d can be obtained by inserting Eq. (54) into anexpression for the NQM based proton mass, m p , NQM , which we will discuss shortly (see e.g., Eq. (57),(59) or (61)). Then, applying the Hellmann-Feynman theorem ∂m p , NQM ∂m q = ∂m p , NQM ∂M u ∂M u ∂m q + ∂m p , NQM ∂M d ∂M d ∂m q = B q ( q = u, d, s ) . (55)An example of the application of the Hellmann-Feynman theorem within the NQM is given in [64].From Eqs. (54) and (55), the coefficients of δv i /v i ( v i = m u , d , s , Λ QCD ) of δM u /M u and δM d /M d canbe obtained as m u M u ∂M u ∂m u = k u m p f T u − k d m p f T d (cid:16) m u m d (cid:17) M u (cid:0) k − k (cid:1) ,m d M u ∂M u ∂m d = k u m p f T d − k d m p f T u (cid:16) m d m u (cid:17) M u (cid:0) k − k (cid:1) , (56) m s M u ∂M u ∂m s = m p f T s M u ( k u + k d ) , Λ QCD M u ∂M u ∂ Λ QCD = 1 − X q = u,d,s m q M u ∂M u ∂m q , where k u , d = ∂m p , NQM ∂M u , d . The v i M d ∂M d ∂v i ( v i = m u , d , s , Λ QCD ) are obtained from the corresponding v i M u ∂M u ∂v i by switching M u ↔ M d and k u ↔ k d .To get k u , d , we consider the following NQM based proton mass formulae as examples. To zerothorder, the proton mass is the sum of the masses of its two constituent u quarks and one constituent d quark m p = 2 M u + M d , (57)so that k u = 2 , k d = 1 . (58)We will use M u = M d = m p in Eq. (56) when Eq. (57) is taken as the NQM based proton massformula.Without some interaction between the constituent quarks, hadrons with the same constituent quarkcompositions would have a same mass, a phenomenon which is not observed in nature. To break themass degeneracy, a spin-spin hyperfine term is introduced [65], and the resulting proton mass is m p = 2 M u + M d + A ′ (cid:18) M − M u M d (cid:19) , (59)where A ′ is a constant usually determined to allow an optimal fit to the baryon octet and decupletmasses [49]. This spin-spin hyperfine term is commonly attributed to one-gluon exchange [65], or,in the chiral quark model [66], it is explained as the interaction between the constituent quarks mediatedby pseudoscalar mesons [67]. Although interpreted with relating to different degrees of freedom (gluonor pseudoscalar mesons) [58], this term nevertheless has a strong interaction origin, and therefore we NUCLEONMAGNETICMOMENTS,CURRENTQUARKMASSESAND Λ QCD A ′ as a hyp Λ QCD , with a hyp a pure dimensionless number. From this formula,we get k u = 2 + A ′ (cid:18) M d M − M (cid:19) , k d = 1 + A ′ M u M . (60)We will use M u = M d = 363 MeV and A ′ = (298 . MeV ) in Eq. (56) when Eq. (59) is taken as theNQM based proton mass formula. Note that we have tuned A ′ a bit compared to the value given in [49]to allow an exact fit to the proton mass.Eq. (59) can be further refined by adding to it the kinetic term of the constituent quarks and a con-stituent quark mass independent term M , which represents the contributions of the confinement potentialand the short-range color-electric interaction [68, 64] m p = 2 M u + M d + A ′′ (cid:18) M − M u M d (cid:19) + B ′ (cid:18) M u + 12 M d (cid:19) + M . (61)From the physical meaning of these two new terms, it may be reasonable to write the constants B ′ and M as a kin Λ QCD and a cce Λ QCD , respectively, since the internal dynamics of a baryon is dominated by thestrong interaction and the confinement is a strong interaction phenomenon. The constant A ′′ needs to bere-fit after introducing the two new terms, and we write it as A ′′ = a ′ hyp Λ QCD . The parameters a kin , a cce and a ′ hyp are pure dimensionless numbers. We find that k u , d from this formula are k u = 2 + A ′′ (cid:18) M d M − M (cid:19) − B ′ M , k d = 1 + A ′′ M u M − B ′ M . (62)We will use M u = M d = 335 MeV, A ′′ = (4 . MeV ) , B ′ = (175 . MeV ) and M = − . MeV in Eq. (56) when Eq. (61) is taken as the NQM based proton mass formula. Note thatwe have tuned M a bit compared to the value given in [64] to allow an exact fit to the proton mass. M u , d expressions with an explicit quark sea – NJL model As can be seen from Eq. (56), the explicit inclusion of the sea quark contribution in the linear formEq. (54) encodes the information of both the NQM based proton mass formula and the f T q ’s. However,different realizations from the NQM alone are also possible. Moreover, although the constituent quarkmasses are usually taken as fitting parameters in the study of hadron properties, it is certainly moreilluminating if some concrete physical origin of these quantities can be given and encoded in their massformulae. As suggested in [66, 69], the constituent quark masses are closely related to spontaneouschiral symmetry breaking. An example of the constituent quark mass formulae applying this idea isgiven by the three flavor Nambu-Jona-Lasinio (NJL) model [70, 71], where the constituent quark massesare obtained from a set of gap equations M u = m u − g s h ¯ uu i − g D (cid:10) ¯ dd (cid:11) h ¯ ss i ,M d = m d − g s (cid:10) ¯ dd (cid:11) − g D h ¯ uu i h ¯ ss i , (63) M s = m s − g s h ¯ ss i − g D h ¯ uu i (cid:10) ¯ dd (cid:11) , ( C ) where h ¯ uu i , (cid:10) ¯ dd (cid:11) and h ¯ ss i are the quark condensates which are the order parameters of the spontaneouschiral symmetry breaking, and they are calculated by one loop integral h ¯ uu i = − iN c Tr Z d p (2 π ) /p − M u + iǫ , NUCLEONMAGNETICMOMENTS,CURRENTQUARKMASSESAND Λ QCD (cid:10) ¯ dd (cid:11) ( h ¯ ss i ) is obtained by changing M u to M d ( M s ). The N c is the number of colors, and we take itto be the real-world value 3. This integration can be performed by introducing a three-momentum cutoff Λ , and the result is h ¯ uu i = − π M u " Λ q Λ + M − M ln Λ + p Λ + M M u ! . (64)The g s and g D in Eq. (63) are the coupling constants of the effective four-point and six-point interactionsof the quark fields in the NJL Lagrangian, and they are fixed, together with m s and the cutoff Λ , by themeson properties as explained in [64, 71]. We simply quote the result given in [64] m s = 135 . MeV , g s Λ = 3 . , g D Λ = − . , Λ = 631 . MeV , (65)which we will use for our calculation. The other parameters we need in order to solve Eq. (63) are the u and d current quark masses, which we take m u = m d = 5 . MeV following [64]. Note that the form ofEq. (63) requires m u = m d if we assume M u = M d . The cutoff Λ characterizes the spontaneous chiralsymmetry breaking scale, while the latter is related to Λ QCD , as we explained in the paragraph belowEq. (48). Therefore, we will write Λ = a c, NJL Λ QCD , g s = a s, NJL Λ − QCD and g D = a D, NJL Λ − QCD , where thecoefficients are pure dimensionless numbers. With these inputs, the constituent quark masses are solvedfrom Eq. (63), with the values M u = M d = 335 MeV and M s = 527 MeV, and we get m u M u ∂M u ∂m u = m d M d ∂M d ∂m d = 0 . , m d M u ∂M u ∂m d = m u M d ∂M d ∂m u = 0 . ,m s M u ∂M u ∂m s = m s M d ∂M d ∂m s = 0 . , Λ QCD M u ∂M u ∂ Λ QCD = Λ
QCD M d ∂M d ∂ Λ QCD = 0 . . (66) We can now calculate the dependence of the nucleon magnetic moments on m u , d , s and Λ QCD , by δg NQM g exp = g NQM g exp δg NQM g NQM , (67)where g exp is the measured value of the g -factor, which equals . for proton, and − . for neu-tron [63]. The first term in the square bracket of δg NQM g NQM (Eq. (40)) is given in section 3.1.1, whilethe second term in that square bracket can be obtained from section 3.1.2, 3.1.3 and 3.1.4 for each ofthe three different constituent quark mass models we have considered. The calculated coefficients of δv i v i ( v i = m u , d , s , Λ QCD ) of δg NQM g exp for proton and neutron are listed in Table 3, where the constituentquark mass formula used for each row is labeled as A, B or C, representing Eqs. (52), (54) or (63),respectively, while the , , or following the label B represents Eq. (57), (59) or (61), respectively.In all of the cases listed in Table 3, we use the f T q given in Eq. (49). Note that for case C, there is aslight inconsistency due to our choice of m u = m d , though this has only a minor numerical effect on theresulting κ ’s.The coefficients in Table 3 show a relatively strong dependence on the constituent quark mass modelsused. Most of the coefficients in A and C are closer and much larger compared to their correspondingvalues in B. While the first and the second terms in Eq. (40) are independent of each other for A and C,the same f T q ’s appear in both terms of Eq. (40) for B, as can be seen from Eq. (56) and thus these twoterms are largely canceled due to a relative sign. We can also see a relatively strong dependence of thecoefficients on the NQM based proton mass formulae when comparing the rows B1, B2 and B3. We have tuned the values of g s and g D relative to the values given in [64] to allow M u = M d = 335 MeV and M s =527 MeV as exact solutions of Eq. (63).
NUCLEONMAGNETICMOMENTS,CURRENTQUARKMASSESAND Λ QCD κ i of δv i v i ( v i = m u , d , s , Λ QCD ) in δg NQM g exp for the proton (left) and the neutron(right); see Eq. (40) for their definition. κ u κ d κ s κ QCD A .
013 0 .
036 0 . − . B1 − . . − . B2 . .
021 0 . − . B3 − . .
022 0 . − . C − . .
029 0 . − . κ u κ d κ s κ QCD A .
021 0 .
020 0 . − . B1 . − .
010 0 0 . B2 .
012 0 . . − . B3 . − . . − . C .
010 0 .
013 0 . − .
40 50 60 70 80 90 100 -0.50.00.5 S p N (MeV) d m u /m u dL QCD / L QCD k i d m d /m d d m s /m s
40 50 60 70 80 90 100 -0.15 -0.10 -0.050.000.050.100.15 S p N (MeV) d m u /m u dL QCD / L QCD k i d m d /m d d m s /m s Figure 1: The dependence of the coefficients in A for the proton on Σ π N (left) and of the coefficients inB3 (right).Furthermore, there is an uncertainty in the coefficients listed in Table 3 due to the uncertainty of the π -nucleon sigma term Σ π N . A discussion of the impact of the uncertainty of Σ π N on the interpretations ofexperimental searches for dark matter can be found in [53]. We plot the dependence of the coefficients inrow A for the proton on Σ π N in the left panel of Fig. 1. A similar plot of the coefficients for the proton inrow B3 is given in the right panel of Fig. 1. As can be seen from these plots, the coefficients of δm s /m s and δ Λ QCD / Λ QCD show a strong dependence on the value of Σ π N . Therefore it is important to pin downthe value of Σ π N if this quantity is used in the study of the current quark mass and Λ QCD dependence ofthe proton g -factor. The same conclusion applies for the neutron g -factor, for which the behavior of theplots are similar to that shown in Fig. 1.In addition to the relatively strong dependence of the κ i on the proton and constituent quark massformula as well as the value of Σ π N we have discussed above, some other comments for this NQM ap-proach in the study of the dependence of the nucleon g -factors on the fundamental constants are in order.Our assumption that the various parameters in the constituent quark mass formulae and the NQM basedproton mass formulae take power law forms for Λ QCD may be valid only in the chiral limit. Therefore,some current quark mass dependence may be lost and the Λ QCD dependence may not be very accu-rately determined from these formulae. To get a more accurate dependence, one may also wish to con-sider relativistic corrections [72] and/or corrections based on higher-dimension terms in the chiral quarkmodel [66] for Eq. (37), and then the dependence on the current quark masses and Λ QCD will changecorrespondingly. Finally, in the above analysis, we did not consider the electromagnetic contribution tothe proton mass or the constituent quark mass formulae, and thus we may have missed some dependenceon the fine structure constant in this approach. NUCLEONMAGNETICMOMENTS,CURRENTQUARKMASSESAND Λ QCD The second approach we consider is χ PT, which provides a systematic method of addressing the lowenergy properties of the hadrons [73, 74]. In contrast to the strong model-dependent NQM approachwe considered in the previous subsection, χ PT can give model-independent calculations of the nucleonmagnetic moments within a perturbative field theory framework in terms of the hadronic degrees offreedom. However, as we will see, our goal of extracting the current quark mass and Λ QCD dependenceis limited by our lack of knowledge of the accurate values of the coupling constants, the so called lowenergy constants (LECs), appearing in the effective Lagrangians of χ PT. These Lagrangians, and theFeynman diagrams generated by them, are organized according to a power counting scheme, and thenumber of LECs we will have to deal with increases as we include higher order contributions to thenucleon magnetic moments.By construction, the LECs in the SU(3) χ PT which we will consider do not depend on the light quark( u , d and s ) masses, and they should in principle be calculable in terms of the heavy quark ( c , b and t )masses and Λ QCD . Without the ability to solve non-perturbative QCD, the LECs are usually determinedby fitting to experimental data for the pertinent physical observables, or estimated theoretically by QCD-inspired models and some other approaches (e.g., the resonance saturation method), and they can alsobe fixed by lattice calculations (for a discussion of the LECs, see for example, [75] and the referencestherein). Most of the LECs are renormalization scale dependent in such a way that they cancel therenormalization scale dependent loop integrals so that the final results for the physical observables arerenormalization scale independent. Furthermore, the values of the LECs are expected to be given bydimensional analysis [66, 76] up to numerical factors of order one. Since the two quantities involvedin such an analysis, namely, the Goldstone boson decay constant (for the meson octet) and the typicalmass of the light but non-Goldstone states, are both pure numbers times Λ QCD in the chiral limit, we willassume that all the LECs under discussion are functions of Λ QCD and the renormalization scale, and bythis assumption we neglect the heavy quark mass dependence in the LECs.For χ PT in the meson-baryon sector, needed for the calculations of nucleon magnetic moments,there exist several renormalization schemes in the literature to ensure consistent power counting which istroubled by the introduction of the baryon mass as a new scale which is non-vanishing in the chiral limit.Among these renormalization schemes, the most studied in the early days in the calculations of octetbaryon magnetic moments is the heavy baryon chiral perturbation theory (HB χ PT) approach [77]. Due toa strong cancellation between the leading order O ( q ) and the next-to-leading order O ( q ) results for thisapproach ( q denotes external momentum in the power counting scheme), one is forced to consider stillhigher order contributions. We will consider the results for this approach to order O ( q ) [78, 79, 80, 81],with (HBwD) and without (HBw/oD) the explicit inclusion of the baryon decuplet states in loops. We willalso consider a result from a more recently developed extended-on-mass-shell (EOMS) renormalizationscheme [82], which gives more convergent results at O ( q ) without [83] or with [84] the inclusion ofdecuplet states in loops. We will restrict our attention to the EOMS without decuplets to avoid theintroduction of several new parameters which do not improve the convergence.At leading order, the octet baryon magnetic moments can be calculated from the Feynman diagramsof chiral order O ( q ) , and the results for both the HB χ PT and EOMS approaches have the same expres-sions as linear combinations of two LECs µ D and µ F , µ (2) B ≡ α B = α DB µ D + α FB µ F , (68)where α D p = 1 / and α F p = 1 for the proton, and α D n = − / and α F n = 0 for the neutron. Note thatwe are writing down the magnetic moments (rather than the anomalous magnetic moments) directly inunits of µ N , and therefore the µ F value we use may differ by compared to the value given in some ofthe references.At O ( q ) and higher order, the results of HB χ PT and EOMS differ. For the HBw/oD approach, we NUCLEONMAGNETICMOMENTS,CURRENTQUARKMASSESAND Λ QCD O ( q ) it is µ (3) B = X X = π,K β XB m p M X πf X , (69)where β π p = − ( D + F ) , β K p = −
23 ( D + 3 F ) , β π n = ( D + F ) , β K n = − ( D − F ) .D and F are dimensionless LECs. We will use the empirical values m p = 938 . MeV, the pion decayconstant f π = 93 MeV and the kaon decay constant f K = 1 . f π in our calculations, but we will considerthese quantities take the forms of their corresponding LECs, namely, the average octet baryon mass, m and the Goldstone boson decay constant F , since the differences between these quantities and theircorresponding LECs give contributions to the octet baryon magnetic moments beyond O ( q ) whichis the highest order we will consider. Therefore, in contrast to the NQM approach we considered inthe previous subsection, we do not need to vary the proton mass in this subsection in calculating thecurrent quark masses and Λ QCD dependence of the nucleon magnetic moments. For the same reason,we take the pion mass, M π , and kaon mass, M K , at their empirical values of MeV and
MeV,respectively, in our calculations, while we only take their lowest order forms M π = (2 B ˆ m ) / and M K = [ B ( ˆ m + m s )] / , where the B is an LEC with mass dimension, when considering light currentquark masses and Λ QCD dependences. We will work in the isospin-symmetric limit in this and the nextsubsection such that m u = m d ≡ ˆ m . The expression for the η mass, M η = [2 / B ( ˆ m + 2 m s )] / is alsoneeded at O ( q ) in the HB χ PT approach and at O ( q ) in the EOMS approach, and we use its empiricalvalue of MeV.For the case HBwD, we have the following terms in addition to Eq. (69) [78, 80, 81], µ (3) B = X X = π,K m p πf X F ( M X , δ, λ ) β ′ XB , (70)where πF ( M, δ, λ ) = − δ ln M λ + √ M − δ [ π/ − arctan ( δ/ √ M − δ )] , M > δ, − √ δ − M ln[( δ + √ δ − M ) /M ] , M < δ, and β ′ π p = − C , β ′ K p = 118 C , β ′ π n = 29 C , β ′ K n = 19 C , where C is a dimensionless LEC, λ is the renormalization scale in dimensional regularization, and δ isthe decuplet-octet mass splitting for which we take to be a number times Λ QCD with a value of
MeV.At O ( q ) in the HB χ PT approach, more LECs appear in the results and the formulae become lengthy.For the case HBw/oD, we take [79] µ (4) B = µ (4 ,c ) B + µ (4 ,d + e + f ) B + µ (4 ,g ) B + µ (4 ,h + i ) B + µ (4 ,j ) B , (71)with µ (4 ,c )p = a + a + 13 a + 13 a − a , µ (4 ,c )n = − a − a − a , There is a misprint in the third term of µ (4 ,d + e + f ) B in [79], where the sign in front should be ‘ − ’. NUCLEONMAGNETICMOMENTS,CURRENTQUARKMASSESAND Λ QCD µ (4 ,d + e + f ) B = X X = π,K δ XB M X π f X ln M X λ + X X = π,K,η η XB M X π f X (cid:18) ln M X λ + 1 (cid:19) − X X = π,K,η φ XB M X π f X (cid:18) M X λ + 1 (cid:19) α B , with δ π p = − µ D − µ F , δ K p = − µ F , δ π n = µ D + µ F , δ K n = µ D − µ F ,η π p = 12 ( D + F ) ( µ D − µ F ) , η K p = − ( 19 D − DF + F ) µ D − ( D − F ) µ F ,η η p = −
118 ( D − F ) ( µ D + 3 µ F ) , η π n = − ( D + F ) µ F ,η K n = ( − D + 23 DF + F ) µ D + ( D − F ) µ F , η η n = 19 ( D − F ) µ D ,φ π p , n = 34 ( D + F ) , φ K p , n = 56 D − DF + 32 F , φ η p , n = 112 ( D − F ) , and µ (4 ,g ) B = X X = π,K γ XB m p M X π f X ln M X λ , with γ π p = 2 a + 2 (cid:18) a + 18 m p (cid:19) , γ K p = a + 4 (cid:18) a + 18 m p (cid:19) ,γ π n = − a − (cid:18) a + 18 m p (cid:19) , γ K n = − a + 2 (cid:18) a + 18 m p (cid:19) , and µ (4 ,h + i ) B = X X = π,K β XB M X π f X (cid:18) M X λ + 1 (cid:19) , and µ (4 ,j ) B = − X X = π,K θ XB m p π f X (cid:18) M X λ + 1 (cid:19) , with θ π p = ( D + F ) (cid:2) M K a + (cid:0) M π − M K (cid:1) a (cid:3) , θ K p = 16 h (3 F + D ) M η + 3( D − F ) M π i a ,θ π n = − ( D + F ) (cid:2) M K a + ( M π − M K ) a (cid:3) , θ K n = ( D − F ) M π a . In the above formulae, the LECs a , , , , (labeled b , , ,D,F in [79]) with their values in units ofGeV − are a = 0 . , a = 0 . , a = 0 . , a = − . and a = − . , where the first threeare estimated by the resonance saturation method which takes into account the contribution from thebaryon decuplet while the other two are determined by fitting to the baryon octet masses, as explained indetail in [79]. We take the value of the η decay constant to be f η = 1 . f π , but we will consider it takingthe form of its corresponding LEC, F , for the same reason explained above for the other Goldstoneboson decay constants. NUCLEONMAGNETICMOMENTS,CURRENTQUARKMASSESAND Λ QCD Λ QCD due to the uncertainties of the values of the LECs appearing in thecounter term Lagrangian, and they are denoted as a , , , , in Eq. (71). These dimensionless numbershave actually absorbed the light current quark masses in the counter term Lagrangian in contrast to theircorresponding true LECs which are independent of the light current quark masses. Therefore theseredefined LECs should contain a factor m s / Λ QCD , if we neglect the contributions from ˆ m as its value ismuch smaller than m s . Two other LECs in the counter term Lagrangian are also present at this order,and they are combined with the two LECs appearing in the O ( q ) result Eq. (68), as µ D,F → µ D,F + 4 B (2 ˆ m + m s )˜ µ D,F , (72)where ˜ µ D,F are LECs appearing in O ( q ) counter term Lagrangian. Then all seven of these redefinedLECs, µ D,F and a , , , , , are used as fitting parameters to perform an exact fit to the seven availableoctet baryon magnetic moments. Since they are used as fitting parameters, and indeed different valuesfor them are obtained with and without the explicit inclusion of baryon decuplet states in loops, as wellas when different values of other LECs are used for the fittings (see the discussion below), it is hard to getan accurate extraction of the light current quark mass and Λ QCD dependence from these redefined LECs.For the light current quark masses dependence, we will only consider the m s dependence for a , , , , ,while we will not try to extract such dependence for the redefined µ D,F (denoted as a , in [80]), sincewe do not know the relative size of the two terms on the right hand side of Eq. (72), where only one of thetwo terms has the light current quark mass dependence, although such dependence in these two redefinedLECs may be not small, as suggested in [79] when comparing the fitting values up to O ( q ) with theones up to O ( q ) . We take the values µ D = 3 . , µ F = 3 . , a = − . , a = − . , a = − . , a = 0 . and a = − . given in [79], where F = 0 . , D = 0 . is used, and the renormalizationscale λ is taken to be . GeV.For HBwD at O ( q ) , we take [80] µ (4) B = µ (4 ,c ) B + X X = π,K,η π f X ( γ ′ XB − φ ′ XB α B ) M X ln M X λ + X X = π,K,η π f X h (˜ γ ′ XB − φ ′ XB α B ) L (3 / ( M X , δ, λ ) + ˆ γ ′ XB L ′ (3 / ( M X , δ, λ ) i , (73)where L (3 / ( M, δ, λ ) = M ln M λ + 2 πδ F ( M, δ, λ ) , and L ′ (3 / ( M, δ, λ ) = M ln M λ + 2 π δ G ( M, δ, λ ) , with πG ( M, δ, λ ) = − δ ln M λ + πM + − M − δ ) / [ π/ − arctan ( δ/ √ M − δ )] , M > δ, − δ − M ) / ln[( δ + √ δ − M ) /M ] , M < δ, There is a misprint in the form of the πG ( M, δ, λ ) for the case M > δ , where the sign in front should be ‘ − ’. NUCLEONMAGNETICMOMENTS,CURRENTQUARKMASSESAND Λ QCD ˜ γ ′ π p = 8027 C µ C , ˜ γ ′ K p = 1027 C µ C , ˜ γ ′ η p = 0 , ˜ γ ′ π n = − C µ C , ˜ γ ′ K n = − C µ C , ˜ γ ′ η n = 0 , ˆ γ ′ π p = 89 C ( D + F ) µ T , ˆ γ ′ K p = 29 C (3 D − F ) µ T , ˆ γ ′ η p = 0 , ˆ γ ′ π n = − C ( D + F ) µ T , ˆ γ ′ K n = − C F µ T , ˆ γ ′ η n = 0 , ˜ φ ′ π p , n = 2 C , ˜ φ ′ K p , n = 12 C , ˜ φ ′ η p , n = 0 . The other coefficients are related to the ones given in Eq. (71), as γ ′ π,KB = δ π,KB + η π,KB , γ ′ ηB = η ηB , φ ′ XB = 3 φ XB , and µ (4 ,c ) B is the same as the case of HBw/oD.In this case, since different LECs are used as inputs for the fittings in comparison to the HBw/oDcase, the resulting fit values of the seven redefined LECs are different, and we take their values from Case(b) in Table II of [80], a = 3 . , a = 2 . , a = − . , a = − . , a = 0 . , a = 0 . and a = − . , corresponding to the LECs inputs F = 0 . , D = 0 . , C = − . , µ T = − . and µ C = 1 . . A renormalization scale λ = 1 GeV is used.For the EOMS approach, to minimize the number of LECs involved and thus perhaps the uncertain-ties introduced by them, we only consider the result given in Eq. (2) to Eq. (5) of [83] which does notinclude the baryon decuplet states in loops. The result is up to O ( q ) , and the loop integrals are finite.The values µ D = 3 . and µ F = 2 . denoted as ˜ b D and ˜ b F in [83], after performing the EOMS scheme,are determined by a fit to minimize the ˜ χ = P ( µ th − µ exp ) as explained in that reference. For otherquantities in the formula, F φ = 1 . f π is the average of the physical values of f π , f K and f η , and westill use M B = 938 . MeV, f π = 93 MeV, and the same values for M π,K,η as specified above. We take D = 0 . and F = 0 . as used in [83] for this EOMS approach.We list the results of the two HB χ PT and the one EOMS approaches in Table 4, where we also needto specify the ratio of m s to ˆ m , for which we use . Note that as we discussed above, we have assumedthat the LECs ( a , , , , , , , correspond to the original LECs before the re-definition) depend only on Λ QCD and the renormalization scale, λ , i.e., they do not depend on the light quark masses and there isno dependence on the renormalization scale in the full result. Therefore, the coefficient of δ Λ QCD / Λ QCD must be equal and opposite to the sum of the light quark mass contributions.Table 4: The coefficients, κ i , of δv i v i ( v i = ˆ m, m s , Λ QCD ) of δg χ PT g exp , defined as in Eq. (40), for the proton(left) and the neutron (right). κ u = 2 κ d κ s κ QCD
HBw/oD − . − .
50 0 . HBwD .
034 0 . − . EOMS − . − .
031 0 .
080 2 κ u = 2 κ d κ s κ QCD − . − .
14 0 . − .
050 0 . − . − .
11 0 .
014 0 . We see from Table 4, the numbers in each column differ considerably for nucleon magnetic momentformulae from different renormalization schemes and depend on the explicit inclusion of baryon decupletstates in loops. As we mentioned in the beginning of this subsection, we believe this discrepancy comesin a large part from our lack of knowledge of the accurate values of the LECs. In particular, many of the NUCLEONMAGNETICMOMENTS,CURRENTQUARKMASSESAND Λ QCD F = 0 . , D = 0 . and C = − . in [80], the resulting values of the a ’s also give an exact fit to the seven available octet baryon magnetic moments, with a prediction forthe ΣΛ transition moment similar to the one given by Case (b). However, one can see that many of thecorresponding a ’s for Case (a) and Case (b) differ greatly, and indeed, for Case (a) we get the coefficientsfrom left to right of Table 4 as . , . and − . for the proton, and − . , . and − . for theneutron, which are different from the results of Case (b). Therefore, it is crucial to pin down the valuesof LECs before one can make a better extraction of the light quark masses and Λ QCD dependence in the χ PT approach.One can also estimate the dependence of m p on the current quark masses and Λ QCD from a formulafor m p within χ PT. Such dependences can be used when one varies the electron-to-proton mass ratio, µ ≡ m e /m p . However, as we explained at the end of the previous subsection, we prefer to use a commonset of values for the coefficients of κ q and κ QCD of δm p /m p . Those values for the isospin-symmetriclimit case are listed at the end of section 3.1.1. χ PT and lattice QCD
As another approach to study hadronic physics, lattice QCD provides a promising way to extract thecurrent quark masses dependence of the nucleon magnetic moments, because one can do explicit cal-culations by assuming a sequence of different current quark masses in lattice computations, although inpractice the computational cost is a limitation. Since most of the current lattice computations are stillusing input current quark masses much larger than their empirical values, an extrapolation of the latticeresults to the physical point is needed. In the extrapolations for the physical observables, terms hav-ing non-analytic behaviors, m / and m q log m q , etc., which are predicted by χ PT and have importantcontributions near the chiral limit, must be considered.An earlier study of this combined lattice and χ PT approach for the nucleon magnetic moments usesan encapsulating form which is the Pad ´ e approximant [85], µ p , n ( M π ) = µ − χ p , n µ M π + cM π , (74)where χ p , n are fixed by the leading non-analytic term given by χ PT, while µ and c are allowed to varyto best fit the lattice data.A later development takes the finite range regulator (FRR) [86] as the regularization method ratherthan the traditional dimensional regularization for the results we discussed in the previous subsection,and the cut-off parameter in the FRR is a mass scale which can be interpreted as the inverse of the sizeof the nucleon.The current quark masses dependence for the nucleon magnetic moments is given in [28], and wesimply quote the result there without going into any detail δg p g p = − . δ ˆ m ˆ m − . δm s m s , δg n g n = − . δ ˆ m ˆ m + 0 . δm s m s . (75)As the same argument we made for the χ PT approach in the previous subsection, all parameters withoutlight quark masses dependence are either pure numbers or are pure numbers time Λ QCD . Therefore, weobtain δg p g p = − . δ ˆ m ˆ m − . δm s m s + 0 . δ Λ QCD Λ QCD ,δg n g n = − . δ ˆ m ˆ m + 0 . δm s m s + 0 . δ Λ QCD Λ QCD . (76) ATOMICCLOCKCONSTRAINTS 21 As we have seen in section 2.4, the frequency shift is related to { g p , g n , b, µ, α } by the relation ˙ ν AB ν AB = λ g p ˙ g p g p + λ g n ˙ g n g n + λ b ˙ bb + λ µ ˙ µµ + λ α ˙ αα , (77)where the coefficients λ are given explicitly in Table 2. Then, in section 3, we have expressed thedependence of the g -factors as δg p g p = κ u p δm u m u + κ d p δm d m d + κ s p δm s m s + κ QCDp δ Λ QCD Λ QCD , (78) δg n g n = κ u n δm u m u + κ d n δm d m d + κ s n δm s m s + κ QCDn δ Λ QCD Λ QCD , (79)where the coefficients κ i have been calculated for different models and collected in Tables 3 and 4 andEq. (76), as well as the dependence of the proton mass δm p m p = f T u δm u m u + f T d δm d m d + f T s δm s m s + f T g δ Λ QCD Λ QCD , (80)where the f T i are given in Eqs. (49) and (51). Also, following [27, 31], b depends on the quark mass and Λ QCD , and there it is found δbb = γ q δ ˆ m ˆ m + γ QCD δ Λ QCD Λ QCD , (81)with γ q = − γ QCD = − . . (82)Assuming for simplicity that all Yukawa couplings are varying similarly, i.e., δh i /h i = δh/h , theexpansions (78), (79), (80) and (81) can be inserted in Eq. (77) to obtain ˙ ν AB ν AB = ˆ λ h ˙ hh + ˆ λ v ˙ vv + ˆ λ QCD ˙Λ QCD Λ QCD + ˆ λ α ˙ αα . (83)The coefficients ˆ λ are easily computed to be given by ˆ λ h = λ g p ( κ u p + κ d p + κ s p ) + λ g n ( κ u n + κ d n + κ s n ) + λ b γ q + λ µ (1 − f T u − f T d − f T s ) (84) ˆ λ v = ˆ λ h (85) ˆ λ QCD = λ g p κ QCDp + λ g n κ QCDn + λ b γ QCD − λ µ f T g (86) ˆ λ α = λ α . (87)The form (83) makes no assumption on unification and only relies on the fact that all Yukawa couplingsare varying in a similar way. It is important to note here that the dimensionality constraint on the f T i , κ i and γ i parameters implies that ˆ λ QCD = − ˆ λ v so that Eq. (83) actually depends only on the combinationof X ≡ hv/ Λ QCD and α as ˙ ν AB ν AB = ˆ λ h ˙ XX + ˆ λ α ˙ αα . (88)This would not be the case if we had not assumed that δh i /h i = δh/h for all Yukawa couplings. ATOMICCLOCKCONSTRAINTS 22Our first hypothesis concerning unification allows one to express the variation of the QCD scale bymeans of Eq. (2) so that ˙ ν AB ν AB = (cid:18) ˆ λ h + 29 ˆ λ QCD (cid:19) ˙ hh + (cid:18) ˆ λ v + 29 ˆ λ QCD (cid:19) ˙ vv + (cid:16) ˆ λ α + R ˆ λ QCD (cid:17) ˙ αα ≡ H S ˙ hh + ˙ vv ! + H α ˙ αα . (89)The second hypothesis on unification assumes that the variation of v and h are related by Eq. (4) so that ˙ ν AB ν AB = (cid:18) ˆ λ h + 29 ˆ λ QCD (cid:19) (1 + S ) ˙ hh + (cid:16) ˆ λ α + R ˆ λ QCD (cid:17) ˙ αα ≡ H S (1 + S ) ˙ hh + H α ˙ αα . (90)The last hypothesis assumes that the variations of h and α are related by Eq. (5) so that ˙ ν AB ν AB = (cid:20) (cid:18) ˆ λ h + 29 ˆ λ QCD (cid:19) (1 + S ) + (cid:16) ˆ λ α + R ˆ λ QCD (cid:17)(cid:21) ˙ αα ≡ ( 12 H S (1 + S ) + H α ) ˙ αα ≡ C α ( R, S ) ˙ αα . (91)The two last equations define the parameter C α ( R, S ) .The forms (89-91) imply increasing assumptions on the unification mechanisms and are thus be-coming more and more model-dependent with the advantage of reducing the number of fundamentalconstants, hence allowing one to draw sharper constraints from the same experimental data.The coefficients introduced above can be easily calculated from Table 2 for the coefficients λ , Ta-bles 3 or 4 or Eq. (76) for the coefficients κ i , Eq. (49) or Eq. (51) for the coefficients f T i , and Eq. (82)for the coefficients γ i . As an example, we provide the value of the coefficients C α assuming S = 160 and R = 30 for the 9 models considered in this article. It is important to stress that this coefficient isalmost always larger than one and typically of order 5 – 30 in absolute value.We can check that the effect of varying the nuclear radius is indeed much smaller than varying theother parameters. This effect can be included by adding a term ǫ r ( ˙ˆ m/ ˆ m − ˙Λ QCD / Λ QCD ) to Eq. (77).Using the values listed in Table IV of [31], we have ǫ r = − . for the Cs-Rb clock system, while ǫ r = − . for the other five clock systems involving Cs. These amount to an adjustment of − . inthe numbers in the first column of Table 5, and − . in the other five columns. We are now in a position to combine our results for the dependence of the nucleon g -factor on funda-mental parameters with the limits imposed from atomic clock measurements. For each experiment, wecan derive a limit on the variation of the fine structure constant under a number of sets of assumptions.For example, assuming first that the only dependence of ν AB on α is related to the coefficient λ α (i.e., we assume that g p , g n , b and µ remain constant), we can use Table 2 to extract a limit on ˙ α/α foreach experiment from ˙ αα = 1 λ α ˙ ν AB ν AB . (92)In contrast, when we take into account the contributions from coupled variations, and we assume therelation between ˙ ν AB /ν AB and ˙ α/α given by Eq. (91) we obtain simply ˙ αα = 1 C α ˙ ν AB ν AB . (93) ATOMICCLOCKCONSTRAINTS 23Table 5: The coefficient C α assuming S = 160 and R = 30 for each of the models for the nucleonmagnetic moment and for the various combinations of clocks discussed in this article. Cs-Rb H-Cs Hg-Cs Yb-Cs Sr-Cs SF -CsA − .
53 13 .
86 17 .
06 12 .
96 13 .
80 4 . B1 − .
26 20 .
16 23 .
36 19 .
26 20 .
10 10 . B2 − .
79 18 .
16 21 .
36 17 .
26 18 .
10 8 . B3 − .
29 18 .
82 22 .
02 17 .
92 18 .
76 9 . C − .
37 15 .
26 18 .
46 14 .
36 15 .
20 5 . HBw/oD .
27 29 .
33 32 .
53 28 .
43 29 .
27 20 . HBwD − .
57 17 .
01 20 .
21 16 .
11 16 .
95 7 . EOMS .
49 20 .
97 24 .
17 20 .
07 20 .
91 11 . χ PT+QCD .
20 21 .
29 24 .
49 20 .
39 21 .
23 12 . Thus, the improvement in the limit from each individual experiment due to the theoretical assumption ofcoupled variations is given by C α /λ α . These factors are tabulated in Table 6 for each experiment andmodel for g p , n .Table 6: The enhancement factor C α /λ α assuming S = 160 and R = 30 for each of the models for thenucleon magnetic moment and for the various combinations of clocks discussed in this article. Cs-Rb H-Cs Hg-Cs Yb-Cs Sr-Cs SF -CsA − .
73 4 .
90 2 .
83 6 .
72 4 .
98 1 . B1 − .
61 7 .
12 3 .
87 9 .
98 7 .
26 3 . B2 − .
86 6 .
42 3 .
54 8 .
94 6 .
53 3 . B3 − .
80 6 .
65 3 .
65 9 .
28 6 .
77 3 . C − .
28 5 .
39 3 .
06 7 .
44 5 .
49 2 . HBw/oD .
32 10 .
36 5 .
39 14 .
73 10 .
57 7 . HBwD − .
48 6 .
01 3 .
35 8 .
34 6 .
12 2 . EOMS .
00 7 .
41 4 .
01 10 .
40 7 .
55 4 . χ PT+QCD .
45 7 .
52 4 .
06 10 .
56 7 .
66 4 . As one can see, there is a strong model-dependence on the resulting limits on ˙ α/α . Overall theenhancements range from ∼ to ∼ . For example, let us consider the case of the Cs-Rb atomic clocksystem. Ignoring the variations in all other constants, this clock would yield a result ˙ αα = (1 . ± . × − yr − . (94)In contrast, coupled variations, according to the factors in Table 6, improve this result by as much as afactor of . using the HBw/oD model for g p , n , yielding ˙ αα = (0 . ± . × − yr − . (95)Cases A and C also make substantial improvements in the limit for the Cs-Rb clock system. On the otherhand, there is no gain for case EOMS, or even a weaker limit if the nuclear radius effect is taken intoaccount. ATOMICCLOCKCONSTRAINTS 24 While the results of individual experiments can be substantially improved by coupled variations, twoclock systems (Dy and Hg-Al) are independent of any assumption on unification and lead to model-independent limits on α . We next combine the available results to obtain a single limit on α for eachchoice of model for g p , n .Each of the eight experimental results used in this article can be written as dd t ln ν AB = η AB ± δ AB , (96)listed in Table 2. From a theoretical point of view, the expression for ν AB depends on a set of con-stants, x , chosen as being independent and on our hypothesis on unification schemes. If we assume d ln ν AB ( x ) /dt − η AB to be Gaussian distributed and all the experiments to be uncorrelated, then thebest-fit for the set of constants x is obtained by maximizing the likelihood, or equivalently by minimizing χ ( x ) = X AB h ˙ ν AB ν AB ( x ) − η AB i δ AB . (97)The 68.27%, 95%, and 99% confidence level (i.e., σ , ∼ σ and ∼ σ ) constraints are then obtained by ∆ χ = (1 , . , . if dim ( x ) = 1 and ∆ χ = (2 . , . , . if dim ( x ) = 2 . Let us start by assuming that { g p , g n , b, µ, α } are independent parameters. One can use the Hg-Al clockto constrain the variation of α and then use the six clock combinations that depend on the five parametersto set a constraint on { g p , g n , b, µ } . However, from Eq. (30), we note that the ratio of the coefficients of δg n /g n and δb/b is g n / ( g n − g p + 1) , which is independent of the clock systems we are considering.Also, from Table 2, we note that the value of λ g p /λ g n for the Cs-Rb clock is very close to that of the otherfive clock combinations. Therefore, for the purpose of constraining the QED parameters, g p , g n and b arenot independent, and we can only constrain their combination, namely, g Cs . The combined constraint on g Cs and µ is depicted on Fig. 2. Note that if a different method in the calculation of g -factors of Rband
Cs, and/or other clock systems, are used, such that the ratio λ g p : λ g n : λ b is not the same fordifferent clock combinations, then g p , g n and b can be taken as independent parameters.As we know from our analysis, such a hypothesis is not correct since the variations are expected tobe correlated but this shows the result one would have derived without any knowledge on QCD. α As in the previous subsection, we can consider first the constraint obtained using the form (91) thatdepends on δα/α alone. Minimizing χ for a single variable is equivalent to taking the weighted mean of η AB /λ α with an uncertainty δ AB /λ α . This result can be compared with that assuming coupled variationsusing the coefficients C α , given in Table 5. In this case, the weighted mean replaces λ α with C α .In order to determine the effect of coupled variations, we compare the constraints arising from thecombination of the eight experiments to the one obtained from the combination of 6 clocks (that isneglecting the Dy and Hg-Al clocks). The results are presented in Table 7 and shall be compared to thesame analysis assuming that only α is varying (i.e., keeping g p , g n , b and µ constant). We find ˙ αα = − (2 . ± . × − yr − (98) ATOMICCLOCKCONSTRAINTS 25 - -
10 0 10 20 - ∆ΜΜ H ´ L ∆ g C s g C s H ´ L Figure 2: Constraints on the variation of parameters { g Cs , µ } assumed to be independent once the con-straint from the variation of α from the Hg-Al clock is taken into account. Solid, dashed and dottedcontours correspond to 68.27%, 95% and 99% C.L.for the combination of the 8 experiments and ˙ αα = − (5 . ± . × − yr − (99)for the combination of the 6 experiments. We also remind the reader that the Hg-Al experiment alone setthe constraint ˙ αα = − (1 . ± . × − yr − , (100)which shows that there is little gain in combining the 8 experiments compared to this experiment alone.When g p , g n , b and µ are allowed to vary in the combination of the 6 clocks, there is a gain of a factorof order 4 so that the constraint obtained from the combination of these 6 clocks assuming unificationbecomes as strong as the constraint obtained from Hg-Al alone. When combining the 8 experiments, thegain is less than a factor of 2, due to the fact that the limit arises mostly from the Hg-Al experiment whichdoes not depend on g p , n . These results are summarized in Table 7 and each result can be compared tothe single Hg-Al result given in Eq. (100). hv As a second application, we can use the constraint (22) arising from the Hg-Al clock to obtain a boundon the time variation of α that is independent of the other constants and then use the 6 other clocks to seta constraint on the combination of parameters hv , assuming the form (89) to set a constraint on δhv/hv alone. This requires the knowledge of the coefficients H S and H α and we assume that R = 30 , but itdoes not depend on the coefficient S .The constraints for each model are summarized on Table 8. It ranges between (cid:12)(cid:12)(cid:12) ( hv ) . hv (cid:12)(cid:12)(cid:12) < . × − yr − and (cid:12)(cid:12)(cid:12) ( hv ) . hv (cid:12)(cid:12)(cid:12) < . × − yr − , respectively for models A and HBw/oD and it turns outthat the model-dependence for this constraint is mild. APPLICATIONTOASTROPHYSICALSYSTEMSANDDISCUSSION 26Table 7: Constraints on the variation of α assuming the unification relation (91) and the values of C α for S = 160 and R = 30 . We compare the constraints obtained from the combination of the 8 clocks andthe constraints obtained from the 6 clocks (i.e. without Dy and Hg-Al). All numbers are in yr − . Model 8 clocks 6 clocksA ( − . ± . × − ( − . ± . × − B1 ( − . ± . × − ( − . ± . × − B2 ( − . ± . × − ( − . ± . × − B3 ( − . ± . × − ( − . ± . × − C ( − . ± . × − ( − . ± . × − HBw/oD ( − . ± . × − ( − . ± . × − HBwD ( − . ± . × − ( − . ± . × − EOMS ( − . ± . × − ( − . ± . × − χ PT+QCD ( − . ± . × − ( − . ± . × − Table 8: Constraints on the variation of hv once the variation of α alone is constrained from the Hg-Alclock. It assumes the unification relation (89). All numbers are in yr − . Model ( hv ) . hv A ( − . ± . × − B1 ( − . ± . × − B2 ( − . ± . × − B3 ( − . ± . × − C ( − . ± . × − HBw/oD ( − . ± . × − HBwD ( − . ± . × − EOMS ( − . ± . × − χ PT+QCD ( − . ± . × − ( hv Λ QCD , α ) As a third application, we use the fact that ˆ λ QCD = − ˆ λ h so that the form (88) allows one to set a constrainton ( hv/ Λ QCD , α ) independent of any hypothesis on unification and thus does not require knowledge ofthe parameters R and S .Figure 3 compares the 99% C.L. constraints obtained from the combination of 6 and 8 experimentsfor each model. Again, we see that the Hg-Al experiment dominates the collective limit. Several different types of observations of astrophysical systems involving quasar absorption spectra aresubject to a similar analysis that has been applied to atomic clocks. Indeed, there are four distinctcombinations of physical parameters which depend on g p . • The comparison of UV heavy element transitions with the hyperfine H I transition allows one toset constraints on x ≡ α g p µ, (101) APPLICATIONTOASTROPHYSICALSYSTEMSANDDISCUSSION 27 - - - - ∆ ln Α H ´ L ∆ l nhv L Q C D H ´ L - - - - ∆ ln Α H ´ L ∆ l nhv L Q C D H ´ L Figure 3: Comparison of the 99% C.L. constraints on ( hv/ Λ QCD , α ) for the 9 models with 8 clocks (left)and 6 clocks (right)since the optical transitions are simply proportional to R ∞ . It follows that constraints on the timevariation of x can be obtained from high resolution 21 cm spectra compared to UV lines, e.g., ofSi II , Fe II and/or Mg II . The recent detection of 21 cm and molecular hydrogen absorption linesin the same damped Lyman- α system at z abs = 3 . towards SDSS J1337+3152 constrains [87]the variation x to ∆ x/x = − (1 . ± . × − , z = 3 . . (102) • The comparison of the H I
21 cm hyperfine transition to the rotational transition frequencies ofdiatomic molecules allows one to set a constraint on y ≡ g p α (103)The most recent constraint [88] relies on the comparison of two absorption systems determinedboth from H I and molecular absorption. The first is a system at z = 0 . in the direction ofTXS 0218+357 for which the spectra of CO(1-2), CO(1-2), C O(1-2), CO(2-3), HCO + (1-2)and HCN(1-2) are available. They concluded that ∆ y/y = ( − . ± . × − , z = 0 . . (104)The second system is an absorption system in the direction of PKS 1413+135 for which the molec-ular lines of CO(1-2), HCO + (1-2) and HCO + (2-3) have been detected. The analysis led to ∆ y/y = ( − . ± . × − , z = 0 . . (105) • The ground state, Π / J = 3 / , of OH is split into two levels by Λ -doubling and each of thesedoubled levels is further split into two hyperfine-structure states. Thus, it has two “main” lines( ∆ F = 0 ) and two “satellite” lines ( ∆ F = 1 ). Since these four lines arise from two differentphysical processes ( Λ -doubling and hyperfine splitting), they enjoy the same Rydberg dependencebut different g p and α dependences. By comparing the four transitions to the H I hyperfine line,one can set a constraint on F ≡ g p ( α /µ ) . . (106)Using the four 18 cm OH lines from the gravitational lens at z ∼ . toward PMN J0134-0931and comparing the H I
21 cm and OH absorption redshifts of the different components allowed oneto set the constraint [89] ∆ F/F = ( − . ± . ± . syst ) × − , z = 0 . , (107) APPLICATIONTOASTROPHYSICALSYSTEMSANDDISCUSSION 28where the second error is due to velocity offsets between OH and H I assuming a velocity disper-sion of 3 km/s. A similar analysis [90] in a system in the direction of PKS 1413+135 gave ∆ F/F = (0 . ± . × − , z = 0 . . (108) • The satellite OH 18 cm lines are conjugate so that the two lines have the same shape, but withone line in emission and the other in absorption. This behavior has recently been discovered atcosmological distances and it was shown [91] that a comparison between the sum and differenceof satellite line redshifts probes the variation of G ≡ g p ( α /µ ) . . (109)From the analysis of a system at z ∼ . towards PKS 1413+135, it was concluded [92] that | ∆ G/G | = (2 . ± . × − , while a newer analysis [93] gave | ∆ G/G | = ( − . ± . × − . (110)It was also applied to a nearby system [94], Centaurus A, to give | ∆ G/G | < . × − at z ∼ . .These constraints are summarized in Table 9.Table 9: Constraints on the variation of different combinations of g p , µ and α from astrophysical obser-vations. Combination λ g p λ µ λ α Constraints (yr − ) redshift x = g p α µ − (1 . ± . × − . y = g p α − . ± . × − . − . ± . × − . F = g p ( α /µ ) . − .
57 3 .
14 ( − . ± . ± . syst ) × − . . ± . × − . G = g p ( α /µ ) . − . ( − . ± . × − . ± . × − . In contrast to our analysis of atomic clocks, we cannot combine the astrophysical observations becausethey have been obtained from different systems at different redshifts and at different spatial locations.However, as we have done previously (but without the g n and b terms), we show in Table 10 the enhance-ment factor for the analysis of the 4 types of combinations of absorption spectra. We emphasize that theenhancement factor is always larger than unity (except for y in the EOMS and χ PT+QCD models). Aslast example of the power of coupled variations, Table 11 compares the constraints on the variation of α that can be obtained under the assumption that g p and µ are constant with the assumption of coupledvariations based on unification. As one can see, in many cases the limits are improved by an order ofmagnitude. In this article, we have discussed the effect of a correlated variation of fundamental constants, focusing onthe gyromagnetic factors g p and g n . These parameters are particularly important to interpret electromag-netic spectra, and thus to derive constraints on the variation of fundamental constants from atomic clock APPLICATIONTOASTROPHYSICALSYSTEMSANDDISCUSSION 29Table 10: Value of the parameter C α for the 4 combinations of constants that can be constrained byastrophysical observations, assuming R = 30 and S = 160 (left) and value of the enhancement factor C α /λ α (right). x y F G A .
10 15 . − . − . B1 .
72 2 . − . − . B2 .
96 6 . − . − . B3 .
52 4 . − . − . C .
21 12 . − . − . HBw/oD . − . − . − . HBwD .
00 8 . − . − . EOMS . − . − . − . χ PT+QCD . − . − . − . x y F G A .
05 7 . − . − . B1 .
36 1 . − . − . B2 .
48 3 . − . − . B3 .
76 2 . − . − . C .
60 6 . − . − . HBw/oD . − . − . − . HBwD .
50 4 . − . − . EOMS . − . − . − . χ PT+QCD . − . − . − . Table 11: Comparison of the constraints obtained from astrophysical systems with and without assump-tion on unification for model A.
Combination independent (yr − ) correlated (yr − ) redshift x ( − . ± . × − ( − . ± . × − y ( − . ± . × − ( − . ± . × − . − . ± . × − ( − . ± . × − . F ( − . ± . × − (3 . ± . × − . . ± . × − ( − . ± . × − . G ( − . ± . × − (6 . ± . × − . ± . × − (0 ± . × − . experiments and from quasar absorption spectra. As discussed, there is an important model-dependencein the computation of the gyromagnetic factors in terms of the quark masses and QCD scale.When applied to the interpretation of atomic clock experiments, we have shown that in general theconstraints on the variation of α are sharper than that under the assumption that g p , g n , b and µ areconstant, but this is not a systematic conclusion as we have exhibited models in which the variation of α stays the same or is even weaker due to cancellations in the sensitivity to α . The constraints on thevariation of α should then be taken with care. In many cases, they may be stronger than reported, butthey may be weaker as well. This points to the need to better understand the fundamental physics neededto calculate baryon magnetic moments. Any limit which depends on g p , n will be subject to the type ofuncertainties discussed here.Fortunately, the tightest constraint arises from the Hg-Al clock experiments, that does not depend on g p , g n , b or µ . As a consequence, we have been able to independently set a bound on the variation of hv from the combination of the other experiments. While this bound is still model-dependent, we haveshown that it is always smaller than (cid:12)(cid:12)(cid:12)(cid:12) ( hv ) . hv (cid:12)(cid:12)(cid:12)(cid:12) < . × − yr − (111)for the models we have considered in this article.Our analysis also applies to astrophysical system and to quasar absorption spectra. We have shownthat the enhancement factor is almost always larger than unity.EFERENCES 30 Acknowledgments
We would like to thank X. Cui, J. Ellis, M. Peskin, M. Srednicki, A. Vainshtein, and M. Voloshin forhelpful discussions. The work of FL and KAO was supported in part by DOE grant DE-FG02-94ER-40823 at the University of Minnesota. JPU was partially supported by the ANR/Thales.
References [1] P. Jordan, Die Naturwissenschaften , 513 (1937);M. Fierz, Helv. Phys. Acta , 128 (1956);C. Brans, and R. Dicke, Phys. Rev. , 925 (1961);R.H. Dicke, in DeWitt, C.M. and DeWitt, B.S., eds., Relativity, Groups and Topology. Relativit´e,Groupes et Topologie, Lectures delivered at Les Houches during the 1963 session of the SummerSchool of Theoretical Physics, University of Grenoble, pp. 165313, (Gordon and Breach, NewYork; London, 1964);J. D. Bekenstein, Phys. Rev. D , 1527 (1982).[2] J. K. Webb, V. V. Flambaum, C. W. Churchill, M. J. Drinkwater, et al. , Phys. Rev. Lett. , 884(1999), [arXiv:astro-ph/9803165];M. T. Murphy et al. , Mon. Not. Roy. Astron. Soc. , 1208 (2001), [arXiv:astro-ph/0012419];J. K. Webb, et al. , Phys. Rev. Lett. , 091301 (2001), [arXiv:astro-ph/0012539];M. T. Murphy, J. K. Webb, V. V. Flambaum, C. W. Churchill, et al. , Mon. Not. Roy. Astron. Soc. , 1223 (2001), [arXiv:astro-ph/0012420].[3] M. T. Murphy, J. K. Webb and V. V. Flambaum, Mon. Not. Roy. Astron. Soc. , 609 (2003),[arXiv:astro-ph/0306483].[4] H. Chand, R. Srianand, P. Petitjean and B. Aracil, Astron. Astrophys. , 853 (2004), [arXiv:astro-ph/0401094];R. Srianand, H. Chand, P. Petitjean and B. Aracil, Phys. Rev. Lett. , 121302 (2004), [arXiv:astro-ph/0402177].[5] R. Quast, D. Reimers and S. A. Levshakov, Astron. Astrophys. , L7 (2004), [arXiv:astro-ph/0311280].[6] M. T. Murphy, J. K. Webb and V. V. Flambaum, Phys. Rev. Lett. , 239001 (2007),[arXiv:0708.3677 [astro-ph]].[7] R. Srianand, H. Chand, P. Petitjean and B. Aracil, Phys. Rev. Lett. , 239002 (2007).[8] J. R. Ellis, S. Kalara, K. A. Olive and C. Wetterich, Phys. Lett. B , 264 (1989).[9] B. A. Campbell and K. A. Olive, Phys. Lett. B , 429 (1995), [arXiv:hep-ph/9411272].[10] P. Langacker, G. Segre and M. J. Strassler, Phys. Lett. B , 121 (2002), [arXiv:hep-ph/0112233];T. Dent and M. Fairbairn, Nucl. Phys. B , 256 (2003), [arXiv:hep-ph/0112279]; X. Calmet andH. Fritzsch, Eur. Phys. J. C , 639 (2002), [arXiv:hep-ph/0112110];X. Calmet and H. Fritzsch, Phys. Lett. B , 173 (2002), [arXiv:hep-ph/0204258];T. Damour, F. Piazza and G. Veneziano, Phys. Rev. Lett. , 081601 (2002), [arXiv:gr-qc/0204094];T. Damour, F. Piazza and G. Veneziano, Phys. Rev. D , 046007 (2002), [arXiv:hep-th/0205111].[11] M. Dine, Y. Nir, G. Raz and T. Volansky, Phys. Rev. D , 015009 (2003), [arXiv:hep-ph/0209134].EFERENCES 31[12] J. R. Ellis, K. A. Olive and Y. Santoso, New J. Phys. , 32 (2002), [arXiv:hep-ph/0202110].[13] V.V. Dixit and M. Sher, Phys. Rev. D (1988) 1097.[14] J.-P. Uzan, Rev. Mod. Phys. , 403 (2003), [arXiv:hep-ph/0205340];J.-P. Uzan, AIP Conf. Proc. , 3 (2005), [astro-ph/0409424];J.-P. Uzan, Space Sci. Rev. , 249 (2010), [arXiv:0907.3081];G.F.R. Ellis and J.-P. Uzan, Am. J. Phys. , 240 (2005), [gr-qc/0305099].[15] J. P. Uzan, Living Rev. Rel. , 2 (2011), [arXiv:1009.5514 [astro-ph.CO]].[16] K. Ichikawa and M. Kawasaki, Phys. Rev. D , 123511 (2002), [arXiv:hep-ph/0203006].[17] C. M. Muller, G. Schafer and C. Wetterich, Phys. Rev. D , 083504 (2004), [arXiv:astro-ph/0405373];T. Dent, S. Stern and C. Wetterich, Phys. Rev. D , 063513 (2007), [arXiv:0705.0696 [astro-ph]].[18] A. Coc, N. J. Nunes, K. A. Olive, J. P. Uzan, et al. , Phys. Rev. D , 023511 (2007), [arXiv:astro-ph/0610733].[19] V. V. Flambaum and E. V. Shuryak, Phys. Rev. D , 103503 (2002), [arXiv:hep-ph/0201303];V. F. Dmitriev and V. V. Flambaum, Phys. Rev. D , 063513 (2003), [arXiv:astro-ph/0209409];V. V. Flambaum and E. V. Shuryak, Phys. Rev. D , 083507 (2003), [arXiv:hep-ph/0212403];V. F. Dmitriev, V. V. Flambaum and J. K. Webb, Phys. Rev. D , 063506 (2004), [arXiv:astro-ph/0310892];J. C. Berengut, V. V. Flambaum and V. F. Dmitriev, Phys. Lett. B , 114 (2010), [arXiv:0907.2288[nucl-th]].[20] S. J. Landau, M. E. Mosquera, C. G. Scoccola and H. Vucetich, Phys. Rev. D , 083527 (2008)[arXiv:0809.2033 [astro-ph]].[21] M. K. Cheoun, T. Kajino, M. Kusakabe and G. J. Mathews, arXiv:1104.5547 [astro-ph.CO].[22] K. A. Olive, M. Pospelov, Y. Z. Qian, A. Coc, et al. , Phys. Rev. D , 045022 (2002), [arXiv:hep-ph/0205269].[23] K. A. Olive, M. Pospelov, Y. Z. Qian, G. Manhes, et al. , Phys. Rev. D , 027701 (2004),[arXiv:astro-ph/0309252].[24] T. Dent, S. Stern and C. Wetterich, Phys. Rev. D , 103518 (2008) [arXiv:0808.0702 [hep-ph]];T. Dent, S. Stern and C. Wetterich, Phys. Rev. D , 083533 (2009) [arXiv:0812.4130 [hep-ph]].[25] M. Nakashima, K. Ichikawa, R. Nagata and J. Yokoyama, JCAP , 030 (2010)[arXiv:0910.0742 [astro-ph.CO]];C. J. A. Martins, E. Menegoni, S. Galli, G. Mangano and A. Melchiorri, Phys. Rev. D , 023532(2010) [arXiv:1001.3418 [astro-ph.CO]].[26] S. Ekstrom, A. Coc, P. Descouvemont, G. Meynet, et al. , Astron. Astrophys. , 62 (2010),[arXiv:0911.2420 [astro-ph.SR]];A. Coc, et al. , Mem. Soc. Astron. Ital. , 658 (2009).[27] V. V. Flambaum, [arXiv:physics/0302015];V. V. Flambaum and A. F. Tedesco, Phys. Rev. C , 055501 (2006), [arXiv:nucl-th/0601050].EFERENCES 32[28] V.V. Flambaum, D.B. Leinweber, A.W. Thomas, and R.D. Young, Phys. Rev. D , 115006 (2004),[arXiv:hep-ph/0402098].[29] S. Bize, P. Laurent, M. Abgrall, H. Marion, et al. , J. Phys. B: At. Mol. Opt. Phys. , S449 (2005),[http://arXiv.org/abs/physics/0502117].[30] T. H. Dinh, A. Dunning, V. A. Dzuba, V. V. Flambaum, Phys. Rev. A79 , 054102 (2009),[arXiv:0903.2090 [physics.atom-ph]].[31] J. C. Berengut, V. V. Flambaum, E. M. Kava, [arXiv:1109.1893 [physics.atom-ph]].[32] J.D. Pretage, R.L. Tjoelker, and L. Maleki, Phys. Rev. Lett. , 3511 (1995).[33] V.A. Dzuba, and V.V. Flambaum, Phys. Rev. A , 034502 (2001).[34] V.A. Dzuba, V.V. Flambaum, and M.V. Marchenko, Phys. Rev. A , 022506 (2003),[http://arXiv.org/abs/physics/0305066].[35] V.A. Dzuba, V.V. Flambaum, and J.K. Webb, Phys. Rev. A , 230 (1999),[http://arXiv.org/abs/physics/9808021].[36] V.V. Flambaum, in Laser Spectroscopy , P. Hannaford, et al.
Eds. (World Scientific, 2004) p. 47,[http://arXiv.org/abs/physics/0309107].[37] M. Fischer, et al. , Phys. Rev. Lett. , 230802 (2004), [http://arXiv.org/abs/physics/0312086].[38] S. Bize, S.A. Diddams, U. Tanaka, C.E. Tanner, et al. , Phys. Rev. Lett. , 150802 (2003), [http://arxiv.org/abs/physics/0212109].[39] T.M. Fortier, N. Ashby, J.C. Bergquist, M.J. Delaney, et al. , Phys. Rev. Lett. , 070801 (2007).[40] E. Peik, B. Lipphardt, H. Schnatz, C. Tamm, et al. , Proc. of the 11th Marcel Grossmann meeting,Berlin, 2006, [http://arXiv.org/abs/physics/0611088].[41] E. Peik, B. Lipphardt, H. Schnatz, T. Schneider, et al. , Phys. Rev. Lett. , 170801 (2004),[http://arXiv.org/abs/physics/0402132].[42] S. Blatt, A.D. Ludlow, G.K. Campbell, J.W. Thomsen, et al. , Phys. Rev. Lett. , 140801 (2008),[http://arXiv.org/abs/0801.1874].[43] V.A. Dzuba, and V.V. Flambaum, Phys. Rev. A , 052515 (2010), [http://arxiv.org/abs/1003.1184].[44] A.T. Nguyen, D. Budker, S.K. Lamoreaux, and J.R. Torgerson, Phys. Rev. A , 022105 (2004),[http://arxiv.org/abs/physics/0308104].[45] A. Cing¨oz, A. Lapierre, A.-T. Nguyen, N. Leefer, et al. , Phys. Rev. Lett. , 040801 (2008),[http://arxiv.org/abs/physics/0609014].[46] T. Rosenband, et al. , Science , 1808 (2008).[47] R.I. Thompson, Astrophys. Lett. , 3 (1975).[48] A. Shelnikov, R.J. Butcher, C. Chardonnet, and A. Amy-Klein, Phys. Rev. Lett. , 150801 (2008),[http://arXiv.org/abs/0803.1829].[49] See, for example, D. Griffiths, Introduction to elementary particles (Weinheim, Germany: Wiley-VCH, 2008).EFERENCES 33[50] H. Fritzsch and G. Eldahoumi, Mod. Phys. Lett. A , 2167 (2009), [arXiv:0906.1139 [hep-ph]].[51] H. Y. Cheng, Phys. Lett. B , 347 (1989).[52] J. R. Ellis, A. Ferstl, and K. A. Olive, Phys. Lett. B , 304 (2000) [arXiv:hep-ph/0001005].[53] J. R. Ellis, K. A. Olive, and C. Savage, Phys. Rev. D , 065026 (2008), [arXiv:0801.3656 [hep-ph]].[54] H. Hellmann, Z. Phys. , 180 (1933);R. P. Feynman, Phys. Rev. , 340 (1939).[55] J. Gasser, Annals Phys. , 62 (1981).[56] R. J. Crewther, Phys. Rev. Lett. , 1421 (1972);M. S. Chanowitz and J. R. Ellis, Phys. Lett. B , 397 (1972);M. S. Chanowitz and J. R. Ellis, Phys. Rev. D , 2490 (1973);S. L. Adler, J. C. Collins, and A. Duncan, Phys. Rev. D , 1712 (1977);J. C. Collins, A. Duncan, and S. D. Joglekar, Phys. Rev. D , 438 (1977);N. K. Nielsen, Nucl. Phys. B (1977) 212.[57] M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, Phys. Lett. B , 443 (1978).[58] L. F. Li and T. P. Cheng, [arXiv:hep-ph/9709293].[59] J. Gasser and H. Leutwyler, Phys. Rept. , 77 (1982).[60] H. Leutwyler, Phys. Lett. B , 313 (1996), [arXiv:hep-ph/9602366].[61] B. Borasoy and U. G. Meissner, Annals Phys. , 192 (1997), [arXiv:hep-ph/9607432].[62] J. Gasser, H. Leutwyler, and M. E. Sainio, Phys. Lett. B , 252 (1991);M. Knecht, PiN Newslett. , 108 (1999), [arXiv:hep-ph/9912443];M. E. Sainio, PiN Newslett. , 138 (2002), [arXiv:hep-ph/0110413].[63] K. Nakamura, et al. , [Particle Data Group], J. Phys. G , 075021 (2010).[64] T. Kunihiro and T. Hatsuda, Phys. Lett. B , 209 (1990);T. Hatsuda and T. Kunihiro, Z. Phys. C (1991) 49;T. Hatsuda and T. Kunihiro, Nucl. Phys. B , 715 (1992).[65] A. De Rujula, H. Georgi, and S. L. Glashow, Phys. Rev. D , 147 (1975).[66] A. Manohar and H. Georgi, Nucl. Phys. B , 189 (1984).[67] L. Y. Glozman and D. O. Riska, Phys. Rept. , 263 (1996), [arXiv:hep-ph/9505422].[68] N. Isgur and G. Karl, Phys. Rev. D , 4187 (1978);N. Isgur and G. Karl, Phys. Rev. D , 2653 (1979) [Erratum-ibid. D , 817 (1981)].[69] J. T. Goldman and R. W. Haymaker, Phys. Rev. D , 724 (1981);M. K. Volkov, Annals Phys. (1984) 282;D. Ebert and H. Reinhardt, Nucl. Phys. B (1986) 188.[70] Y. Nambu and G. Jona-Lasinio, Phys. Rev. , 345 (1961);Y. Nambu and G. Jona-Lasinio, Phys. Rev. , 246 (1961).EFERENCES 34[71] S. Klimt, M. F. M. Lutz, U. Vogl, and W. Weise, Nucl. Phys. A , 429 (1990);U. Vogl, M. F. M. Lutz, S. Klimt, and W. Weise, Nucl. Phys. A , 469 (1990).[72] H. Georgi and A. Manohar, Phys. Lett. B , 183 (1983).[73] S. Weinberg, Physica A , 327 (1979);J. Gasser and H. Leutwyler, Annals Phys. , 142 (1984);J. Gasser and H. Leutwyler, Nucl. Phys. B , 465 (1985);J. Gasser, M. E. Sainio, and A. Svarc, Nucl. Phys. B , 779 (1988);A. Krause, Helv. Phys. Acta (1990) 3.[74] For a pedagogical review, see, for example, S. Scherer, Adv. Nucl. Phys. , 277 (2003),[arXiv:hep-ph/0210398].[75] V. Bernard, Prog. Part. Nucl. Phys. , 82 (2008), [arXiv:0706.0312 [hep-ph]].[76] H. Georgi and L. Randall, Nucl. Phys. B , 241 (1986);H. Georgi, Phys. Lett. B , 187 (1993), [arXiv:hep-ph/9207278].[77] E. E. Jenkins and A. V. Manohar, Phys. Lett. B , 558 (1991).[78] E. E. Jenkins, M. E. Luke, A. V. Manohar, and M. J. Savage, Phys. Lett. B , 482 (1993)[Erratum-ibid. B , 866 (1996)], [arXiv:hep-ph/9212226].[79] U. G. Meissner and S. Steininger, Nucl. Phys. B , 349 (1997), [arXiv:hep-ph/9701260].[80] L. Durand and P. Ha, Phys. Rev. D , 013010 (1998), [arXiv:hep-ph/9712492].[81] S. J. Puglia and M. J. Ramsey-Musolf, Phys. Rev. D , 034010 (2000), [arXiv:hep-ph/9911542].[82] T. Fuchs, J. Gegelia, G. Japaridze, and S. Scherer, Phys. Rev. D , 056005 (2003), [arXiv:hep-ph/0302117].[83] L. S. Geng, J. Martin Camalich, L. Alvarez-Ruso, and M. J. Vicente Vacas, Phys. Rev. Lett. ,222002 (2008), [arXiv:0805.1419 [hep-ph]].[84] L. S. Geng, J. Martin Camalich, and M. J. Vicente Vacas, Phys. Lett. B , 63 (2009),[arXiv:0903.0779 [hep-ph]].[85] D. B. Leinweber, D. H. Lu, and A. W. Thomas, Phys. Rev. D , 034014 (1999), [arXiv:hep-lat/9810005];E. J. Hackett-Jones, D. B. Leinweber and A. W. Thomas, Phys. Lett. B , 143 (2000), [arXiv:hep-lat/0004006].[86] J. F. Donoghue, B. R. Holstein, and B. Borasoy, Phys. Rev. D , 036002 (1999) [arXiv:hep-ph/9804281];D. B. Leinweber, A. W. Thomas, and R. D. Young, Phys. Rev. Lett. , 242002 (2004), [arXiv:hep-lat/0302020].[87] R. Srianand, N. Gupta, P. Petitjean, P. Noterdaeme, et al. , Mon. Not. R. Astron. Soc. , 1888(2010), [arXiv:1002.4620].[88] M.T. Murphy, J.K. Webb, V.V. Flambaum, M.J. Drinkwater, et al. , Mon. Not. R. Astron. Soc. ,1244 (2001), [astro-ph/0101519].EFERENCES 35[89] N. Kanekar, C.L. Carilli, G.I. Langston, G. Rocha, et al. , Phys. Rev. Lett. , 261301 (2005),[astro-ph/0510760].[90] J. Darling, Astrophys. J. , 58 (2004), [astro-ph/0405240].[91] J.N. Chengalur, and N. Kanekar, Phys. Rev. Lett. , 241302 (2003), [astro-ph/0310764].[92] N. Kanekar, J.N. Chengalur, and T. Ghosh, Phys. Rev. Lett. , 051302 (2004), [arXiv:astro-ph/0406121].[93] N. Kanekar, et al. , Astrophys. J. , L23 (2010).[94] N. Kanekar, Mod. Phys. Lett. A23