Enhanced effect of quark mass variation in 229Th and limits from Oklo data
aa r X i v : . [ nu c l - t h ] O c t Enhanced effect of quark mass variation in
Th and limits from Oklo data
V. V. Flambaum , , and R. B. Wiringa Argonne Fellow, Physics Division, Argonne National Laboratory, Argonne, Illinois 60439 Physics Division, Argonne National Laboratory, Argonne, Illinois 60439 School of Physics, University of New South Wales, Sydney 2052, Australia Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada (Dated: November 2, 2018)The effects of the variation of the dimensionless strong interaction parameter X q = m q / Λ QCD ( m q is the quark mass, Λ QCD is the QCD scale) are enhanced about 1 . · times in the 7.6 eV“nuclear clock” transition between the ground and first excited states in the Th nucleus andabout 1 · times in the relative shift of the 0.1 eV compound resonance in Sm. The bestterrestrial limit on the temporal variation of the fundamental constants, | δX q /X q | < · − at1.8 billion years ago ( | ˙ X q /X q | < . · − y − ) , is obtained from the shift of this Sm resonancederived from the Oklo natural nuclear reactor data. The results for Th and
Sm are obtained byextrapolation from light nuclei where the many-body calculations can be performed more accurately.The errors produced by such extrapolation may be smaller than the errors of direct calculations inheavy nuclei. The extrapolation results are compared with the “direct” estimates obtained using theWalecka model. A number of numerical relations needed for the calculations of the variation effectsin nuclear physics and atomic spectroscopy have been obtained: for the nuclear binding energy δE/E ≈ − . δm q /m q , for the spin-orbit intervals δE so /E so ≈ − . δm q /m q , for the nuclearradius δr/r ≈ . δm q /m q (in units of Λ QCD ); for the shifts of nuclear resonances and weakly boundenergy levels δE r ≈ δX q /X q MeV.
PACS numbers: PACS: 06.20.Jr, 42.62.Fi, 23.20.-g
INTRODUCTION
Unification theories applied to cosmology suggest thepossibility of variation of the fundamental constants inthe expanding Universe (see, e.g., the review [1]). A re-view of recent results can be found, e.g., in Ref. [2]. InRef. [3] it was suggested that there may be a five ordersof magnitude enhancement of the variation effects in thelow-energy transition between the ground and the firstexcited states in the
Th nucleus. This transition wassuggested as a possible nuclear clock in Ref. [4]. Indeed,the transition is very narrow. The width of the excitedstate is estimated to be about 10 − Hz [5]. The latestmeasurement of the transition energy [6] gives 7 . ± . . ± . ± − . Several experimental groups havealready started working on this possibility [9]. However,a recent paper [10] claims that there is not any enhance-ment of the effects of the variation of the fundamentalconstants in this transition. The main aim of the presentnote is to demonstrate that the enhancement exists. Wealso estimate the relative shift of the 0.1 eV compoundresonance in Sm to obtain new limits on the varia-tion of the fundamental constants from the Oklo naturalnuclear reactor data [11, 12, 13].We can measure only the variation of dimensionless pa-rameters which do not depend on which units we use. Inthe Standard Model, the two most important dimension-less parameters are the fine structure constant α = e / ¯ hc and the ratio of the electroweak unification scale deter-mined by the Higgs vacuum expectation value (VEV) tothe quantum chromodynamics (QCD) scale Λ QCD (de-fined as the position of the Landau pole in the loga-rithm for the running strong coupling constant, α s ( r ) ∼ constant / ln (Λ QCD r/ ¯ hc )). The variation of the HiggsVEV leads to the variation of the fundamental masseswhich are proportional to the Higgs VEV. The presentwork considers mainly effects produced by the variationof X q = m q / Λ QCD where m q = ( m u + m d ) / X q may be 1–2 orders of mag-nitude larger than the variation of α [14]. Note that inthe present work we do not consider effects of variationof the strange quark mass since they have larger uncer-tainty and should be treated separately. These effectswere estimated in Refs. [3, 15].The results depend on the dimensionless parameter X q = m q / Λ QCD . In all calculations it is convenient toassume that Λ
QCD is constant and calculate the depen-dence on the small parameter m q . In other words, wemeasure all masses and energies in units of Λ QCD andwill simply restore Λ
QCD in the final results. Note thatwhen a relative effect of the variation is enhanced it doesnot matter what units we use. The variation of the ratioof different units may be neglected anyway.
THORIUM
To explain the origin of the enhancement we shouldpresent the small 7.6 eV interval between the groundand excited states in the
Th nucleus as a sum of afew components which nearly cancel each other and havedifferent dependence on the fundamental constants. Ifone performs the calculations exactly, it does not mat-ter how we select these components. However, in prac-tice the calculations are always approximate, therefore,a reasonable selection of the components will determineour final accuracy. For example, to study dependenceon α we should separate the Coulomb energy from theremaining contributions to the energy. To study depen-dence on X q = m q / Λ QCD it is convenient to separate outthe spin-orbit interaction energy: ω = E b + E so = 7 . . (1)Here E b is the difference in bulk binding energies of theexcited and ground states (including kinetic and poten-tial energy but excluding the spin-orbit interaction) and E so is the difference in the spin-orbit interaction ener-gies V ls h l · s i in the excited and ground states. We makethis separation because we expect E b and E so to havea very different dependence on X q = m q / Λ QCD , as dis-cussed below. In
Th the strength of the spin-orbitinteraction is estimated to be V ls = − .
85 MeV from Ta-ble 5-1 of Ref. [16]. The difference of h l · s i between theexcited and ground states can be easily calculated usingthe expansion of the wave functions over Nilsson orbitalspresented in Table 4 of Ref. [17]: E so ≈ . V ls ≈ − . / + and ground state[633]5 / + , E so = 2 V ls .) Then Eq. (1) gives us E b ≈− E so ≈ δωω ≈ E so ω ( δE so E so − δE b E b ) = 1 . · ( δE so E so − δE b E b ) . (2)Qualitatively, we expect E b and E so to have a ratherdifferent dependence on X q . In the Walecka model(which was used in Ref. [3] to estimate the enhance-ment factor) there is a significant cancellation betweenthe σ and ω meson contributions to the mean-field po-tential and the total binding energy E , while the σ and ω mesons contribute with equal sign to the spin-orbit interaction constant V ls [18]. A similar argumentmay be made from the variational Monte Carlo (VMC)calculations with realistic interactions used in Ref. [19]to evaluate binding energy dependence on X q . Thesecalculations use nucleon-nucleon potentials that fit NN scattering data together with three-nucleon potentialsthat reproduce the binding energies of light nuclei, Thebinding energy is the result of a significant cancellationbetween intermediate-range attraction due to two-pionexchange and short-range repulsion arising from heavy vector-meson exchange. However, spin-orbit splitting be-tween nuclear levels has been found to be a coherentaddition of short-range two-nucleon l · s interaction andmultiple-pion exchange between three or more nucleons[20]. Thus if meson masses move in the same directiondue to an underlying quark mass shift, contributions frompion exchange and heavy vector-meson exchange will can-cel against each other in the binding energy, but reinforceeach other in spin-orbit splittings. Binding energies
The binding energy per nucleon and the spin-orbit in-teraction constant have a slow dependence on the nucleonnumber A . The total binding is dominated by the bulkterms, so we make the reasonable assumption that thevariation of the bulk energy with X q is the same for thetwo levels in Th and thus the variation of the difference δE b /E b ≈ δE/E . Moreover, the common factors (like A − / in the spin-orbit constant V ls [16]) cancel out inthe relative variations δE so /E so and δE b /E b . Therefore,it may be plausible to extract these relative variationsfrom the type of calculations in light nuclei performedin Ref. [19]. The advantage of the light nuclei is thatthe calculations can be performed quite accurately, in-cluding different many-body effects. Their accuracy hasbeen tested by comparison with the experimental datafor the binding energies and by comparison of the resultsobtained using several sophisticated interactions (AV14,AV28, AV18+UIX – see [19]). As the first step, the varia-tions of the nuclear binding energies have been expressedin terms of the variations of nucleon, ∆, pion and vector-meson masses. The dependence of these masses on quarkmasses have been taken from Refs. [21, 22]. The resultsfor the relative variations of the total binding energiesare presented in Table I (in the present work we add He, He, and Be to this table). We see that all the resultsare close to the average value δE/E ≈ − . δX q /X q .The maximal deviations are for He, which is especiallytightly bound, and for He, which is a resonant state.
Spin-orbit intervals
To find the dependence of the spin-orbit constant V ls on m q / Λ QCD we calculate the spin-orbit splitting be-tween the p / and p / levels in He, He, Li, and Be in the present work. We use the Argonne v two-nucleon and Urbana IX three-nucleon (AV18+UIX) in-teraction which provides our best results for small nuclei(see Ref. [19] for details and references). In all calcula-tions it is convenient to keep Λ QCD = constant, i.e., mea-sure the quark mass m q in units of Λ QCD . We restoreΛ
QCD in the final answers. As the first step we calcu-late the binding energies of the ground and excited states
TABLE I: Dimensionless derivatives K = δE/EδX q /X q of the binding energy over X q = m q / Λ QCD . H H He He He He Li He Li Be Be Be − . − . − . − . − . − . − . − . − . − . − . − . shown in Table II and their dependence on the nucleon,∆, pion, and vector-meson masses, ∆ E ( m H ) = δE/Eδm H /m H ,shown in Table III. To find the dependence of these ener-gies on the quark mass, we utilize the results of a Dyson-Schwinger equation (DSE) study of sigma terms in light-quark hadrons [21]. Equations (85-86) of that work givethe rate of hadron mass variation as a function of theaverage light current-quark mass m q = ( m u + m d ) / δm H m H = σ H m H δm q m q , (3)with σ H /m H values of 0.498 for the pion, 0.030 for the ρ -meson, 0.043 for the ω -meson, 0.064 for the nucleon,and 0.041 for the ∆. The values for the ρ and ω -mesonswere reduced to 0.021 and 0.034 in subsequent work [22].We use an average of the ρ and ω terms of 0.030 for ourshort-range mass parameter m V .It is convenient to present the result for the variationof the spin-orbit splitting in the following form: δE so = δE / − δE / = E / δE / E / − E / δE / E / . (4)Accidentally, the calculated spin-orbit constant in He isthe same as in
Th, V ls = − .
83 MeV (the p / - p / splitting in He is 1 . V ls ). The spin-orbit constant in Be is larger than in
Th, in accord with the expecteddependence A − / (see e.g. Ref. [16]). The spin-orbitinterval sensitivity coefficients K so defined from δE so E so = K so δm q m q (5)for the quark mass variation in He, He, Li, and Beare − . − . − .
58, and − .
22, respectively. The He, He, and Be values are all very similar, as all thesenuclei are essentially one nucleon outside a 0 + core. The Li value is anomalously large because its ground andfirst excited states are primarily a triton outside an al-pha core, so although δE so is comparable to Be, E so is very small and not typical of the single-particle spin-orbit interaction we seek. Excluding the Li result givesus an average value of K so = − .
22 to use in
Th. Notethat the estimate based on the Walecka model, outlinedin Sec. V below, gives a very similar value K so = − . Frequency shift
Substituting δE so /E so = − . δX q /X q and δE b /E b = − . δX q /X q into Eq. (2) we obtain the fol- lowing energy shift for the 7.6 eV transition in Th: δω = 1 . δX q X q MeV . (6)This corresponds to the frequency shift δν = 3 · δX q /X q Hz. The width of this transition is 10 − Hzso one may hope to get the sensitivity to the variationof X q about 10 − per year. This is 10 times betterthan the current atomic clock limit on the variation of X q , ∼ − per year (see e.g. Ref. [2]).The corresponding relative energy shift is δωω = 1 . · δX q X q . (7)This enhancement coefficient may be compared with thecoefficient 0 . · from Ref. [3] and 0 . · from Ref. [23].The calculations in Ref. [23] have been done using therelativistic mean field theory (extended Walecka model)and some basic ideas from Ref. [3]. Thus, in this work weobtain an even larger enhancement! Here we present therelative variations from Refs. [3, 23] for the new measuredvalue 7.6 eV of the frequency ω (the old value was 3.5eV, and we multiplied the numbers from [3, 23] by (3.5eV)/ ω ). The difference between the results of differentapproaches looks pretty large. However, this is only areflection of the current accuracy of all three calculations.The present aim is to show that the enhancement doesexist.Note that because of the huge enhancement it does notmatter what units one will use to measure the frequency ω . In the calculations above we assumed that ω is mea-sured in units of Λ QCD . However, the variation of theratio of any popular frequency standard to Λ
QCD doesnot have such enhancement and may be neglected.
Coulomb energy and effect of α variation We also would like to comment about the possible en-hancement of α variation. Ref. [10] claims that this en-hancement is impossible since the ground and excitedstates differ in the neutron state only and the neutronis neutral. Therefore, the ground and excited stateshave the same Coulomb energy and the interval does notchange when α varies. We do not agree with this conclu-sion. Indeed, the total Coulomb energy of the Th nu-cleus is 900 MeV (see, e.g., [16]) which is 10 time largerthan the energy difference ω =7.6 eV. Therefore, to have TABLE II: Experimental and calculated energies for the ground ( p / ) and first excited ( p / ) states of A =5,7,9 nuclei in MeV. He( − ) He*( − ) He( − ) He*( − ) Li( − ) Li*( − ) Be( − ) Be*( − )AV18+UIX − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . E ( m H ) = δE/Eδm H /m H of the binding energy to the different hadron masses and thesensitivity K after folding in the DSE values of δm H /m H . He( − ) He*( − ) He( − ) He*( − ) Li( − ) Li*( − ) Be( − ) Be*( − ) m N + δ N δ ∆ − . − . − . − . − . − . − . − . m π (+TNI) − . − . − . − . − . − . − . − . m V K = δE/Eδm q /m q − . − . − . − . − . − . − . − . an enhancement it is enough to change the proton den-sity distribution (deformation parameter) by more than10 − . Any change in the neutron state influences the nu-clear mean field and proton distribution. For example,neutron removal changes the Coulomb energy of Thby 1.3 MeV [16]. This gives us an upper estimate (anda natural scale) for the change of the Coulomb energy inthe 7.6 eV
Th transition. One should expect a frac-tion of MeV change in any neutron transition in heavynuclei. According to [17] the weight of admixed octupolevibrations to the 7.6 eV state exceeds 20%. Octupole vi-brations involve both protons and neutrons. Therefore,the proton density distribution in the excited state is dif-ferent from the ground state and this difference is only anorder of magnitude smaller than the difference in neutrondistribution.The existence of the enhancement was confirmed bythe direct calculation in Ref. [23]. The authors performedthe calculation of the change of the nuclear mean fieldacting on neutrons induced by the change of the pro-ton density due to the variation of α . They obtainedthe enhancement coefficient 4 · . This corresponds tothe Coulomb energy difference 0.03 MeV. As it was men-tioned above, an additional enhancement may come fromthe change of the nuclear deformation. Anyway, there isno doubt that the enhancement of the sensitivity to α variation in Th does exist.
SHIFT OF THE RESONANCE IN SAMARIUMAND LIMITS FROM THE OKLO NATURALNUCLEAR REACTOR DATA
In Refs. [24, 25] we derived a simple formula to esti-mate the shift of the resonance or weakly bound energylevel due to the variation of the fundamental constants.Let us assume a Fermi gas model in a square well nuclearpotential of the radius R and depth V . The energy of a single-particle energy level or resonance is determined as E r ≈ h p m i − V . (8)The momentum p in the square well is quantized, p ≈ constant /R . Therefore, E r = K mR − V . (9)For a resonance or a weakly bound level E r ≈ E r ≪ V ) and the constant K ≈ mR V . Then we have δE r = − K mR ( δmm + 2 δRR ) − δV ≈ − V ( δmm + 2 δRR + δV V ) . (10)This equation is also valid for a compound state withseveral excited particles. Indeed, the position of the com-pound state or resonance relative to the bottom of thepotential well is determined mainly by the kinetic energywhich scales as 1 /R (both the Fermi energy and sumof the single-particle excitation energies scale this way).The shift of the resonance due to the residual interac-tion between excited particles ( ∼ . V ≈ V is approximately the same in light and heavynuclei. The radius of the well R ≈ . A / r , therefore,the relative variation δR/R = δr /r is the same too.Thus, the resulting shift of the resonance both in lightand heavy nuclei is given by Eq. (10) and we may extrap-olate the accurate result for light nuclei to the resonancein Sm.In Table IV we present binding energies of the valencenucleon, S = − E (in MeV), and shift of the energy level(resonance), δEδm q /m q = − δSδm q /m q , due to the variation ofthe quark mass (in units MeV δX q /X q ) in light nucleiwith A = 5 , , , ,
9. In the derivation of Eq. (10) it wasassumed that the valence nucleon is localized inside the
TABLE IV: He He Li He Li Be Be Be S expt − .
89 1.86 4.59 − .
43 7.25 5.61 18.90 1.67 S calc − . − .
30 2.96 − .
19 5.11 3.52 16.89 − . − δS expt δm q /m q − δS calc δm q /m q potential well. This is not the case for He where thevalence nucleon is localized mainly outside the narrowpotential well produced by the He core. As a result thepotential < V > averaged over the valence neutron wavefunction 1 p / is significantly smaller than the depth ofthe potential V . This explains why the shift in He (pro-portional to < V > - see Eq. (10) and Ref. [25]) is muchsmaller than the shift in other nuclei. Another extremecase is Be where | E r | is too large and the condition E r ≪ V is not fulfilled. The results for other nuclei arereasonably close to the average value δE r ≈ δX q X q MeV . (11)We assume this shift for the 0.1 eV resonance in Sm.This value does not contradict the order-of-magnitudeestimates in Refs. [2, 24, 25]. Finally, we can add tothis shift the contribution of α variation from Refs. [26]( δE r = − . ± . δα/α ). The total shift of theresonance in Sm is δE r = 10 ( δX q X q − . δαα ) MeV . (12)Now we can can extract limits on the variation of X q from the measurements of δE r . Pioneering work in thisarea was done in Ref. [26]. We will use recent measure-ments [11, 12, 13] where the accuracy is higher. Ref. [13]has given | δE r | <
20 meV. Then Eq. (12) gives | δX q X q − . δαα | < · − . (13)Ref. [12] has given -73 < δE r <
62 meV. This gives | δX q X q − . δαα | < · − . (14)Ref. [11] has given -11.6 < δE r < | δX q X q − . δαα | < . · − . (15)The limits on δE r have been been presented with 2 σ range. Note that Ref. [11] has presented also the second,non-zero solution (it exists since the resonance has twotails): -101.9 < δE r < -79.6 meV. However, Ref. [13]tentatively ruled out this solution based on the data forthe shift of a similar resonance in the Gd nucleus. Based on the results above we conclude that | δX q X q | < · − (for simplicity, we omit the small contribution of α variation here). Assuming linear time dependence duringthe last 1.8 billion years we obtain the best terrestriallimit on the variation of the fundamental constants | ˙ X q X q | < . · − y − . (16) VARIATION OF NUCLEAR RADIUS
Variation of the nuclear radius is needed to calculate ef-fects of the fundamental constant variation in microwaveatomic clocks where the transition frequency depends ona probability of the electron to be inside the nucleus. In-deed, the hyperfine interaction constant in heavy atomshas some sensitivity to the nuclear radius (including theCs hyperfine transition which defines the unit of time, thesecond, and is used as a reference in numerous atomic andmolecular clock experiments). This dependence was alsorequested by S. Schiller who proposed new experimentswith hydrogen-like ions to search for the variation of thefundamental constants [27].In Table V we present a comparison of calculated andmeasured charge nuclear radii for the stable A = 2 , A ≥ δm q = 0, and then not allowed to varyas the energy was evaluated for different δm H . Conse-quently the “size” of the trial wave function was essen-tially unchanged. For the radius determination, we mustallow this size to vary. We do this by multiplying a set ofvariational parameters (those to which the radius is mostsensitive) by a scale factor, and then carefully reminimizethis scale factor for each δm H . This allows us to deter-mine ∆ r ( m H ) = δr/rδm H /m H . The ∆ E ( m H ) = δE/Eδm H /m H reported in Ref. [19] are unchanged in this new mini-mization. TABLE V: Experimental and calculated point proton rmsradii for stable A = 2 − H H He He He Li Li Be BeAV18+UIX 1.967 1.58 1.77 1.45 1.92 2.46 2.34 2.45 2.40Expt. 1.953 1.59 1.75 1.45 1.93 2.39 2.25 2.38
This procedure works well for the A = 3 , r ( m H ) is presented in Table VI, along with the to-tal sensitivity to the quark mass, K r ≡ δr/rδm q /m q , obtainedby folding in the DSE values for δm H /m H . However, be-cause our trial functions for A = 6 − Li is asymptotically an alpha and adeuteron bound by 1.47 MeV, Li is asymptotically analpha and a triton bound by 2.47 MeV, and Be is asymp-totically an alpha and a He bound by 1.59 MeV. ( Heis asymptotically a three-body α + n + n cluster and Beis an α + α + n cluster, so they cannot be treated thisway.) For a quark mass shift δm q /m q = ± .
01, we knowthe total energy shift expected from our previous calcula-tions. We subtract that portion attributable to the alphaand deuteron or trinucleon subclusters, and use the re-maining energy shift to adjust the asymptotic separationenergy of our trial function. This allows the size of boththe subclusters and the well binding them to vary. For A = 6 ,
7, we have carried out this calculation for the totalsensitivity K r only, and not for the individual ∆ r ( m H );these results are also given in Table VI. TABLE VI: Dimensionless derivatives of point proton rmsradii ∆ r ( m H ) = δr/rδm H /m H and the sensitivity with respect to m q after folding in the DSE values of δm H /m H . H H He He Li Li Be m N + δ N − . − . − . − . δ ∆ m π (+TNI) 2.57 1.80 1.77 1.11 m V − . − . − . − . δr/rδm q /m q The average value of K r is about 0.3, which may serveas an estimate of the sensitivity for all nuclei. There aresignificant deviations from this value for the very weaklybound deuteron H and very strongly bound He; thelatter is probably a solid lower bound.The dependence of the nuclear radius on fundamen-tal constants manifests itself in microwave transitions inatomic clocks which are used to search for the variationof the fundamental constants (see e.g. Refs. [2, 27]). Thedependence of the hyperfine transition frequency ω h onnuclear radius r in atoms with an external s -wave elec-tron is approximately given by the following expressions(in units of Λ QCD ): δω h ω h = K hr δrr = K hr K r δm q m q ≈ . K hr δm q m q , (17) K hr ≈ − (2 γ − δ h − δ h , (18) δ h ≈ · − Z / ) γ − , (19)where γ = (1 − Z α ) / . For the Cs atom microwavestandard the nuclear charge Z = 55 and K hr = − . + microwave clock Z = 80 and K hr = − . He radiuson α : δr/rδα/α = 0 . α variation should be larger. ESTIMATES IN WALECKA MODEL
It is instructive to compare the results obtained bythe extrapolation from light nuclei with some “direct”calculations. In this section we estimate the variations ofthe resonance positions and spin-orbit splittings in heavynuclei using the Walecka model [28] where the strongnuclear potential is produced by scalar and vector mesonexchanges: V = − g s π e − rm S r + g v π e − rm V r . (20)Averaging Eq. (20) over the nuclear volume we can findthe depth of the potential well [24] V = 34 πr (cid:18) g s m S − g v m V (cid:19) . (21)Here 2 r = 2 . δV V ≈ − . δm S m S + 5 . δm V m V − δr r . (22)Here we have used g s m S / g v m V = 266 . / . .
364 fromRef. [18]. There is an order of magnitude enhancement ofthe meson mass variation contributions due to the can-cellation of the vector and scalar contributions in the de-nominator V . Eq. (10) for the variation of the resonanceposition becomes δE r ≈ V (7 . δm S m S − . δm V m V − δm N m N + δr r ) . (23)We do not know the variation of r in the Walecka model,therefore, to make a rough numerical estimate we ne-glect this term. As above we take dependence of the nu-cleon and meson masses on the current light quark mass m q = ( m u + m d ) / δm ω m ω = 0 . δm q m q , δm N m N = 0 . δm q m q , δm σ m σ = 0 . δm q m q , δm π m π = 0 . δm q m q .The vector meson in the Walecka model is usually iden-tified with the ω -meson so δm V m V = 0 . δm q m q . The scalarmeson exchange, in fact, imitates both the σ meson ex-change and two-pion exchange. Even if we neglect thetwo-pion exchange in zero approximation, there is vir-tual σ decay to two π . These virtual decays (loops on σ line in the NN-interaction diagrams with intermediate σ ) very strongly modify the σ propagator and change itslarge distance asymptotics from e − m σ r to e − m π r [29].The mixing between σ and two π in m S should increasethe sensitivity coefficient for the variation of m S . For anestimate we take an intermediate value between the neu-tron and vector meson mass sensitivity, δm S m S ∼ . δm q m q .(Note that the positive contribution of δr r in Eq. (23)produces an effect similar to that of an increase of δm S m S .)Then Eq. (23) gives δE r ∼ δX q X q MeV . (24)This rough estimate agrees with the result extrapolatedfrom light nuclei. Note, however, that the accuracy ofthis estimate is very low due to the cancellations of dif-ferent terms.The scalar and vector mesons contribute with equalsign to the spin-orbit interaction constant V ls [18]. Also,the spin-orbit interaction is inversely proportional to thenucleon mass m N squared. Thus, we have V ls ∝ m N (cid:18) g s m S + g v m V (cid:19) , (25) δE so E so = − δm N m N + 0 . δm S m S + 0 . δm V m V ) ≈ − . δm q m q . (26)This estimate is close to the result ( − . δm q m q ) obtainedby the extrapolation from light nuclei. Note, however,that here we neglected the effect of variation of r whichprobably should increase the absolute value of the sensi-tivity coefficient. CONCLUSION
At the moment one can hardly calculate the sensitivitycoefficient for the dependence of the strong interaction onthe quark mass m q with an accuracy better than a fac-tor of 2. Moreover, it is hard to identify this dependencein phenomenological interactions which are used for thecalculations in heavy nuclei. For example, it is not ob-vious that the scalar and vector mesons in the Waleckamodel are actually equivalent to free σ and ω mesons inparticle physics. Therefore, to test conclusions obtainedusing the Walecka model, we explored a complementaryapproach. We performed the calculations in light nucleiwhere the interactions are well-known and the accuracyof the calculations is high. The binding energy per nu-cleon E b , the spin-orbit interaction constant V ls and thenuclear radius r have a slow dependence as a function ofthe nucleon number A . Moreover, the common factors(like A − / in the spin-orbit constant V ls and A / in thenuclear radius) cancel out in the relative variations δr/r , δV ls /V ls and δE b /E b . Therefore, we can extract these rel-ative variations from the calculations in light nuclei anduse them in heavy nuclei. The errors produced by suchextrapolation may be smaller than the errors of directcalculations in heavy nuclei. So far, this extrapolationand direct calculations using Walecka model give com-parable values of the enhancement factors in Th and
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