Effect of quark-mass variation on big bang nucleosynthesis
aa r X i v : . [ nu c l - t h ] J u l Effect of quark-mass variation on big bang nucleosynthesis
J. C. Berengut and V. V. Flambaum
School of Physics, University of New South Wales, Sydney 2052, Australia
V. F. Dmitriev
Budker Institute of Nuclear Physics, 630090, Novosibirsk-90, Russia (Dated: 14 July 2009)We calculate the effect of variation in the light-current quark mass, m q , on standard big bangnucleosynthesis. A change in m q at during the era of nucleosynthesis affects nuclear reaction rates,and hence primordial abundances, via changes the binding energies of light nuclei. It is foundthat a relative variation of δm q /m q = 0 . ± .
005 provides better agreement between observedprimordial abundances and those predicted by theory. This is largely due to resolution of the existingdiscrepancies for Li. However this method ignores possible changes in the position of resonances innuclear reactions. The predicted Li abundance has a strong dependence on the cross-section of theresonant reactions He ( d, p ) He and t ( d, n ) He. We show that changes in m q at the time of BBNcould shift the position of these resonances away from the Gamow window and lead to an increasedproduction of Li, exacerbating the lithium problem.
I. INTRODUCTION
Measurements of the primordial baryon-to-photon ra-tio η from the cosmic microwave background fromWMAP [1], coupled with precise measurements of theneutron half-life [2], have made big bang nucleosynthesis(BBN) an essentially parameter-free theory [2, 3, 4]. Inthis paradigm excellent agreement has been obtained be-tween predicted and observed abundances of deuteriumand He (see, e.g. the Particle Data Group review [2] andreferences therin). However there is some disagreementfor Li, the only other element for which the abundancehas been measured to an accuracy at which fruitful com-parison with theory can be made. While the “lithiumproblem” has been known for some time, it has been ex-acerbated by recent measurements of the He( α, γ ) Bereaction [5]. Standard BBN theory with η providedby WMAP 5 overproduces Li by a factor of 2.4 – 4.3(around 4 – 5 σ ) [4].One possible solution to the lithium problem is thatthe physical constants of the early Universe may havebeen slightly different. In fact, such variations in thephysical laws can be well-motivated theoretically in anexpanding Universe; see [6] for a review. Ref. [7] consid-ered variation of the deuterium binding energy B d duringprimordial nucleosynthesis. BBN has a high sensitivityto B d since its value determines the temperature at whichdeuterium can withstand photo-disintegration and hencethe time at which nucleosynthesis begins. Their best-fit result ∆ B d /B d = − . ± .
005 resolved then-extantdiscrepancies between theory and observation in both Liand η (or alternatively, in Li and He with η fixed byWMAP).More recently, Ref. [8] examined the response of BBNto variation of several physical parameters, includingbinding energies, in a linear approximation. These werecoupled with calculated dependences of binding energieson m q in [9], which found that the Li abundance dis-crepancy could be resolved by a variation in light-quark mass of δm q /m q = 0 . ± . He and d abundances were found to be relatively insensitive to m q and so the existing agreement between theory andobservation in these elements was maintained.In this paper we re-examine the dependence of light-element production on variation of the dimensionless pa-rameter X q = m q / Λ QCD where m q is the light-quarkmass and Λ QCD is the pole in the running strong-coupling constant. We follow [9] and assume that Λ
QCD is constant, calculating the dependence on the small pa-rameter m q . This is not an approximation. Rather itonly means that we measure all dimensions ( m q , crosssections, etc) in units of Λ QCD . Therefore δm q /m q should be understood as δX q /X q . We take into accountseveral effects that were not previously considered, mostimportantly the nonlinear dependence on m q and varia-tion of resonance positions. II. VARIATION OF BINDING ENERGIES
The energy released in each reaction, Q , is determinedby the masses of the reactants and products, which inturn are determined by the nuclear binding energies. Asnoted in [8], the Q -values affect the forward (exothermic)reaction rates via phase space and radiative emission fac-tors. For radiative capture reactions at low energy E the Q -dependence is σ ( E ) ∝ E γ ∼ ( Q + E ) . (1)For low-energy reactions with two nucleons in the exitchannel the dependence is proportional to the outgoingchannel velocity, v ∼ ( Q + E ) / . When the outgoing par-ticles are charged, the Gamow factor of the exit channelcan also contribute: σ ( E ) ∼ ( Q + E ) / e − √ E g / ( Q + E ) . (2)The Gamow factor appears because of the Coulomb bar-rier to the reaction; E g = 2 π Z Z α µc where α is the TABLE I: ∂ ln Y a /∂ ln B D , the dependence of nuclear abun-dances, Y a , on deuterium binding energy under differentassumptions:1. Variation of virtual level not considered, h σv i ∼ Q / . Q changed only for p ( n, γ ) d .2. Variation of virtual level not considered; effect of B d included in all reactions (similar to theory of [8]).3. p ( n, γ ) d changed according to (5), including variationof the virtual level; effect of B d on other reactions ignored(similar to theory of [7]).4. p ( n, γ ) d changed according to (5); effect of B d included inall reactions.Method d He He Li Li1. − . − .
75 0 . − .
17 10 . − . − .
08 0 . − .
58 9 . − . − .
29 0 . − .
23 17 . − . − .
62 0 . − .
64 16 . fine-structure constant, Z and Z are the charge num-bers of the products, and µ is the reduced mass of theproducts. At BBN temperatures we can usually assumethat E ≪ Q . Expanding in Q , σ = σ " s E g Q ! δQQ + ... (3)and we see that the Gamow term in (2) is generallysmall (it was neglected in [8]). However it can be im-portant for some reactions, for example in Be ( n, p ) Li, p E g /Q = 2 .
17, i.e. it triples the effect of δQ on thereaction rate.The reverse reaction rates are simply related to the for-ward rates via statistical factors. From detailed balanceone finds h σv i rev h σv i fwd ∼ e − Q / T (4)and we see that the reverse reactions also provide sensi-tivity to Q .An exception to the rule (1) is found in the reaction p ( n, γ ) d , an important reaction because d is a precursorto all further nucleosynthesis. This reaction is sensitivenot only to Q but also to the position of the virtual levelwith energy ǫ ν = 0 .
07 MeV. The sensitivity of this reac-tion to Q was calculated in [7] h σv i ∼ " / r Qǫ ν ! δQQ . (5)Note that [8, 9] did not take variation of the virtual levelinto account. In Table I we show the linear dependence ofabundances on the deuterium binding energy with differ-ent theories of variation. It shows the effect of variationof the virtual level, as well as the effect of including B d variation on other Q -values and reaction rates. We denote the sensitivity of nuclear binding energiesto the light-current quark mass m q by K = δE/Eδm q /m q . (6)Values of K for several light nuclei were presented inRefs. [9, 10]. We use the “best values” from these pa-pers, given by the AV18+UIX nuclear Hamiltonians,with hadron mass variations calculated in terms of the m q using the Dyson-Schwinger equation calculation of[11]. From these one calculates the m q -dependence of the Q values, and therefore the reaction rates, and thereforethe primordial abundances of light elements in BBN.In Fig. 1 we present our predicted values of He, d , and Li with different values of light-quark mass.Details of the calculations and explanation of obser-vational abundances are presented in the appendices.Comparing the observed and predicted abundances fromthe figures we obtain for He, d , and Li respec-tively, δm q /m q = − . ± . . ± . . ± . δm q /m q = 0 . ± . . (7)It is seen that the He abundance has a low sensitivityto m q ; furthermore we show in Fig. 1 the conservativeobservational error bounds provided by [12]. Therefore,it is worth pointing out that the more tightly constrainedabundance, Y p = 0 . ± . m q is important, particularly for Li. In fact,if we assume a linear response, as was done in [8, 9], weinstead obtain δm q /m q = 0 . ± . K (Equa-tion 6) the final result (7) should be interpreted as δm q /m q = k · (0 . ± . k ∼ k is approximately a factor of two. III. RESONANCES
Of the most important reactions in BBN, the mirrorreactions He ( d, p ) He (Reaction 1) t ( d, n ) He (Reaction 2)are the only reactions where the cross-section is dom-inated by a fairly narrow resonance. Therefore, onecan hope for sensitivity of primordial abundances to theposition of these resonances. (Note that the reaction Be ( n, p ) Li is also dominated by a near-threshold res-onance, however in this case the resonance is a ratherbroad and hence strong sensitivity can hardly be ex-pected.) -0.01 0 0.01 0.02 0.03024682.02.53.03.54.00.2400.2450.2500.2550.260
FIG. 1: Calculated He, d , and Li abundances vs. rela-tive change in light quark mass m q / Λ QCD (solid lines). Theranges showed by the dashed lines are 1 σ errors in the theory,assuming the relative errors are constant (i.e. these do nottake into account any error in the K factors of Eq. 6). Theshaded areas show 1 σ ranges of observed abundances (detailsin Appendix A). Both of these reactions have the cross-sections with thegeneral form σ ( E ) = e − √ E g /E E P ( E )( E − E r ) + Γ r / E g is the Gamow energy of the reactants, E r and Γ r are resonance parameters, and P ( E ) is a poly-nomial chosen to fit the measured reaction cross-section.In this work we use the cross-section fits of Ref. [14], which give E (1) r = 0 .
183 MeV, Γ (1) r = 0 .
256 MeV and E (2) r = 0 . (2) r = 0 . E r → E r + δE r , due to a variation of the fundamentalconstant m q . Reaction 1 will be affected in the followingway. The resonance is an excited state of Li; that is, acompound nucleus with three protons and two neutrons:we call this state Li ∗ . Similarly there is a state He ∗ forreaction 2. Then E (1) r = E Li ∗ − E He − E d (9) E (2) r = E He ∗ − E t − E d (10)and so E Li ∗ = − .
76 MeV and E He ∗ = − .
66 MeV.The change in the resonance position due to a variationin m q is therefore δE (1) r = δE Li ∗ − δE He − δE d (11)= (cid:0) K Li ∗ E Li ∗ − K He E He − K d E d (cid:1) δm q m q (12)with the K defined by (6).Changes to the cross-section of reaction 1 affects theprimordial abundances of He and Be, while changesin reaction 2 affect abundances of t and Li. Since t and He are not well constrained observationally, we choose tofocus on Li. In Fig. 2 we present Li abundance againstvariation of light quark mass δm q /m q at η = 6 . × − ,the WMAP5 value. For such a value of η , the majorityof Li is created as Be (which β -captures to Li) via thereaction He ( He , γ ) Be.We need to find K Li ∗ (and similarly K He ∗ ). One as-sumption is that the mass-energy of the resonance varieswith the mass-energy in the incoming channel [8]; in thiscase the resonance does not shift. This assumption cor-responds to K Li ∗ = − .
54 and K He ∗ = − .
44. Itcorresponds to the solid line in Fig. 2.A more reasonable guess is to assume that the vari-ation of the resonant state Li ∗ will be approximatelythe same as that of the ground state Li. This can beseen by considering the resonance and the ground stateconfigurations as residing in the same potential. The sen-sitivity of the ground state He to m q has been calculated K He = − .
24 [10]; K Li was not calculated explicitly,but its value will be very close to that of He. Our as-sumption of equal variation of the ground and excitedstate then gives K Li ∗ = − .
35 (13) K He ∗ = − .
19 (14)This assumption corresponds to the dashed line in Fig. 2.The equal-variation assumption in the previous para-graph represents an upper limit on the relationship be-tween the ground and excited state. In reality thepotential-dependence of the states may be different, inwhich case the shift of the Li (or He) resonance may be -0.01 0 0.01 0.02 0.030246810
FIG. 2: Calculated Li abundance vs. relative change in lightquark mass m q / Λ QCD . Solid line: no shifts in resonance po-sitions included (same as solid line in Fig. 1); dashed line:resonance shifts according to assumption that resonant statevaries as much as the ground state (equations 13 and 14); dot-dashed line: an averaged value of resonance-position sensitiv-ity used. The shaded area shows the 1 σ range of abundances. smaller than the shift of the ground state. On the otherhand a minimum value of K for the resonance statesis that of the ground state, K Li ∗ = K Li = − . K Li ∗ = − .
29 and K He ∗ = − .
21; this is the dot-dashed line in Fig. 2.Ultimately however, we require a nuclear calculation ofsensitivity, of the kind presented in Refs. [9, 10].The effect of δE (1) r on BBN can be understood inthe following way. When the cross-section is convolvedwith a Maxwellian distribution, the exponential termgives rise to the “Gamow window” at energy E /E g =( kT / E g ) / . This reaction is most active at kT ≈ . E =0 .
180 MeV. This is remarkably close to the resonanceenergy for this reaction E r = 0 .
183 MeV. Thereforemovement of the resonance position in either directionwill reduce the cross-section for this reaction at the rel-evant temperatures. In turn this reduces the amount of He that is destroyed via reaction 1, leaving more to reactwith He to produce Be. On the other hand the effectof this reaction on d and He abundances is minimal.The effect of δE (2) r is very similar: it reduces theamount of t destroyed in reaction 2, leaving more tri-tium to react with He to produce Li directly. De-spite this production channel being suppressed at high
FIG. 3: Calculated He abundance vs. relative change inlight quark mass m q / Λ QCD . Solid line: no shifts in resonancepositions included; dashed line: resonance shifts according toassumption that resonant state varies as much as the groundstate (equations 13 and 14); dot-dashed line: an averagedvalue of resonance-position sensitivity used. η , the effect of δE (2) r is still important for Li produc-tion because the relative effect of the variation is larger: δE (2) r / Γ (2) r > δE (1) r / Γ (1) r . The trends seen in Fig. 2 arethe same even at low η since both reaction pathways be-have in much the same way to variation in m q .From Fig. 2 we see that taking shifts in the resonancepositions into account can destroy the agreement betweentheory and observation previously obtained by varying m q . In the case where the shifts in the ground and reso-nant states vary by the same amount (dashed line), the Li discrepancy actually gets worse with variation in lightquark mass. On the other hand the milder “averaged K ”response (dot-dashed line) still significantly challengesthe conclusions of Section II. It is not appropriate todirectly compare primordial He abundances with obser-vations because of the complexity of the stellar evolutionof this isotope [15], however we note that primordial Heproduction could also be greatly increased by movementof these resonances (Fig. 3).
IV. CONCLUSION
We have shown in Section II that a variation in thelight-quark mass during the era of big-bang nucleosyn-thesis of δm q /m q = 0 . ± .
005 provides better agree-ment between theory and the observed primordial abun-dances. This is largely because it resolves the existingdisagreement in Li abundances [4].However this conclusion is threatened when movementof the resonance positions in the reactions He ( d, p ) Heand t ( d, n ) He is taken into account. These reactionsstrongly affect Li production during BBN; furthermorethey are already “on resonance” meaning that movementof the resonance position in either direction increases Li.Our estimates suggest that the He ∗ and Li ∗ resonancesmay be very sensitive to variation of m q / Λ QCD . There-fore it is very important that the sensitivity of these res-onances to fundamental constants be studied in more de-tail using nuclear models.This work is supported by the Australian ResearchCouncil. JCB thanks Daniel Wolf Savin for hospitalityduring the early stages of this work.
APPENDIX A: OBSERVATIONALABUNDANCES
Our observational abundances largely follow the rec-ommendations of the Particle Data Group review [2].The deuterium abundances are derived from several stud-ies of isotope-shifted Ly- α spectra in quasar absorptionsystems. Combined these give d/H = (2 . ± . × − where the errors have been increased to account for thescatter between different systems. He is observed in H II regions of low-metallicity dwarfgalaxies. A very conservative estimate of observed Heabundance comes from [12] Y p = 0 . ± . . The error here is significantly larger than other extrapo-lations to zero metallicity, e.g. [13, 16].Primordial Li abundance is determined from metal-poor Pop II stars in our galaxy. Lithium abundancedoes not vary over many orders of magnitude of metal-licity in such stars; this is the Spite plateau [17]. Recentstudies give abundances of (1 . − . ± . × − [18],(1 . ± . × − [19], and (1 . − . × − [20].Significantly higher results were obtained with differentmethods of obtaining effective temperature of the stars[21] since the derived lithium abundance is very sensitive to temperature. However no evidence for high tempera-tures was found in the studies [18, 20].On the other hand measurements of Li abundancein the globular cluster NGC 6397 give values of (2 . ± . × − [22], (1 . ± . × − [23], and (1 . ± . × − [24]. The M 92 globular cluster yields avalue of (2 . ± . × − [25]. For a more detailedreview and discussion of systematics, see, e.g. [2, 4, 18].In this paper we use the conservative range of Li /H = (1 . ± . × − , which was also adopted in [8], although we note thatsome of the studies listed above give ranges as high as2 . × − . APPENDIX B: COMPUTER CODE, REACTIONRATES, AND THEORETICAL UNCERTAINTIES
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