Dependence of hadronic properties on Quark Masses and Constraints on their Cosmological Variation V.V. Flambaum1 and E.V. Shuryak2 1 School of Physics, The University of New South Wales, Sydney NSW 2052, Australia and Institute for Atomic and Molecular Theory, Harvard-Smithsonian Center for Astrophysics, 60 Garden st, Cambridge, MA 02138-1516, USA. 2 Department of Physics and Astronomy, State University of New York, Stony Brook NY 11794-3800, USA (February 1, 2008) We follow our previous paper on possible cosmological variation of weak scale (quark masses) and strong scale, inspired by data on cosmological variation of the electromagnetic fine structure constant from distant quasar (QSO)absorption spectra. Inthis work we identify the strange quark massmsasthemostimportantquantity,andthesigmamesonmassastheingredientofthenuclear forcesmostsensitivetoit. Asaresult,weclaimsignificantlystrongerlimitsonratioofweak/strong 3 Nscuacleleo(Wsynt=hemsiss/(Λ|δQWC/DW) |va=2/3 (5) −4π r 4π r val ∂m | | | s 2 since with probability 1/3 there is a strange pair. A K¯K states, while σ does mix with them strongly. An mixing between different scalar mesons f (σ f (600), admixture of virtual K¯K,ηη pairs can be viewed as an 0 0 ≡ f (980),f (1370))wouldfurtherchangethevalencecon- additional contribution to the strange sea, on top of the 0 0 tributions downward. However, based on large gap be- strangecontentofthenon-strangeconstituentquarks(7). tween sigma and other states, and also based on bet- The σ mixing with continuum of pseudoscalars is de- ter studied ηη′ mixing, one might think that the relative scribedbythestandardmassoperatorgivenbytheusual change due to mixing with next f states is not signifi- loop diagram 0 cant. d4k λ2(k,Q) Let us now consider the contribution of the so called Σ(Q2)= (10) 2π4[(k+Q/2)2 m2+iǫ][(k Q/2)2 m2+iǫ] strange sea, virtual s¯s pairs, which are always present Z − − − evenin a completely non-strangehadronslike a nucleon. where λ is the KKσ,ηησ orππσ couplings. Its realpart Therelativelywellstudiedcaseisthenucleonmasssensi- describes the shift of m due to repulsion from these σ tivity to ms, already discussed in [3]. Let briefly remind states: the imaginary part (for pions only) gives the that KN scattering data imply that width. The sign of the shift due to K¯K is obviously negative, since m < 2m . The effect of ππ has con- σ K ∂m N = 1.5 (6) tributions of both signs. For constant coupling the shift ∂ms | | ≈ is logarithmically divergent, in reality it has to be reg- ulated by formfactors in the vertexes. The total shift and thus about 1/5 of the nucleon mass comes from the is negative and large, of the order of very large width strange sea (m 120 MeV). Similarmatrsix≈elementsforσ,ω mesonsisnotpossible Γσ ∼ 300MeV. Note that this large negative shift is partlythereasonwhytheσmassissosmall. However,in to obtainexperimentally, althoughit canbe done onthe this paper we focus on the dependence on quark masses. lattice. Toestimateitwewilladoptasimpleconstituent Assuming that the main dependence comes from masses quark picture, assuming additivity of the strange sea. If of Goldstone bosons, m ,m , we differentiate the mass so,the derivativesanalogousto (6)for allmesons should π K operatoroverthesemassesandobtaintheconvergentre- be 2/3 of it, or sult. Thus one can ignore formfactors and extract the effective coupling constant out of the integral. ∂m mesons ∂m |sea ≈1 (7) For the derivative at Q = mσ = 0.5GeV we get the s following numerical value for the shifts As we will see later, the exact value of the common sea ∂Σ contribution is not actually important, since its contri- ∂m2 =0.0229GeV−2λ2σKK (11) butions to σ and ω mesons tend to cancel each other K nearly exactly when we calculate variation of N N in- teraction. Whatmattersisthedifferencebetwe−entheir ∂Σ =0.019GeV−2λ2 (12) ∂m2 σηη strange seas, to be discussed in the next section. η It is convenient to present the effect of the possible The couplings are not experimentally known, so we rely quarkmassvariationontheσ massinthefollowingform on the SU(3) symmetry and relate them to λ , which σππ is in turn related to sigma meson width δm δm m +m δm δm δm σ s u d q s q =0.4( + )=0.4 +0.04 (8) mσ ms ms mq ms mq Γ = 3 λ2σππ 1 4m2π (13) where we have used m = 120MeV. We see that the σ 216πmσs − m2σ s relativechangeofthestrangequarkmassproducesmuch Taking Γ =250MeV we obtain λ2 = 5 Gev2. The fac- σ σππ larger effect than the relative change of the light quark tor 3/2 account for π+π−,π0π0 modes. However in the mass. This is similar to the case of the nucleon mass mass shift there are contribution of K+K−,K¯0K0,ηη variation channels which we would count as 5/2. Substituting numbersandusingstandardGell-Mann-Oaks-Rennerex- δm δm δm N =0.19 s +0.045 q (9) pressionsform ,m (m2 m Λ ),weobtainthead- mN ms mq ditional sensitiηvityKof the∝sigmsaQmCaDss shift arising from the mixing effects δm δm m2 ∂Σ m2 ∂Σ δm C. K¯K,ηη mixing with σ σ = s 2 K +0.5 η s0.14 (14) mσ ms " 2m2σ ∂m2K 2m2σ ∂m2η#≈ ms As the second approximation, we will discuss loop ef- We included K+K−,K¯0K0 modes with the coefficient 1 fects, or mixing with 2-meson states. Those are also and ηη with 1/2: the latter only contribute about 1/5of completely different for σ and ω mesons. As we men- the final answer. We ignored even smaller contribution tioned earlier, the latter practically does not mix with of the η′η′ loop. 3 D. The total sensitivity of mσ as compared to mN The sign difference between these two derivatives and very large derivative value is due to fine tuning between Together with the one estimated in the previous sec- omegaandsigmaterms,whichareseparatelymuchlarger tion, it leads to total than the sum. The next step one can do analytically is to add a sim- δmσ δms δms plest square potential to the hard core. The energy of a (0.24+0.16+0.14) =0.54 (15) m ≈ m m shallow level in such potential is equal to [11]: σ s s where 3 terms are the contributions of the common π2(U U )2 0 strangesea(7),valencestrangeness(5)andtheloopmix- Ed = Qd = − , (21) − −16 U 0 ing (14), respectively. In the same units the sensitivity of the nucleon mass is π2 U = . (22) 0 8ma2 δm δm N s 0.19 (16) m ≈ m Here U and a are the depth and width of the potential N s well (a=c b, where c and b are outer and inner radii), We conclude that the sigma mass is about 3 times more m=m /2−isthe reducedmass. Selectingthe widthand N sensitive to the variationof the strangequarkmass than depth of the well to be a= 1.6fm,U =40.2MeV,U = 0 the nucleon mass. 52.6MeV,we got We alsoneed the sensitivity ofω mesonto the strange massvariation. Thismesondoesnothavevalencestrange δQd 81.6e−mσbδmσ (23) quarks and practically does not mix with φ, K and η Q ≈− m d σ mesons. Therefore, only the strange see contributes: δmω 0.15δms (17) δQd 87.4e−mωbδmω (24) mω ≈ ms Qd ≈ mω By changing the core radius from b = 0 to b = 0.4 fm onecanvarythe answerby aboutfactor3. Simple expo- III. THE MODIFICATION OF THE DEUTERON nential dependence on the core radius b appears because BINDING of translational invariance of 1d Scredinger equation for rψ(r). A. Preliminary analytic estimates Another effect one should consider is the modification of the nucleon mass: its contribution to modification of the deuteron binding is Simpleanalyticestimateforsensitivityofthedeuteron binding to sigma and omega mass modification is ob- δM p2 N tained by the differentiation of the potential over the δQd = (25) M |2M | massandaveragingtheresultingexpression exp( mr) N N ∼ − over the radial wave function. which leads to The simplest short range approximation leads to the δQ U +U δm δm free motion wave function = 0 N 7.7 N. (26) Q U U m ≃ m 0 N N √2κ − ψ(r)= exp( κr) (18) Although the sensitivity to the nucleon mass is much r − weaker than that for mesons, it is still quite strong: we This wave function tends to infinity at small distances. attribute it to the fact that the small deuteron binding However,therealdeuteronwavefunctionshouldbesmall energy is in turn the delicate balance between larger ki- there because ofthe repulsivecoreinthe potentialV(r). netic and the potential energies. Therefore, we introduced a small cut-off radius b in the integration over r. We estimated b from the condition V(b) = 0 which gives b = 0.45 fm. We get the following B. Using the Walecka potential shift of the deuteron binding The estimates of the preceding section are given for δQ m δm g2 2κ δm d = σ σ s e−(2κ+mσ)b 75 σ (19) orientation only, and in fact one of course have to solve Q −Q m 4π2κ+m ≈− m d d σ σ σ numerically the radialSchreodinger equation and obtain A variation of m gives the correct wave function. The one can either average ω the potential derivative over it or simply vary all masses δQ m δm g2 2κ δm involvedexplicitly. Wedidthelatteranddeterminedthe d = ω ω v e−(2κ+mω)b 80 ω (20) Qd Qd mω 4π2κ+mω ≈ mω sensitivity of Qd to the sigma, omega and nucleon mass. 4 Strictly speaking, at this point it is no longer possible zero [12]. Today the lowest resonance energy is only to limit ourselves to Walecka model, with the coupling E =0.0973 0.0002eV is large compared to its width, 0 ± constants (3), since it does not describe correctly the so the neutron capture cross section σ 1/E2. The ∼ 0 deuteronbinding. Infact,byignoringallspin-dependent data constrain the ratio of this cross section to a non- forces one cannot even separate the spin-singlet and the resonance one (which was used to measure number of spin-triplet states. The tensor forces, attributed to pion neutrons emitted by the reactor). It therefore implies † and rho exchanges, are needed for this task. Instead of that these data constrain the variation of the following doing so, we have chosen to modify a bit the strength of ratio δ(E /E ) where E 1MeV is a typical single- 0 1 1 ∼ the omega term, reducing g2 by a factor 0.953 as com- particleenergyscale,whichmaybe viewedastheenergy v paredto (3) and obtaining the correctdeuteronbinding. of some 1-body “doorway” state. Our results are∗ A generic expression for the level energy in terms of fundamental parameters of QCD can be written as fol- δQ δm δm d 48 σ 26 s (27) lows Q ≈− m ≈− m d σ s δE =A δΛ +B δm +B δm +Cδα Λ (33) 0 QCD q q s s QCD ∗ ∗ ∗ ∗ δQ δm δm d ω s 50 7.5 (28) where A,B,C are some coefficients. The first term is Q ≈ m ≈ m d ω s the basic QCD term, while others are corrections due to modificationofthequarkmassesandtheelectromagnetic δQd δmN δms α. 6 1.1 (29) Q ≈ m ≈ m In this section we provide new estimate of the B . d N s s More specifically, we estimate the variation of the res- One can see that the first two derivatives are more sen- onanceenergyresultingfromamodificationofthesigma sitive to exact shape of the wave function: they agree mass. The energyofthe resonanceE =E S 0 excitation n qualitatively but notquantitatively withthe analytic es- − consists of excitation energyof a compound nucleus, mi- timates above. We will not show here such details as ex- nus the neutron separation energy S . This, in turn, is n act and approximate wave functions, but just comment a depth of the potential well V minus the neutronFermi that the difference between those explain the difference energy ǫ , S = V ǫ . The latter scales like 1/R2 if F n F in the integrals. − the radiusofthe wellis changed. The kinetic partof the Summing all the contributions we find excitation energy E scales in the same way. So, excitation δQ δm d 17 s (30) ¯h2 Qd ≈− ms E0 =Eexcitation Sn =Eexcitation+ǫF V =K V (34) − − MR − Using limits on Big Bang Nucleosynthesis from [3] whereKisanumericalconstantwhichcanbefoundfrom δQd <0.1 (31) the present time condition E0 ≈ 0. The shift of the resonance then is | Q | d one gets the final limit on the m variation to be ¯h2 δM 2δR δM 2δR δV s δE = K + δV = V + + (35) 0 − MR M R − − M R V δ(m /Λ ) (cid:18) (cid:19) (cid:18) (cid:19) s QCD <0.006 (32) (m /Λ ) Using eq. (2) we can find the depth of the potential well (cid:12) s QCD (cid:12) (cid:12) (cid:12) ‡ (cid:12) (cid:12) (cid:12) (cid:12) 3 g2 g2 V = s v (36) IV. OKLO 4πr3 m2 − m2 0 (cid:18) σ ω(cid:19) Inthis sectionweextractlimitsonδms followingfrom Here r0= 1.2 fm is an inter-nucleon distance. Numerical data on natural nuclear reactor in Oklo active about 2 estimates shows that the contribution of the variationof bn years ago. The most sensitive phenomenon (used r0 (andthe variationofR=A1/3r0 ineq. (35))is notas previously for limits on the variations of the electromagnetic α) is disappearance of certain isotopes (espe- cially Sm149) possessing a neutron resonance close to †Of course, underassumption that the sameresonance was thelowest one at thetime of Oklo reactor. ‡Note that the suppression of N-N wave function at small ∗Inthederivativeovertheomegamasswehavedividedback separation dueto therepulsive core reducesthe depthof the bythefactor0.953, restoringtheoriginalstrengthofthevec- effectivepotentialV. However,thiseffectisnotsoimportant tor term. in the ratio δVV. 5 important as the direct contribution of the m variation Thislimitcanbeimprovedafteranaccuratecalculation. σ in the equation above. This gives us Acknowledgments We thank J.Bjorken, G.Brown and V.Dmitriev for helpful discussions. One of the au- δV δm δm δm = 8.6 σ +6.6 ω = 3.5 s (37) thors (VF) is thankful to the State University of New V − mσ mω − ms YorkandInstituteforTheoreticalAtomicandMolecular Physics, Harvard-Smithsonian Center for Astrophysics δm for hospitality and support. This work is also supported δE =1.7 108eV s (38) 0 × × m by the Australian Research Council. ES work is par- s tially supported by the US-DOE grant No. DE-FG02- We used V= 50 MeV in eq. (35). Comparison of this 88ER40388. result with the the observational limits claimed in [12] δE <0.02 eV gives a very strong limit 0 | | δ(m /Λ ) s QCD <1.2 10−10 (39) | (m Λ ) | × s QCD at time 1.8 billion years ago. ≈ [1] J.D.Bjorken, Cosmology and the standard model. hep- Note that the authors of the last work in [12] found th/0210202 and references therein. alsothenon-zerosolutionδE = 0.097 0.008eV.This 0 [2] J. K. Webb , V.V. Flambaum, C.W. Churchill, M.J. − ± solutioncorrespondstothesameresonancemovedbelow Drinkwater, and J.D. Barrow, Phys. Rev. Lett., 82, thermal neutron energy. In this case 884-887, 1999. J.K. Webb, M.T. Murphy, V.V. Flam- baum, V.A. Dzuba, J.D. Barrow,C.W. Churchill, J.X. δ(m /Λ ) s QCD = (0.56 0.05) 10−9 (40) Prochaska, andA.M.Wolfe, Phys.Rev.Lett.87, 091301 (msΛQCD) − ± × -1-4 (2001). M. T. Murphy, J. K. Webb, V. V. Flam- baum, V. A. Dzuba, C. W. Churchill, J. X. Prochaska, The error here does not include the theoretical uncer- J. D. Barrow and A. M. Wolfe, Mon.Not. R. Astron. tainty. Soc. 327, 1208 (2001) ; astro-ph/0012419. M.T. Mur- The production of nuclei with A > 5 during BBN is phy, J.K. Webb, V.V. Flambaum, C.W. Churchill, and strongly suppressed because of the absence of stable nu- J.X. Prochaska. Mon.Not. R. Astron. Soc. 327, 1223 cleiwithA=5. 5Heisunstablenucleuswhichisseenasa (2001); astro-ph/0012420. M.T. Murphy, J.K. Webb, resonanceinn-αelasticscattering. Thegroundstatelies V.V. Flambaum, C.W. Churchill, J.X. Prochaska, and at 0.89 MeV above neutron threshold. The variation of A.M. Wolfe. Mon.Not. R. Astron. Soc. 327,1237 (2001); the strangequarkmassmayinfluence the positionofthe astro-ph/0012421. resonance making, for example, 5He stable. Stable 5He [3] V. V. Flambaum and E. V. Shuryak, Phys. Rev. D 65, at the time of BBN would change strongly the primor- 103503 (2002) [arXiv:hep-ph/0201303]. dial abundances of light elements. The estimate similar [4] V.F. Dmitriev, V.V. Flambaum, astro-ph/0209409. to that we made for Sm nucleus gives us a limit [5] S.R.Beane and M.J.Savage, nucl-th/0208021. [6] T. Schafer and E. V. Shuryak,Rev. Mod. Phys. 70, 323 δ(m /Λ ) s QCD (1998) [arXiv:hep-ph/9610451]. see also a comment on > 0.006 (41) (msΛQCD) − scalars by thesame authors, hep-lat/0005025 [7] B. D. Serot and J. D. Walecka, Adv. Nucl. Phys. 16, 1 This limit corresponds to δE > - 0.89 MeV at the time (1986). of BBN. [8] G. Chanfray, M. Ericson and P. A. Guichon, Phys. Rev. We obtained limits on variation of ms/ΛQCD during C 63, 055202 (2001) [arXiv:nucl-th/0012013]. the interval between the Big Bang and present time and [9] E. V.Shuryakand G. E. Brown, arXiv:hep-ph/0211119. on shorter time scale from Oklo natural nuclear reactor [10] Review of Particle Properties, Phys.Rev.D66 (2002) which was active 1.8 billion years ago. It is also possi- 010001 ble to obtain limits on the intermediate time scale. One [11] L.D. Landau,E.M. Lifshits. QuantumMechanics, Theo- possibility is related to position of the resonance in 12C retical Physics, Volume3 (Nauka,Moscow,1974). during production of this element in stars. This famous [12] A.I.Shlyakhter, Nature 264 (1976) 340; T.Damour resonance at E=380 KeV is needed to produce enough and F.J.Dyson, Nucl.Phys.B 480 (1996) 37. Y.Fujii, carbon and create life. According to Ref. [13] the posi- A.Iwamoto, T.Fukahori, T. Ohnuki, M. Nakagawa, H. tionofthisresonancecannotshiftbymorethan60KeV Hidaka,Y.Oura,P.Moller.Nucl.Phys.B573(2000)377. (onecanalsofindinRef.[13]thelimitsonthestrongin- [13] H. Oberhummer,R.Pichler, A. Csoto, nucl-th/9810057. teractions and other relevant references). We have made a very rough estimate of the limit on the strange quark massvariationwhichcanbeobtainedviam -mechanism: σ δ(m /Λ ) s QCD <0.001 (42) | (m Λ ) | s QCD 6

Dependence of hadronic properties on Quark Masses and Constraints on their Cosmological Variation

Similar Dependence of hadronic properties on Quark Masses and Constraints on their Cosmological Variation