Deducing Quark Rest Masses with Phenomenological Formulae

合集下载

Phase diagram of dense neutral three-flavor quark matter

ular and gapless color-ﬂavor-locked (gCFL) phases [15] are favored. At nonzero temperature, T = 0, some other phases were proposed as well [16]. In this paper, we study the phase diagram of dense neutral three-ﬂavor quark matter as a function of the strange quark mass at zero and ﬁnite temperature. In order to allow for the most general ground state, we employ a nine-parameter ansatz for the gap function. The eﬀects of the strange quark mass are incorporated in our model by a shift of the chemical potential of strange quarks, i 2 µi s → µs − ms /(2µ) where i = r, g, b is the color index, ms is the strange quark mass, and µ is the quark chemical potential. This shift reﬂects the reduction of the Fermi momenta of strange quarks due to their mass. Such an approach is certainly reliable at small values of the strange quark mass. We assume that it is also qualitatively correct at large values of the strange quark mass. We note that the analysis of this paper is restricted to locally neutral phases only. This automatically excludes, for example, mixed [18] and crystalline [19] phases. Also, in the mean ﬁeld approximation utilized here, we cannot get any phases with meson condensates [20]. This paper is organized as follows. In Sec. II we present the details of our approach which is based on the Cornwall-Jackiw-Tomboulis (CJT) formalism [22]. There, we also derive the gap equations and the neutrality conditions, and obtain an expression for the pressure. In the next two sections, the gap equations and the neutrality conditions are studied by using numerical methods. The results at zero temperature are presented and discussed in Sec. III. There, the appearance of the CFL and gCFL phases is established in the case of small and moderately large values of the strange quark mass, respectively. The results at nonzero temperature are discussed in Sec. IV. Also, the phase diagram of quark matter in the T –m2 s /µ plane, as well as in the T –µ plane is presented. Finally, Sec. V concludes this paper with a summary of the results. Our units are = c = kB = 1. The metric tensor is gµν = diag (1, −1, −1, −1). Four-vectors are denoted by capital Latin letters, e.g., K µ = (k 0 , k) where k is a three-vector with absolute value k = |k| and direction ˆ = k/k . We use the imaginary-time formalism, i.e., the k

Deconfinement and color superconductivity in cold neutron stars

a r X i v :a s t r o -p h /0504564v 2 15 S e p 2005Deconﬁnement and color superconductivity in cold neutron starsG.Lugones ∗and I.Bombaci †Dipartimento di Fisica “Enrico Fermi”Universit`a di Pisa and INFN Sezione di Pisa,Largo Bruno Pontecorvo 3,56127Pisa,Italy We study the deconﬁnement transition of hadronic matter into quark matter in neutron star conditions in the light of color superconductivity.Deconﬁnement is considered to be a ﬁrst order phase transition that conserves color and ﬂavor.It gives a short-lived (τ∼τweak )transitory colorless-quark-phase that is not in β-equilibrium.We deduce the equations governing deconﬁnement when quark pairing is allowed and ﬁnd the regions of the parameter space (pairing gap ∆versus bag constant B )where deconﬁnement is possible inside cold neutron stars.We show that for a wide region of (B,∆)a pairing pattern is reachable within a strong interaction timescale,and the resulting “2SC-like”phase is preferred energetically to the unpaired phase.We also show that although β-stable hybrid star conﬁgurations are known to be possible for a wide region of the (B,∆)-space,many of these conﬁgurations could not form in practice because deconﬁnement is forbidden,i.e.the here studied non-β-stable intermediate state cannot be reached.I.INTRODUCTIONA general feature of degenerate Fermi systems is that they become unstable if there exist any attractive inter-action at the Fermi surface.As recognized by Bardeen,Cooper and Schrieﬀer (BCS)[1]this instability leads to the formation of a condensate of Cooper pairs and the appearance of superconductivity.In QCD any attractive quark-quark interaction will lead to pairing and color su-perconductivity,a subject already addressed in the late 1970s and early 1980s [2,3]which came back a few years ago since the realization that the typical superconducting gaps in quark matter may be larger than those predicted in these early works (∆as high as ∼100MeV)[4].The phase diagram of QCD has been analyzed in the light of color superconductivity and model calculations suggest that the phase structure is very rich at high densities.Depending on the number of ﬂavors,the quark masses,the interaction channels,and other variables many possi-ble β-stable color superconducting phases of quark mat-ter are possible [5,6,7,8,9].There is at present some indication that the quark gluon plasma might have been produced in laboratory [10].However,it is not yet established whether decon-ﬁnement happens in nature in the high density,low tem-perature regime that is relevant for neutron stars.Un-fortunately,ﬁrst principle calculations are not available in this region of the QCD phase diagram.In turn we shall base our analysis on phenomenological considera-tions which could delineate at least a broad brush pic-ture of the physics involved.Matter in compact stars should be electrically neutral and colorless in bulk.Also,any equilibrium conﬁguration of such matter should re-main in β-equilibrium.Satisfying these requirements im-pose nontrivial relations between the chemical potentials of diﬀerent quarks.Moreover,such relationssubstan-2analyzed.To the best of our knowledge,all previous works about color superconductivity in compact stars have dealt with matter in β-equilibrium.This is the situation expected to appear in strange stars or hybrid stars as soon as they set-tle in a stable conﬁguration.However,during the decon-ﬁnement transition in neutron stars,matter is transito-rily out of equilibrium with respect to weak interactions.In fact,the transition from β-stable hadron matter to quark matter in cold neutron stars should occur trough a quantum nucleation process [12,13,15,16,17,18].Quantum ﬂuctuations could form bothvirtual drops of unpaired quark matter (hereafter the Q ∗unp phase)or vir-tual drops of color-superconducting quark matter (Q ∗∆phase).In both cases,the ﬂavor content of the quark matter virtual drop must be equal to that of the conﬁned β-stable hadronic phase at the same pressure (the central pressure of the hadronic star).In fact,since quark decon-ﬁnement and quark-quark pairing are due to the stronginteraction,the oscillation time ν−10of a virtual quark droplet in the potential energy barrier separating the hadronic from the quark phase,is of the same order of the strong interaction characteristic time (τstrong ∼10−23s).The latter is many orders of magnitude smaller than the weak interaction characteristic time (τweak ∼10−8s).Thus,quark ﬂavor must be conserved forming a virtual drop of quark matter [17,18,19,20,21].Which one ofthe two kind of droplets (Q ∗unp or Q ∗∆)will nucleate de-pends on the value of the corresponding Gibbs free energy per baryon (g unp ,g ∆).In fact,the latter quantity enters in the expression of the volume term of the energy barrier separating the conﬁned and deconﬁned phases (see e.g.eq.(7)in [18],where the Gibbs free energy per baryon is denoted by µi ,i =Q ∗,H ).Clearly,when g ∆<g unp the nucleation of a Q ∗∆drop will be realized.The direct formation by quantum ﬂuctuations of a drop of β-stable quark matter (Q phase)is also possi-ble in principle.However,it is strongly suppressed with respect to the formation of the non β-stable drop by afactor ∼G 2N/3Fermi being N the number of particles in the critical size quark drop.This is so because the formation of a β-stable drop will imply the almost simultaneous conversion of ∼N/3up and down quarks into strange quarks.For a critical size β-stable nugget at the center of a neutron star it is found N ∼100−1000,and there-fore the factor is actually tiny.This is the same reason that impedes that an iron nucleus converts into a drop of strange quark matter,even in the case in which strange quark matter had a lower energy per baryon (Bodmer-Witten-Terazawa hypothesis).Because of this reason it is assumed that a direct transition to β-stable quark mat-ter is not possible [33].Therefore,the β-stable state Q could be reached only after the β-decay of the interme-diate state Q ∗.This is in agreement with many other previous works,see e.g.[17,18,19,20,21].In this context,one question addressed by the present work is whether the system settles in a paired or in an un-paired state just after the deconﬁnement.On the otherhand,notice that although the above mentioned non-β-stable quark phase is very short-lived,it constitutes an unavoidable intermediate state that must be reached be-fore arriving to the ﬁnal β-stable conﬁguration,e.g.CFL quark matter (c.f.[22,23]).The second question we shall address is whether this intermediate phase can eventually preclude the transition to the ﬁnal β-stable state in spite of the latter having a lower energy.This is because the Q -phase can be formed only after the nucleation of a real(i.e critical size)drop of Q ∗unp or Q ∗∆matter,and its sub-sequent “long term”(t ∼τweak ∼10−8s)weak decay process.II.DECONFINEMENT OF HADRONIC MATTER INTO COLOR SUPERCONDUCTINGQUARK MATTERGiven the uncertainties in the nature of matter at high densities,the analysis is based on the extrapolation to higher densities of an hadronic model valid around the nuclear saturation density ρ0,and the extrapolation to ρ0of a quark model that is expected to be valid only for ρ→∞.Within this kind of analysis the (in general)dif-ferent functional form of both EOSs,induces the phase transition to be ﬁrst order.Notice that from lattice QCD calculations there are indications that the transition is ac-tually ﬁrst order in the high-density and low-temperature regime,although this calculations involve temperatures that are still larger than those in neutron stars,and do not include the eﬀect of color superconductivity [26].Deconﬁnement is analyzed here as a ﬁrst order phase transition that conserves the ﬂavor abundances in both phases.Therefore,it gives a transitory colorless-quark-phase that is not in β-equilibrium.For describing the just deconﬁned quark phase we shall model it as a free Fermi mixture of quarks and leptons and we will subtract the pairing and the vacuum energy.The thermodynamic potential can be written asΩ=Ωfree Q +ΩgapQ +ΩL +B,(1)withΩfree Q=f,c1π2∆2¯µ2,(3)3PressureG i b b s e n e r g y p e r p a r t i c l e G / n BFIG.1:Schematic comparison of the free energy of hadronic matter (H ),non-β-stable ”just-deconﬁned”matter (Q ∗),and β-stable quark matter (Q )for diﬀerent cases.In the case of panel a)the transition can never occur inside neutron stars in spite of the ﬁnal state (Q)having a lower energy per baryon.As explained in the text,a direct transition to Q is strongly suppressed.Since Q ∗has a larger energy per baryon than H for all pressures below the central pressure of the maximummass hadronic star (P maxc ),deconﬁnement cannot occur even if the Q phase has a lower energy.In panels b)and c)the phase Q ∗has a lower energy per baryon than H for somepressures below (P maxc ).Therefore,deconﬁnement is possibleif pressures between P 0and P maxc are reached inside a given neutron star.The diﬀerence between panels b)and c)is the energy per particle at zero pressure (indicated with a dot for Q and with an asterisk for H ).In b)quark stars are the so called strange stars,because they can be made up of quark matter from the center up to the surface (P=0).In case c)they are hybrid stars,because at zero pressure H has a lower energy than Q .For a ﬁxed hadronic equation of state,these possibilities correspond to diﬀerent values of the parameters of the quark model (the vacuum energy density B and the superconducting gap ∆).which results from E cf =3π2+2A3π2(5)for quarks that do not pair.The number density of each ﬂavor in the quark phase is given byn f =cn fc ,(6)and the baryon number density n B byn B =13 cn c =14Y H f=Y Qff=u,d,s,L(8)being Y H f≡n H f/n H B and Y Q i≡n Qf /n Q B the abundancesof each particle in the hadron and quark phase respec-tively(we shall omit the super-indexes H and Q in the following).In other words,the just deconﬁned quark phase must have the same“ﬂavor”composition than the β-stable hadronic phase from which it originated. Additionally,the deconﬁned phase must be locally col-orless;therefore,it must be composed by an equal num-ber or red,green and blue quarks:n r=n g=n b(9) being n r,n g and n b the number densities of red,green and blue quarks respectively,given by:n c= f n fc.(10) Color neutrality can be automatically fulﬁlled by im-posing that eachﬂavor must be colorless separately,i.e. n ur=n ug=n ub,n dr=n dg=n db,and n sr=n sg=n sb. But in general this conﬁguration will not allow pairing with a signiﬁcant gap.As already stated,for quarks hav-ing diﬀerent color andﬂavor the pairing gap may be as large as100MeV,while for particles having the same ﬂavor the gap is found to be about two orders of magni-tude smaller(see[25]and references therein).Pairing is allowed even in the caseδµ>∆but the corresponding gaps are small[6,28].Therefore,in order to allow pair-ing between quarks with a non negligible gap,the Fermi momenta of at least u r and d g quarks must be equal(the choice of these two particular colors andﬂavors is just a convention).This implies the equality of the correspond-ing number densitiesn ur=n dg.(11) The above condition represents a state that fulﬁlls all the physical requirements of the deconﬁned phase(e.g.is color and electrically neutral),and should be the actual state(forτ≪τweak)if it has the lowest free energy per baryon.Energy must be paid in order to equal at least two Fermi seas,but in compensation the pairing energy is recovered.The gained energy depends on the value of the pairing gap∆and,at least for suﬃciently large∆, is expected to be larger than the energy invested to force a pairing pattern.Also,notice that color conversion of quarks allows the adjustment of the Fermi seas within a givenﬂavor in a very short timescale(∼τstrong),i.e. several orders of magnitude faster thanβ-equilibration (τstrong≪τweak).We emphasize that this phase is not inﬂavor equilib-rium.After a weak interaction timescale this transitorypaireddbdgdrubugurncfunpaireddrdbdgubugurncf010*******-40-2020406080B = 100 MeV fm-3∆ = 150∆ = 100∆ = 50∆ = 0g∆-gunp[MeV/particle]P [MeV fm-3]FIG.2:Deconﬁnement of pure neutron matter.Upper panel: Sketch of the lowest energy conﬁguration of the paired and un-paired phases just after the deconﬁnement.Lower panel:The diﬀerence in the Gibbs energy per particle between paired and unpaired quark matter.For positive values of g∆−g unp the preferred phase just after the deconﬁnement is the unpaired one while for negative values it is the paired one.pairing pattern will be abandoned by the system in favor of the lowest-energyβ-stable conﬁguration.Depending on the density,the lowest energy state may be LOFF, gapless2SC,gapless CFL,standard CFL,(to name just some possibilities)as extensively discussed in the litera-ture.III.APPLICATION TO SIMPLIFIEDEQUATIONS OF STATEA.Deconﬁnement of pure neutron matterA simple solution can be found in the case of the de-conﬁnement of pure neutron matter,since strange quarks and electrons are not present in the hadronic gas.First, we apply the colorless conditions andﬂavor conserva-tion introduced in the previous section,in order deter-mine the abundances of each quark species.In order to allow pairing of at least two diﬀerent quarks species in the just deconﬁned phase,we impose the condition of Eq.(11),i.e.n ur=n ing one of the color-5 less conditions(n r=n g)it is found that n dr=n ug,implying that these quarks can also pair with a sig-niﬁcant gap.From the remaining colorless condition(n r=n b)it is found n ur+n dr=n ub+n db.The con-dition ofﬂavor conservation states n d=2n u;therefore,n dr+n dg+n db=2(n ur+n ug+n ub).Introducing theratio x=n ug/n ur weﬁnd from the above equations:n ub=0(12)n db12π2(1+x2/3)∆2µ2ur+.(18)1+xMinimizing g∆=g∆(P∆,x)with respect to x it is foundthat the minimum correspond to x= 1.Therefore,quarks u r−d g and u g−d r pair in a“2SC-like”patternlike the one shown in Fig.2.In order to determine whether the system settles in apaired or in an unpaired conﬁguration we compare theGibbs free energy per baryon of the above conﬁgurationwith the Gibbs free energy per baryon of an unpairedquark gas(both evaluated at the same pressure,and withthe sameﬂavor composition).For unpaired quark mat-ter the ground state of the colorless mixture(compatiblewithﬂavor conservation)is shown in Fig.2,and is de-scribed by n dr=n dg=n db=2n ur=2n ug=2n ub=26 conservation in the following simple formn d=ξn u,(20)beingξ≡Y H d/Y H u(c.f Eq(8)).In the case of the n-p-e−gas,the parameterξcan be expressed in terms of the proton fraction Y p=n p/n B of nuclear matter as ξ=(2−Y p)/(1+Y p).It is easy to check thatξ=2 corresponds to the deconﬁnement of pure neutron mat-ter,ξ=1to symmetric nuclear matter andξ=0.5to the unrealistic case of pure proton matter.We emphasize that in the case ofβ-stable n-p-e−system,ξis a function of density(or pressure)that depends only on the state of the hadronic matter that deconﬁnes.Therefore,ﬂavor conservation states that n dr+n dg+ n db=ξ(n ur+n ug+n ub).In addition,the condition of Eq.(8)applied to electrons yields:3n e=2n u−n d,(21) which conﬁrms thatﬂavor conservation automatically guarantees electric charge conservation.Finally,we im-pose that1)n dr=n ug in order to allow for paring be-tween quarks d r with u g,and2)n ur=n dg in order to allow for paring between quarks u r with d g. Introducing the ratio x=n ug/n ur,and using Eqs.(20)and(21)weﬁnd the particle number densities of eachﬂavor and color in the paired phase as a function of one of the particle densities(e.g.n ur),the parameter ξthat depends only of the state of the hadronic phase, and the free parameter x:n ug=x n ur(22)n ub=(1+x)2−ξ1+ξn ur(26)Note that the free parameter x that can be elimi-nated by minimizing the Gibbs energy per baryon g∆= ( fc n fcµfc+µe n e)/n B with respect to x at constant pressure P∆.The minimization gives x=1,which means that the conﬁguration of the paired phase that is energet-ically preferred is the one having n dr=n ug=n ur=n dg, as sketched in the upper panel of Fig. 2.In Fig.3we compare the Gibbs energy per baryon of the“2SC-like”paired phase and unpaired matter for diﬀerent values of the bag constant B and the pairing gap∆.Compar-ing with the pure neutron matter case of Fig.2it can be noticed that an increase in the proton fraction of the hadronic phase favors the formation of a paired quark phase after deconﬁnement.ncfsbsgsrunpaireddrdbdgubugurncfsbsgsrpaireddbdgdrubugur010*******-20-10102030ξ = 1.6∆ = 50∆ = 100∆ = 0η = 0.3g∆-gunp[MeV/particle]P [MeV fm-3]FIG.4:Deconﬁnement ofβ-stable hadronic matter including strange hadrons.Upper panel:Sketch of the particle num-ber conﬁguration just after deconﬁnement.The most general paired conﬁguration compatible with the condition n ur=n dg, n ug=n dr,ﬂavor conservation,and charge neutrality is the one sketched here.Lower panel:The diﬀerence in the Gibbs energy per baryon between both phases.For positive values of g∆−g unp the preferred phase just after the deconﬁnement is the unpaired one while for negative values it is the paired one. The curves correspond to B=60MeV fm−3(dashed line), B=100MeV fm−3(solid line),and B=140MeV fm−3 (doted line).We employed m s=150MeV.IV.DECONFINEMENT OF COLD HADRONICMATTERIn the following we analyze the deconﬁnement of a gen-eral hadronic system including strange hadrons and then we apply the results to a realistic EOS in order to study deconﬁnement inside cold neutron stars.A.Deconﬁnement of a general hadronic equationof state•Flavor conservation:After deconﬁnement the parti-cle densities of quarks u,d and s are the same as in the hadronic phase and can be determined by Eqs.(8).An-other equivalent way of expressing theﬂavor conservation7condition is in terms of two parametersξandη:n d=ξn u.(27)n s=ηn u.(28) whereξ≡Y H d/Y H u andη≡Y H s/Y H u depend only on the composition of the hadronic phase.These expressions are valid for any hadronic EOS.For hadronic matter contain-ing n,p,Λ,Σ+,Σ0,Σ−,Ξ−,andΞ0,we haveξ=n p+2n n+nΛ+nΣ0+2nΣ−+nΞ−2n p+n n+nΛ+2nΣ++nΣ0+nΞ0.(30)As typical values,we notice thatη=0corresponds to zero strangeness,and that at the center of the maxi-mum mass star(calculated with the hadronic equation of state of Glendenning and Moszkowski GM1[31])we haveξ=1.15andη=0.85.Notice thatξandηdeter-mine univocally the number of electrons present in the system through electric charge neutrality of the decon-ﬁned phase:3n e=2n u−n d−n s.(31)•Pairing condition:As made in the previous section for the n-p-e−gas,we impose that1)n dr=n ug in order to allow for paring between quarks d r with u g,and2) n ur=n dg in order to allow for paring between quarks u r with d g.•Color neutrality:The condition n r=n g leads immediately to n sr=n sg.Also,n r=n b leads to 2n ur+n sr=n ub+n db+n sb.The above conditions lead to the pairing pattern schematically shown in the upper panel of Fig.4.Note that these conditions still leave a degree of freedom that can beﬁxed by introducing an additional parameter h relating the particle number densities of two arbitrary quark species.Therefore,it is possible to impose the equality of two arbitrary Fermi seas in order to allow pairing between them.We have analyzed the10possible combinations and veriﬁed that6of them lead to a nega-tive value of the particle number density of at least one quark species.The other4possibilities allow pairing of particles that don’t have diﬀerent color andﬂavor,allow-ing pairing with a negligible gap.For this reason,it is more convenient to introduce h≡n sb/n sr,and minimize the free energy with respect to h.Using the above Eqs.weﬁnd the following linear set of equations:2n ur+n sr=n ub+n db+n sb(32)2n ur+n db=ξ(2n ur+n ub)(33)2n sr+h n sr=η(2n ur+n ub),(34)from which we obtain the number densities of each quark species in the paired phase as functions of only four quan-tities:n ub=24+η+2h−ηh−2ξ−hξ2−η+h+ηh+2ξ+hξn ur(36) n sb=6ηh2−η+h+ηh+2ξ+hξn ur(38) n e=2(2+h)(2−η−ξ)12π2+µ4eπ2¯µ2∆2−B,(40) g∆= fc n fcµfc n B,(41)where k fc=(µ2fc−m2fc)1/2,¯µ=µur,the chemical po-tentialsµfc are obtained fromn fc=µ3fcπ2f c=u r,u g,d r,d g(42)µfc=(3π2n fc)1/3f c=u b,d b(43)µfc=[(3π2n fc)2/3+m2s]1/2f c=s r,s g,s b(44) The minimization of g∆with respect to h gives h= 1and therefore the number densities are given by the following equations:n ub=4−2ξ1+ξn ur(46)n sb=2η1+ξn ur.(48)with n ug=n dr=n dg=n ur and n sg=n sr=n sb. Equations(40)-(48)constitute the equations of state for just deconﬁned quark matter.In the lower panel of Fig. 4we show∆g for particular values of the parameters (ξ=1.6,η=0.3).8∆ [M e V f m -3]B [Mev fm -3]FIG.5:The cparameter space ∆vs.B indicating the re-gions for which deconﬁnement is possible inside the maximum mass neutron star with the GM1EOS [31](M max =1.8M ⊙).We also indicate whether the ﬁnal state reached after β-equilibration of the just-deconﬁned phase has energy per baryon less or greater than the neutron mass (i.e.leads to the formation of strange stars or hybrid stars respectively).We adopted m s =150MeV for the strange quark mass.If (B,∆)fall inside the dashed region,deconﬁnement is not pos-sible even at the center of the maximum-mass star with this EOS.For (B,∆)inside the grey region the just-deconﬁned unpaired phase has always less energy per baryon than the just-deconﬁned paired phase.For (B,∆)in the white region the just-deconﬁned phase is always paired quark matter.The regions met at a point of coordinates (B ∗,∆∗)indicated with an asterisk and shown in Table I for diﬀerent values of the strange quark mass m s .The maximum of the grey region is indicated with a dot,and the corresponding value ∆max is shown in Table I for diﬀerent m s .032334781003003780150275498520024168979the maximum of the grey region(which is the same in Figs.5and6)and give the corresponding value∆max in Table I.We have also included in the parameter space the curve separating the regions in whichβ-stable quark matter has an energy per baryon smaller than the neutron mass from the region in whichǫ/n B(P=0)>m n(for simplicity, pairedβ-stable quark matter is assumed in all cases to be CFL).To the left of this curve theﬁnal state after β-equilibration is absolutely stable quark matter leading to the formation of strange stars.To the right,β-stable quark matter is restricted to the core of neutron stars (hybrid stars).The position of this curve also depends on the value of m s.In Figs.5and6it is shown for m s=150MeV(for more details the reader is refereed to [29]).V.DISCUSSIONIn this paper we have analyzed the deconﬁnement tran-sition from hadronic matter to quark matter,and investi-gated the role of color superconductivity in this process. We have deduced the equations governing deconﬁnement when quark pairing is allowed and,employing a realis-tic equation of state for hadronic matter,we have found the regions of the parameter space B versus∆where the deconﬁnement transition is possible inside neutron stars. The main results are shown in Figs.5and6and were explained in the last section.In the following we discuss some implications for neutron star structure.Stars containing quark phases fall into two main classes:hybrid stars(where quark matter is restricted to the core)and strange stars(made up completely by quark matter).This structural characteristic depends on whether the energy per baryon ofβ-equilibrated quark matter at zero pressure and zero temperature is less than the neutron mass(the so called“absolute stability”con-dition).In the absence of pairing,quark matter inβ-equilibrium has an energy per baryon(at P=0)smaller than the neutron mass only if B is in the range57MeV fm−3<∼B<∼90MeV fm−3.Within this range of B, unpairedβ-stable quark matter is the so called strange quark matter,and it is possible the existence of stars made up entirely by the quark phase.For B>∼90MeV fm−3unpairedβ-stable quark matter at P=0and T=0 decays into hadrons,and therefore it can be present only in the core of neutron stars.The size of the core(if any) depends on the value of B:the larger the value of B, the smaller the size of the quark matter core(for a given neutron star mass).Pairing enlarges substantially the region of the param-eter space whereβ-stable quark matter has an energy per baryon smaller than the neutron mass[29,30].Al-though the gap eﬀect does not dominate the energetics, being of the order(∆/µ)2∼a few percent,the eﬀect is substantially large near the zero-pressure point(which determines the stability and also the properties of the outer layers and surface of the star).As a consequence,a “CFL strange matter”is allowed for the same parameters that would otherwise produce unbound strange matter without pairing[29].The line separating strange mat-ter from non-absolutely stable quark matter is shown in dotted line in Figs.5and6,according to[29]. Concerning just deconﬁned quark matter(i.e.not in β-equilibrium)it has been already shown that the transi-tion to unpaired quark matter is not possible in a1.6M⊙neutron star if the Bag constant is B>∼126MeV fm−3, because the transition pressure is never reached inside the star,even in the proto-neutron star phase[21].The results when pairing is allowed have been shown in the previous section,where we have shown the“deconﬁne-ment”parameter space for the maximum mass neutron star with the GM1EOS(1.8M⊙),and for a1.6M⊙neutron star.As it is evident from Figs.5and6,de-conﬁnement is facilitated for large∆(i.e.it is possible for a larger range of B).This result can be roughly un-derstood if we think paired matter as unpaired matter with an eﬀective bag constant depending on the chemi-cal potential(or on density):B eﬀ(∆,µ)=B−A10pairs(B,∆)fall comfortably inside the dashed region of Fig.6where deconﬁnement is not allowed.For the same value of B,heavier stars(∼1.8M⊙)could deconﬁne, since(B,∆)would be inside the grey region of Fig.5, but the resulting conﬁguration would be not structurally stable and would form a black hole(c.f.[32]).Although the EOSs are diﬀerent in[32]and in the present work, this should not aﬀect this generic trend.Notice that qualitatively similar results have been found in[18,21] for unpaired quark matter.A stated in the Introduction,the transition from nu-clear matter to quark matter proceeds by bubble nucle-ation.However,notice that for largeB the results with typical surface tensionσ=10−30MeVfm−2do not dif-fer much from the case in bulk[17,18].This means that we don’t expect that the dashed region of Figs.5and 6will change signiﬁcantly when including surface eﬀects.Anyway,even if the surface tension were very large,the here presented bulk case is still relevant because it gives a lower limit for the transition:i.e.,if deconﬁnement is not possible in bulk,it will be even more diﬃcult when including surface eﬀects.In other words,the dashed line of Figs.5and6could move to the left in a more re-ﬁned study,but not to the right.A complete study of the astrophysical implications is in progress and will be published elsewhere.VI.ACKNOWLEDGEMENTSG.L.wants to thank FAPESP for support during an early phase of this work.We thank Jorge Horvath and Ettore Vicari for stimulating discussions.[1]J.Bardeen,L.N.Cooper and J.R.Schrieﬀer,Phys.Rev.108,1175(1957).[2]B.Barrois,Nucl.Phys.B129,390(1977)[3]D.Bailin and A.Love,Phys.Rep.107,325(1984),andreferences therein.[4]M.G.Alford,K.Rajagopal and F.Wilczek,Phys.Lett.B422,247(1998);R.Rapp,T.Sch¨a fer,E.V.Shuryak and M.Velkovsky,Phys.Rev.Lett.81,53(1998);M.G.Alford,Ann.Rev.Nucl.Part.Sci.51,131(2001). [5]M.G.Alford,K.Rajagopal,S.Reddy and F.Wilczek,Phys.Rev.D64,074017(2001).[6]M.Huang and I.Shovkovy,hep-ph/0311155and refer-ences therein.[7]K.Rajagopal and F.Wilczek,hep-ph/0011333[8]S.B.Ruester,I.A.Shovkovy,D.H.Rischke,Nucl.Phys.A743(2004)127-146[9]G.Nardulli,Riv.Nuovo Cimento25,1(2002)[10]M.Gyulassy and L.McLerran,Nucl.Phys.A750,30-63(2005);E.Shuryak,J.Phys.G30,S1221(2004) [11]I. A.Shovkovy,S. B.R¨u ster and D.H.Rischke,arXiv:nucl-th/0411040(2004)[12]I.M.Lifshitz and Y.Kagan,Soviet Phys.JETP35,206(1972)[13]C.Alcock,E.Farhi,and A.Olinto,Astrophys.J.310,261(1986)[14]J.E.Horvath and H.Vucetich,Phys.Rev.D59,023003(1998)[15]J.E.Horvath,Phys.Rev.D49,5590(1994)[16]F.Grassi,Astrophys.J.492,263(1998)[17]K.Iida and K.Sato,Phys.Rev.D58,2538(1998)[18]I.Bombaci,I.Parenti,I.Vida˜n a,Astrophys.J.614,314(2004)[17,18,19,20,21][19]M.L.Olesen and J.Madsen,Phys.Rev.D49,2698(1994)[20]G.Lugones and O.G.Benvenuto,Phys.Rev.D58,083001(1998)[21]O.G.Benvenuto and G.Lugones,Mon.Not.R.A.S.304,L25(1999)[22]Z.Berezhiani,I.Bombaci,A.Drago,F.Frontera and A.Lavagno;Astrophys.J.586,1250(2003).[23]A.Drago,vagno and G.Pagliara;Phys.Rev.D69,057505(2004)[24]E.J.Ferrer,V.de la Incera and C.Manuel,e-PrintArchive:hep-ph/0503162[25]M.Alford and K.Rajagopal,JHEP06,031(2002)[26]F.Csikor,G.I.Egri,Z.Fodor,S.D.Katz,K.K.Szab´o,A.I.T´o th,JHEP0405,046(2004)and references therein.[27]see M.Buballa,hep-ph/0402234and references therein.[28]I.Shovkovy and M.Huang,Phys.Lett.B564,205(2003)[29]G.Lugones and J.E.Horvath,Phys.Rev.D66,074017(2002)[30]G.Lugones and J.E.Horvath,Astronomy and Astro-physics,403,173(2003)[31]N.K.Glendenning and S.A.Moszkowski,Phys.Rev.Lett.67,2414(1991).[32]M.Alford and S.Reddy,Phys.Rev.D67,074024(2003)[33]Notice that the nucleation of an initial quark dropletmight be induced in principle by external inﬂuences such as high energy cosmic rays or neutrinos[13].However,es-timates of the production rates of quark droplets by neu-trino sparking[14]show that this mechanism is not likely to drive a neutron to quark conversion for realistic values of the minimum center of mass energy necessary to pro-duce a quark-gluon plasma in heavy ion collisions.Ultra high energy neutrinos would be also harmless because the outer crust acts as a shield due to the huge cross section[14].In this paper we are assuming that the conversionmust proceed trough an intermediate“twoﬂavor”phase, but other possibilities cannot be deﬁnitely excluded.。

LATTICE QCD AND THE STANDARD MODEL

a rXiv:h ep-ph/954249v16Apr1995FERMILAB-CONF-95/067-T hep-ph/9504249LATTICE QCD AND THE STANDARD MODEL ∗ANDREAS S.KRONFELD Theoretical Physics Group,Fermi National Accelerator Laboratory,P.O.Box 500,Batavia,IL 60510,USA ABSTRACT Most of the poorly known parameters of the Standard Model cannot be determined without reliable calculations in nonperturbative ttice gauge theory provides a ﬁrst-principles deﬁnition of the required functional integrals,and hence oﬀers ways of performing these calculations.This paper reviews the progress in computing hadron spectra and electroweak matrix elements needed to determine αS ,the quark masses,and the Cabibbo-Kobayashi-Maskawa matrix.1IntroductionMany contemporary reviews of elementary particle physics start by celebrating (or lamenting!)the success of the Standard Model.Indeed,with some nineteen∗parameters the SU(3)×SU(1)×U(1)gauge theory explains an enormous array of experiments.Even a terse compendium1of the experiments is more than big enough toﬁll a phone book.A glance at Table1shows,however,that roughly half of the parameters are not so well determined.To test the Standard Model stringently, and thus to gain an inkling of what lies beyond,we must learn the values of these parameters more precisely.Except for the mass of the Higgs boson(or any other undiscovered remnant of electroweak symmetry breaking),the poorly known parameters all involve quarks. Other than top,2which decays too quickly for conﬁnement to play a role,the masses of the quarks are a bit better than wild guesses.The information on the Cabibbo-Kobayashi-Maskawa(CKM)quark-mixing matrix is spotty,especially when one relaxes the assumption of three-generation unitarity,as shown in Table2.They are poorly determined simply because experiments measure properties not of quarks, but of the hadrons inside which they are conﬁned.Of course,everyone knows what to do:calculate with QCD,the part of the Standard Model that describes the strong interactions.But then,the strong coupling is known only at the5%level;not bad, but nothing like theﬁne structure or Fermi constants.Moreover,the binding of quarks into hadrons is nonperturbative—the calculations cannot be done on the back of an envelope.The most systematic technique for understanding nonperturbative QCD is lat-tice gauge theory.The lattice provides quantumﬁeld theory with a consistent and mathematically well-deﬁned ultraviolet regulator.Atﬁxed lattice spacing,the quan-tities of interest are straightforward(combinations of)functional integrals.These integrals can be approximated by a variety of techniques borrowed from statistical mechanics.Especially promising is a numerical technique,the Monte Carlo method with importance sampling,which has become so pre-eminent that the young and uninitiated probably haven’t heard of any other.Results from lattice-QCD Monte Carlo calculations have begun to inﬂuence Ta-ble1.The world average for the SU(3)gauge couplingαS includes results from lattice calculations of the quarkonium spectrum,4,5,6and at the time of this con-ference an even more precise result had appeared.7The same calculations are also providing some of the best information on the charm8and bottom9masses.This is an auspicious beginning.Over the next several years the lattice QCD calculations will mature.They will help to determine the other unknowns—light quark masses and the CKM matrix.The third column of Tables1and2lists relevant quantities or processes,and the rest of this talk explains how the programﬁts together.Table1:Parameters of the standard model and lattice calculations that will help de-termine them.Numerical values taken from the1994Review of Particle Properties,1 except m t(Ref.2),sinδ,andθQCD.The strong couplingαS refers to the gauge couplingsαem1/137.036105G F 1.166GeV−2αS0.116±0.005∆m1P–1S;scalingelectroweak massesm Z91.19GeVm H>58GeVlepton massesm e0.51100MeVmµ105.66MeVmτ1777MeVquark massesm u2–8MeV m2π,m2Km d5–15MeV m2π,m2Km s100–300MeV m2Km c 1.0–1.6GeV m J/ψm b 4.1–4.5GeV mΥm t174±10+13−12GeVCKM matrixs120.218–0.224K→πeνs230.032–0.048B→D∗lνs130.002–0.005B→πlνsinδ=0B K,B B,B BsQCD vacuum angleθQCD<10−9d nSect.2gives the non-expert some perspective on the conceptual and numerical strengths and weaknesses of lattice QCD.Sect.3reviews1)the status of the light hadron spectrum and the propects for extracting m u,m d,and m s;and2)results for the quarkonium spectrum,which yieldαS,m b,and m c.Sect.4outlines lattice QCD calculations of electroweak,hadronic matrix elements that are needed to pin down the unitarity triangle of the CKM matrix.There are,of course,many other inter-esting applications to electroweak phenomenology;for more comprehensive reviews2Table2:Ranges for CKM matrix elements|V qr|assuming unitarity but not three generations.Numerical values taken from the1994Review of Particle Properties. In three generations|V ud|=s12,|V cb|=s23,and|V ub|=s13,to excellent approxi-mation.parameter value or range related lattice calculationsall degrees of freedom in a block of size a4.The limit“a→0”can be obtained not only literally,but also by improving the action of the blockedﬁelds.These observations apply to any cutoﬀscheme for quantumﬁeld theory.A nice introduction to the renormalization-group aspects is a summer-school lecture by Lepage.11In lattice gauge theory a is nothing but the spacing between lattice sites. If there are N on a side,L=Na.For given N one can compute the integrals numerically.With the107–1010-dimensional integrals that arise,the only viable technique is a statistical one:Monte Carlo with importance sampling.To compute masses the observable O=Φ(t)Φ†(0),whereΦ(t)is an operator at time t with theﬂavor and angular-momentum quantum numbers of the state of interest.One can construct such operators using symmetry alone.The radial quantum number would require a solution of the theory,but that’s what we’re after.The“two-point function”Φ(t)Φ†(0) = n| 0|Φ|n |2e−m n t,(2)where the sum is over the radial quantum number.The exponentials are a happy consequence of the weight e−S in eq.(1).It is advantageous because at long times t only the lowest-lying state survives.In a numerical calculation masses are obtained byﬁtting two-point functions,once single-exponential behavior is veriﬁed.Since Φis largely arbitrary,some artistry enters:if single-exponential behavior sets in sooner,the statistical quality of the mass estimate is better.To compute a matrix element of part of the electroweak Hamiltonian,H,the observable O=Φπ(tπ)H(t h)Φ†B(0)for the transition from hadron“B”to hadron “π.”At long times t h and tπ−t h the“three-point function”Φπ(tπ)H(t h)Φ†B(0) ≈ 0|Φπ|π e−mπ(tπ−t h) π|H|B e−m B t h B|Φ†B|0 ,(3) plus excited-state contributions.If,as in decays of hadrons to leptons,the hadronic ﬁnal-state is the vacuum,a two-point function will do:H(t)Φ†B(0) = n 0|H|B n e−m n t B n|Φ†B|0 .(4)The desired matrix elements π|H|B and 0|H|B can be obtained from eq.(3) and(4),because the masses andΦ-matrix elements are obtained from eq.(2).To obtain good results from eqs.(2)–(4),it is crucial to devise nearly optimal op-erators in the two-point analysis.Consumers of numerical results from lattice QCD should be wary of results,still too prevalent in the literature,that are contaminated by unwanted states.In the numerical work that mostly concerns us here,the integrals are computed at a sequence ofﬁxed a’s and L’s.One adopts a standard mass,say mρ,and deﬁnes(amρ)lQCDa=to obtain the latice spacing in physical units,and other quantities are predicted viam B=(am B)lQCDmρ=limL→∞lima→0am B(a,L)But theβfunction of quenched and genuine QCD diﬀer,as one sees in perturbation theory,so one cannot expect agreement at all scales.As with any model,only inspecial cases can one argue that these eﬀects are correctable or negligible;thesecases will be highlighted in the rest of the talk.3From Hadron Spectra to the QCD Parameters3.1Light hadrons and light-quark massesOver the past few years a group at IBM has carried out a systematic calculationof the light-hadron spectrum using the dedicated supercomputer GF11.13They havenumerical data for5diﬀerent combinations of(a,L).At L≈2.3fm there arethree lattice spacings varying by a factor of∼2.5.At the coarsest lattice spacing(a−1≈1.4GeV)there are three volumes,up to almost2.5fm.A variety of quark masses are used,and the physical strange quark is reached by interpolation,whereasthe light(up and down)through extrapolation.The mass dependence is assumedlinear,as expected from weakly broken chiral symmetry,and the data substantiate the assumption.The units(ttice spacing)has beenﬁxed with mρand the quark masseswith mπand m K.Theﬁnal results,after extrapolation to the continuum limit andinﬁnite volume,are shown in Fig.1for two vector mesons and six baryons.(Thequark-mass interpolation could reach only the combination mΞ+mΣ−m N.)Despite the quenched approximation the agreement with experiment is spectacular.Fig.1also includes results from the same investigation for decay constants.The agreement of fπ/mρand f K/mρis not as good as for the masses.Because ofthe quenched approximation,this is not entirely unexpected.Recall the argumentconcerning distance scales and eﬀective theories in sect.2.1.The binding mechanismresponsible for the masses encompasses distances out to the typical hadronic radius. The decay constant,on the other hand,is proportional to the wavefunction at the origin and thus is more sensitive to shorter distance scales.One sees better agreement when forming the ratio f K/fπ,which—recall eq.(5)and subsequent discussion—is like retuning to the shorter distance.One would like to use the hadron masses to extract the quark masses.Becauseof conﬁnement,the quark mass more like a renormalized coupling than the classicalconcept of mass.Calculations like the one described above yield immediately thebare mass of the lattice theory.More useful to others would be the(m d+m u),2∆m2du=m2d−m2u,and m s.Ratios of the light-quark masses are currently best estimated using chiral perturbation theory.18To set the overall scale requires a dynamical calculation in QCD.In lattice QCD,ˆm and m s can be extracted from6K* φ N Ξ+Σ-N ∆ Σ* Ξ* Ω 10f π 10f K f K /f πm X/mρFigure 1:The spectrum and decay constants of the light hadrons.Error bars are from lattice calculations in the quenched approximation,13,14and •denotes experiment.the variation in the square of the pseudoscalar mass between m 2πand m 2K .Themost diﬃcult quark-mass combination is∆m 2du ,which causes the isospin-violatingend of the splittings in hadron multiplets.Since chiral perturbation theory provides a formula for ∆m 2du /m 2s with only second-order corrections,it is likely that the best determination of ∆m 2du will come from combining the formula with a lattice QCD result for m s .Using the compliation of quenched and unquenched results of Ukawa,19Mack-enzie 20has estimated ˆm MS (1GeV)∼65MeV.Thesymbol ∼stresses the lack of error bar.This is outside the ranges of 3.5–11.5MeV and 100–300MeV indicated in Table 1.A more recent analysis of the strange quark ﬁnds m3.2Quarkonia,αS,and heavy-quark massesQuarkonia are bound states of a heavy quark and heavy anti-quark.Threefamilies of states exist,charmonium(ηc,J/ψ,etc),bottomonium(ηb,Υ,etc),andthe as yet unobserved B c(b¯c and¯bc bound states).Compared to light hadrons,these systems are simple.The quarks are nonrelativistic,and potential models givean excellent empirical description.But a fundamental treatment of these systemsrequires nonperturbative QCD,ttice QCD.†Potential models can be exploited,however,to estimate lattice artifacts,and in the quenched approximation they canbe used to make corrections.Many states have been observed in the lab,providingcross-checks of the methodology of uncertainty estimation.Once the checks are satisfactory,one can use the spectra to determineαS,m c,and m b.One can also have some conﬁdence in further applications,such as the phenomenology of D and B mesons discussed in sect.4.For charm,and especially for bottom,the quark mass is close to the ultravioletcutoﬀ,1/a orπ/a,of present-day numerical calculations.Originally lattice gaugetheory was formulated assuming m q a≪1,so quarks m q a∼1require some reassess-ment.There are four ways to react.The patient,stolid way is to wait ten years,until computers are powerful enough to reach a cutoﬀof20GeV—not very inspir-ing.The naive way is to extrapolate from smaller masses,assuming the m q a≪1interpretation of the lattice theory is adequate;history shows that naive extrapola-tions can lead to naive and,thus,unacceptable error estimates.The insightful wayis to formulate an eﬀective theory for heavy quarks with a lattice cutoﬀ;22,23thisis the computationally most eﬃcient approach,and when the eﬀectiveness of the heavy-quark expansion is a priori clear,it is the method of choice.The compulsiveway to examine a wide class of lattice theories without assuming either m q a≪1orm q≫(ΛQCD,a−1);by imposing physical normalization conditions on masses and matrix elements,it is possible to interpret the correlation functions at any valueof m q a.24The underlying reason is that the lattice theory is completely compatiblewith the heavy-quark limit,so the mass-dependent interpretation connects smoothlyonto both the insightful method for m q a≫1and the standard method for m q a≪1.Fig.2shows the charmonium spectrum,on a scale appropriate to the spin-averaged spectrum.Light quark loops are quenched in these calculations.25,26Theagreement with experimental measurements is impressive,but Fig.2barely displaysthe attainable precision.Fig.3shows theﬁne and hyperﬁne structure of the P states, now for bottomonium.25(The1P1state h b has not been observed in the lab;the h c has been seen.)The authors of Ref.25also have results with the virtual quark loops from two light quarks,i.e.up and down are no longer quenched,but strange still is.The agreement is comparable.27To obtain these results only two parameters have been adjusted.The standardFigure2:A comparison of the charmonium spectrum as calculated in lattice QCD, using two diﬀerent methods.Ref.25:◦,Ref.26:2.From Ref.6.mass in eq.(5)is∆m1P–1S,the spin-averaged splitting of the1P and1S states,which is insensitive to the quark mass.By the renormalization group,this is equivalent to eliminating the bare gauge coupling,or to determiningΛQCD.The bare quark mass is adjusted to obtain the spin average of the1S states that is measured in the lab. Otherwiseﬁgs.2and3represent predictions of quenched QCD.The success of these calculations permits one to extract the basic parameters,αS and m q.There are four steps:pute the charm-and bottomonium spectra with n f,MC=0,2or3ﬂavorsof virtual quark loops.(n f,MC=0corresponds to the quenched approxima-tion;n f,MC=2quenches just the strange quark;n f,MC=3would be the real world.)2.With perturbation theory,convert the bare lattice couplingα(n f,MC)0to thequark-potential(V)orMS scheme.The natural scale for this conversion is near(but not quite29)π/a.3.Unless n f,MC=3,correct for the quenched approximation.9−40−2020MeVχb 1r MS (π/a )and am460MeV ,(8)where the numerator is the 1P–1S splitting in lattice units.Steps1and 4are explained above.Step 2requires one-loop perturbation theory,suitably optimized.29Step 3is crucial,because without it the results have no busi-ness in Table 1.Consider ﬁrst αS ,and recall the idea of treating the quenched approximation as an eﬀective theory.One sees that the couplings are implicitly matched at some scale q Q characteristic of quarkonia.So the matching hypothesis,supported by ﬁgs.2and 3,asserts α(n f,MC )S (q Q )=α(3)S (q Q ).(9)Potential models tell us that 200<q c <800MeV and 200<q b <1400MeV.Step 3yields α(n f,MC )S (π/a ),so one can use the two-loop perturbative renormalization group to run from π/a to q Q .The perturbative running is an overestimate if q Q is10taken at the lower end of these ranges.‡This argument was used for the original lattice determinations of the strong coupling,4,5and its reliability was conﬁrmed in n f,MC=2calculations.28Currently the most accurate result is from Ref.7,α(3)V(8.2GeV)=0.196±0.003,(10)based on n f,MC=0and n f,MC=2results,with an extrapolation in n f.The V scheme is preferred for the matching argument,not only for physical reasons,but also because of its empirical scaling behavior.29The scaling behavior implies that one can run with the two-loop renormalization group to high scales and convert to other schemes.For comparison to other determinations,eq.(10)corresponds toαMS.To determine the quark mass the one applies the same renormalization-group argument.But,quark masses don’t run below threshold!9Hence,for heavy quarks§m(n f,MC) Q (m Q)=m(n f,MC)Q(q Q)=m(3)Q(q Q)=m(3)Q(m Q).The only corrections areperturbative,from lattice conventions toMS convention may be more appropriate.Eq.(12b)corresponds to mb,‡For light hadrons,qlight∼ΛQCD,so there would be no perturbative control whatsoever.§For light quarks the threshold is deep in brown muck,and all bets are oﬀ.11North American,Japanese,and European taxpayers provide us with lots of money for the relevant experiments,because they want to know the CKM factors.But unless we calculate the inherently nonperturbative QCD factor,they will be sorely disappointed.It is convenient to start with the assumption of three-generation unitarity.ThenV ud V∗ub+V cd V∗cb+V td V∗tb=0,(14) an equation that prescribes a triangle in the complex plane.Dividing by V cd V∗cb and writing V ud V∗ub/V cd V∗cb=¯ρ+i¯η,one sees that unitarity predictsV td V∗tb= G2F q2dq2Figure4:The Isgur-Wise functionξ(ω)(essentially the form factor A1of the text) from lattice QCD and CLEO.The kinematic variableω=v a·v b=1−(q2max−q2)/2m B m D∗.From Ref.36.corrections31(estimated to be small)and known radiative corrections.Other form factors,which are phase-space suppressed near q2max,are also related by heavy-quark symmetry to A1(q2).Hence,eq.(16)provides an essentially model-independent32 way to determine|V cb|.The diﬃculty with the model-independent analysis is that the decay rate vanishes at q2max.To aid experimentalists’extrapolation to that point,several groups33,34,35 have used quenched lattice QCD to compute the slope of A1.A typical analysis is toﬁt the slope to lattice-QCD numerical data,and thenﬁt the normalization to CLEO’s experimental data,as shown in Fig.4.For example,Simone of the UKQCD Collaborationﬁnds36|V cb|=0.034+3+2−2−2 1.49ps.(17) Theﬁrst error is experimental;the second is from the lattice-QCD slope.Unfor-tunately,it is not clear how to correct for the quenched approximation,and the13associated uncertainty has not been estimated.Moreover,consistency checks of varying lattice spacing,volume,etc,are still in progress.Nevertheless,the overall consistency with experiment,shown in Fig.4,is encouraging.|V ub|can be obtained from the semi-leptonic decays B→ρlνand B→πlν. Expanding in q2near q2max=(m B−mπ)2,the diﬀerential decay rate for B→πlνreadsdΓ24π3 mπ2G2F m2Wm2K f2KˆB K|V ud V us|2|V cb|2×3¯η |V cb|2(1−¯ρ)y tη2f2(y t)+y c(η3f3(y t)−η1) ,(19) where y q=m2q/m2W.This formula assumes three-generation unitarity and neglects the deviation of|V cs|and|V tb|from unity.Theηi and f i multiplying the CKM factors arise from box diagrams and their QCD corrections.37The nonperturbative QCD factor is8m s.The23%uncertainty is an estimate of O(m s−m d)bining the errors and converting to the renormalization-group invariant that appears in eq.(19),one ﬁnds38ˆBK=(αThis result places a high standard on calculations ofˆB K,whether by lattice QCD or any other method.Would-be competitors must not only reach10%uncertainties, they must do so with an error analysis as thorough and forthright as Ref.38.Mixing in the B0–¯B0system is also sensitive to V td.In the Standard Model the mass splitting is given byx d=∆m Bd16π2m BdηB f2(y t) 83m2Bqf2BqˆBB q,whichis the B q–¯B q transition matrix element of a∆B=2operator.The calculation of the decay constant f B has received a great deal of attention over the last several years,42but the matrix element needed here,8MS =1.3±0.2,mt,¶Nonperturbative QCD is needed to extract these results!15-1-0.500.51¯ρ¯00.51¯η¯Figure 5:Constraints on (¯ρ,¯η)from |εK |(solid hyperbolae),|V ub /V cb |(dashed circles with origin (0,0)),and x d /x s (dash-dotted circles with origin (1,0)),and contempo-rary uncertainties.What if lattice QCD calculation and the experiments improve?Consider for the masses m c,MS=175±5,(27)for the hadronic matrix elementsˆBK =0.825±0.027,|f B d /f B s |=0.90±0.02,|B B d /B B s |=1.0±0.1,(28)in particular eliminating almost all the statistical error in ˆBK ;for “experimental”CKM results |V cb |=0.035±0.002,|V ub /V cb |=0.080±0.004“low,”=0.091±0.004“high,”(29)and for neutral B mixing measurementsx d =0.72±0.04,x s =18±2.(30)Fig.6shows how this 5–10%level of precision improves the limits on (¯ρ,¯η).The16-1-0.500.51¯ρ¯00.51¯η¯Figure 6:Constraints on (¯ρ,¯η)from |εK |(solid hyperbolae),low |V ub /V cb |(dashed circles with origin (0,0))or high |V ub /V cb |(dotted circles with origin (0,0)),and x d /x s (dash-dotted circles with origin (1,0)),and improved (5–10%)uncertainties.wildest guess remains x s ,so ignore the dashed-dotted curves momentarily.The region allowed by the hyperbolic band from εK and the circular band from |V ub /V cb |shrinks if |V ub |is too small.The tension between these two constraints is partly a consequence of the low value of |V cb |suggested by eq.(17).Increasing |V cb |brings the hyperbolic band down more rapidly than it shrinks the circular band.If the real-world values of |V cb |and |V ub /V cb |allow a sizable region,as for the dot-ted circles in Fig.6,neutral B mixing becomes crucial.The constraint becomes more restrictive as x s increases.Unfortunately,the experimental measurement becomes more diﬃcut as x s increases.If it proves impossible to obtain useful information on x s ,one can return to eq (22),and focus on x d alone.The lattice-QCD calculations of 8seem fairly estimated.The quenched approximation is another matter.In quarko-nia,one can correct for it with potential models,yielding determinations ofαS andthe charm and bottom masses.The quenched error in B K is also thought to beunder control,and—taking the error bars at face value—B K is no longer the limit-ing factor in the|εK|constraint on the unitarity triangle.A second class consists of f B,the semi-leptonic form factors of K and D mesons(not discussed in thistalk,but see Ref.42),and the Isgur-Wise function.These quantities are essentialfor direct determinations of theﬁrst two rows of the CKM matrix.The quenched-approximation calculations are in good shape,but the the corrections to it cannotbe simply estimated.A third class consists of the semi-leptonic decay B→πlνandneutral B mixing,for which only exploratory work has appeared.Nevertheless,all QCD quantities discussed here will follow a conceptually clear path to ever-more-precise results.The next ten years or so will almost certainly wit-ness computing and other technical improvements that will allow for wide-ranging calculations without the quenched approximation.By then the most eﬃcient tech-niques for extracted the most relevant information will have been perfected.AcknowledgementsThe Adriatic glistened in the moonlight as it lapped against the quay.In the bar of the Hotel Neptun a winsome lounge singer cooed“...strangers in the night, exchanging glances,....”I looked up at the waiter and said,“Molim pivo,”when a man strolled into the bar,clapped me on the back,and cried,“Ay,you kook,what’s new?”In a diﬀerent place and a diﬀerent time he had rescued my sanity,if not my life.I peered into his eyes and replied,“It’s the same old story,always the same. And even when it changes,it always ends the same way:Fermilab is operated by Universities Research Association,Inc.,under contract DE-AC02-76CH03000with the U.S.Department of Energy.”References1.L.Montanet,et al(Particle Data Group),Phys.Rev.D50(1994)1173.2.F.Abe,et al(CDF Collaboration),Phys.Rev.Lett.73(1994)225;Phys.Rev.D50(1994)2966.These papers claim“evidence for,”not“observation of”the top quark.Since the Adriatic Meeting both CDF(F.Abe,et al,Fermi-lab report FERMILAB-PUB-95-022/E,hep-ex/9503002)and D0(S.Abachi, et al,Fermilab report FERMILAB-PUB-95-028/E,hep-ex/9503003)report observation,without changing the mass signiﬁcantly.3.A.S.Kronfeld and P.B.Mackenzie,Annu.Rev.Nucl.Part.Sci.43(1993)793.4.A.X.El-Khadra,G.Hockney,A.S.Kronfeld,and P.B.Mackenzie,Phys.Rev.18Lett.69(1992)729.5.G.P.Lepage and J.Sloan,Nucl.Phys.B Proc.Suppl.34(1994)417.6.A.X.El-Khadra,Nucl.Phys.B Proc.Suppl.34(1994)141;Ohio State Uni-versity report OHSTPY-HEP-T-94-020(hep-ph/9412279),to appear in the proceedings of Physics in Collision(1994).7.C.T.H.Davies,K.Hornbostel,G.P.Lepage,A.Lidsey,J.Shigemitsu,and J.Sloan,Phys.Lett.B345(1995)42.8.A.X.El-Khadra and B.P.Mertens,Ohio State University report OHSTPY-HEP-T-021,(hep-lat/9501022).9.C.T.H.Davies,K.Hornbostel,ngnau,G.P.Lepage,A.Lidsey,C.J.Morn-ingstar,J.Shigemitsu,and J.Sloan,Phys.Rev.Lett.73(1994)2654.10.S.R.Sharpe,University of Washington report UW-PT-94-15(hep-ph/9412243);A.Soni,Brookhaven National Laboratory report BNL-61378.11.G.P.Lepage,“What is Renormalization?”in From Actions to Answers,editedby T.DeGrand and D.Toussaint(World Scientiﬁc,Singapore,1990).12.M.L¨u scher,Commun.Math.Phys.104(1986)177,105(1986)153.13.F.Butler,H.Chen,J.Sexton,A.Vaccarino,and D.Weingarten,Phys.Rev.Lett.70(1993)729;Nucl.Phys.B430(1994)179.14.F.Butler,H.Chen,J.Sexton,A.Vaccarino,and D.Weingarten,Nucl.Phys.B421(1994)217.15.R.Groot,J.Hoek,and J.Smit,Nucl.Phys.B237(1984)111;M.F.L.Golterman and J.Smit,Phys.Lett.140B(1984)392.16.C.J.Morningstar,Phys.Rev.D48(1993)2265.17.A.S.Kronfeld and B.P.Mertens,Nucl.Phys.B Proc.Suppl.30(1993)49518.J.Gasser and H.Leutwyler,Phys.Rept.87(1982)77.An update is H.Leutwyler,in Perspectives in the Standard Model,edited by R.K.Ellis,C.T.Hill and J.D.Lykken(World Scientiﬁc,Singapore,1992).awa,Nucl.Phys.B Proc.Suppl.30(1993)3.20.P.B.Mackenzie,Nucl.Phys.B Proc.Suppl.34(1994)400.21.C.R.Allton,M.Ciuchini,M.Crisafulli,E.Franco,V.Lubicz,and G.Mar-tinelli,Nucl.Phys.B431(1994)667.22.G.P.Lepage and B.A.Thacker,Nucl.Phys.B Proc.Suppl.4(1987)199;B.A.Thacker and G.P.Lepage,Phys.Rev.D43(1991)196;W.E.Caswell and G.P.Lepage,Phys.Lett.167B(1986)437;G.P Lepage,L.Magnea,C.Nakhleh,U.Magnea,and K.Hornbostel,Phys.Rev.D46(1992)4052.1923.E.Eichten,Nucl.Phys.B Proc.Suppl.4(1987)170;E.Eichten andF.Feinberg,Phys.Rev.Lett.43(1979)1205;Phys.Rev.D23(1981)2724;E.Eichten and B.Hill,Phys.Lett.B234(1990)511;B243(1990)427.24.A.S.Kronfeld,Nucl.Phys.B Proc.Suppl.30(1993)445;P.B.Mackenzie,Nucl.Phys.B Proc.Suppl.30(1993)35;A.X.El-Khadra,A.S.Kronfeld,and P.B.Mackenzie,to appear.25.C.T.H.Davies,K.Hornbostel, ngnau,G.P.Lepage, A.Lidsey,J.Shigemitsu,and J.Sloan,Phys.Rev.D50(1994)6963.26.A.X.El-Khadra,Nucl.Phys.B Proc.Suppl.30(1993)449;A.X.El-Khadra,G.M.Hockney,A.S.Kronfeld,P.B.Mackenzie,T.Onogi,andJ.N.Simone,in progress.27.J.Sloan,Florida State University report FSU-SCRI-94C-129(hep-lat/9412095).28.S.Aoki,M.Fukugita,S.Hashimoto,N.Ishizuka,H.Mino,M.Okawa,T.Onogi,and awa,Phys.Rev.Lett.74(1995)22.29.G.P.Lepage and P.B.Mackenzie,Phys.Rev.D48(1993)2250.30.A.J.Buras,utenbacher,and G.Ostermaier,Phys.Rev.D50(1994)3433.31.M.Luke,Phys.Lett.B252(1990)447.32.M.Neubert,Phys.Lett.B264(1991)455.33.C.Bernard,Y.Shen,and A.Soni Phys.Lett.B317(1993)164.34.S.P.Booth,et al(UKQCD Collaboration),Phys.Rev.Lett.72(1994)462.35.J.E.Mandula and M.C.Ogilvie,hep-lat/9408006.36.J.N.Simone(UKQCD Collaboration),Nucl.Phys.B Proc.Suppl.34(1994)486.37.T.Inami and C.S.Lim,Prog.Theo.Phys.65(1981)297;(E)65(1981)772.38.S.R.Sharpe,Nucl.Phys.B Proc.Suppl.34(1994)403;G.W.Kilcup,S.R.Sharpe,R.Gupta,and A.Patel,Phys.Rev.Lett.64(1990)25.39.N.Ishizuka,M.Fukugita,H.Mino,M.Okawa,Y.Shizawa,and awa,Phys.Rev.Lett.71(1993)24.40.G.W.Kilcup,Phys.Rev.Lett.71(1993)1677.41.S.R.Sharpe,Phys.Rev.D41(1990)3233;C.W.Bernard and M.F.L.Golterman,Phys.Rev.D46(1992)853.42.Some recent reviews with useful bibliography include Ref.10and:R.Sommer,CERN report CERN-TH-7498-94(hep-lat/9411024);20。

A Comprehensive Four-Quark Interpretation of D_s(2317), D_s(2457) and D_s(2632)

a r X i v :h e p -p h /0407062v 1 6 J u l 2004A Comprehensive Four-Quark Interpretation of D s (2317),D s (2457)and D s (2632)Yu-Qi Chen 1and Xue-Qian Li 21.Institute of Theoretical Physics,Chinese Academy of Sciences,100080,Beijing,China.2.Department of Physics,Nankai University,300071,Tianjin,China.Abstract The recently observed new member of the charm-strange family D s (2632)which has a surprisingly narrow width is challenging our theory.D s (2317)and D s (2457)which were observed earlier have similar behaviors and receive various theoretical explanations.Some authors use the heavy hadron chiral eﬀective theory to evaluate heavy-light quark systems and obtain a reasonable evaluation on the masses of D s (2317)and D s (2457).An alternative picture is to interpret them as four-quark or molecular states.In this work,we are follow-ing the later and propose a unitive description for all the three new members D s (2632),D s (2317)and D s (2457)and at least,so far our picture is consistent with the data.1.Introduction Recently,the SELEX collaboration reported a new observation of a narrow resonance D s (2632)with a mass of 2632±1.6MeV /c 2and total width <17MeV [1].This resonance is the heaviest member,so far,of the charm-strange family.D s (2632)and the other two resonances D s (2317)and D s (2457),which were observed earlier[2,3,4],constitute an exotic group which is obviouslydistinguished from the ordinary D s (1968)and D ∗s (2112)by the masses,narrow total widths and decay modes.The total decay widthes of these states seem to be too small to be understoodin the regular q ¯q ′structure.If the newly observed resonances D s (2632)were of a simple quark structure of c ¯s ,one would expect that the heavier the resonance is,the shorter its lifetime should be,but the observation seems not to follow this pattern.These unusual features challenge our present theory.There have been some theoretical models proposed to interpret the structures of D s (2317)and D s (2457)which are most likely to possess spin-parity as 0+and 1+respectively [5,6].For example,Cahn and Jackson [7,8]studied the new forms of the spin-orbit and tensor forces in the heavy-light quark system and applied the technique to the D s (2317)meson.In the special models,a mass diﬀerence was predicted which is close to the experimental data.Especially,two groups independently work on the spectra by using the heavy hadron chiral eﬀective theory[9,10].In their work,they study the heavy-light quark hadrons where the doublet [0+,1+]stands as a chiral partner of the doublet [0−,1−].Thus the mass diﬀerence between the two doublets can be accounted by the Godberg-Trieman relation and is roughly m N /3where m N is the mass of nucleon.It is consistent with the present experimental data.Instead,the authors of other theoretical papers suggested that the newly observed D s (2317)and D s (2457)are not in the simple c ¯s composition,but possibly are four-quark states,or mixtures of four-quark states and c ¯s [11,12,13].The discovery of D s(2632)which is650MeV heavier than the ground state D s,indicates that all three D s(2317)D s(2457)and D s(2632)might constitute a new group which is diﬀerent from the normal members of the charm-strange family.Thus we are motivated to search for a unitive picture to describe all of the three new resonances.In this work we follow the four-quark picture and see if it canﬁt the data.Concretely,it is postulated that all the three resonances are four-quark states and have special spin-parity structures with narrow total widths.Since all of the resonances are isospin singlets,and possibly are four-quark states,we would determine their SU(3)quark structures according to the general principle.We stress that in this picture, we postulate that all the quarks are in S-waves.Since c¯s is an iso-singlet,thus to constitute a four quark state which is still an iso-singlet,the extra q¯q constituent must be an iso-singlet.As well known,in the SU(3)framework,an iso-singlet can have two independent components,i.e. u¯u+d¯d and s¯s.Because the orbital angular momenta for all constituents are zero,the dynamics (which is still not very clear so far)permits one of the two components to dominate in a physical resonance.We propose that in D s(2317)and D s(2457),the u¯u+d¯d component dominates, namely they are of structure c¯s(u¯u+d¯d),while for D s(2632),the s¯s component dominates,so the composition is mainly c¯s s¯s which is exactly an isospin partner of the former one.Of course, QCD dynamics allows reaction s¯s→u¯u+d¯d,which can occur by exchanging only one gluon if they are in color octet,and then might mix the two structures.But generally as the system is in a color-octet,the interaction between the quark and anti-quark is repulsive and the annihilation process is strongly suppressed by the wavefunction.Thus the mixing between the two states can be,in general,neglected.2.The D s(2317)and D s(2457)states.As proposed,D s(2317)and D s(2457)would have the quark structure c¯s(u¯u+d¯d)which guarantees the iso-singlet requirement.Generally,they can be in either(c¯s)8(u¯u+d¯d)8or (c¯s)1(u¯u+d¯d)1where the subscripts1and8refer to the color singlet and octet respectively, and may also be a mixture of the two states.With a color re-combination,the former one can be transformed into a(c¯u(¯d))1⊗(u(d)¯s)1structure,which is a DK molecule.The molecular structure was proposed to interpret some mesons such as f0(980)and a0(980)[15],several states of charmonium spectroscopy[16]andχ(3.86)[17].Here we will show that D s(2317) and D s(2457)can be accommodated in the molecular structure.Actually,D s(2317)may be a molecule of DK mesons,whereas D s(2457)is a D∗K molecule.The data support the picture of a molecular structure.The masses of D s(2317)and D s(2457) are very close and slightly below the thresholds ofM D0+M K+∼M D++M K0∼(2358∼2367)MeV,and M D∗+M K∼(2501∼2508)MeV. This is consistent with the molecular picture[15].When the mass is near the threshold,c and ¯u(¯d)quarks tend to form a D meson while u(d)and¯s quarks tend to form a K meson.The two clusters may be separated by a sizable distance.Moreover,it is noticed that the mass diﬀerences follow the relationsM D∗−M D≈M D∗s −M Ds≈M Ds(2457)−M Ds(2317)∼140MeV.(1)These relations can be understood in the molecular picture and imply that D s(2317)and D s(2457)are indeed molecular states of DK and D∗K in S-wave respectively.In a system where a heavyﬂavor is involved,the mass diﬀerence arises from spin-ﬂip of the charm quarkand is suppressed by1/m c according to the heavy quark eﬀective theory(HQET)[20].In the molecular states DK or D∗K the K-meson is pseudoscalar which is blind to the charm spin,the energy diﬀerence of D s(2317)and D s(2457)should completely come from the mass diﬀerence of D and D∗.Thus the relations(1)is nicely understood.The main decay modes of D s(2317)is D sπ.Because of the constraint of theﬁnal state phase space,D s(2317)→D+K is forbidden.The only available channel isD s(2317)→D+s+π0,(2) where theﬂavor is conserved.However,this decay mode does not conserve isospin which must be retained in the processes governed by strong interaction.A possible explanation is that the reaction is realized through D s(2317)decay into D s and a virtualη.With a virtualη−πmixing it eventually comes to D s+π[13].Certainly,this process is highly suppressed.This explains why the decay width is so narrow.For D s(2457),its main channel is D s(2457)→D∗s+π0,which is also an isospin violated process and suﬀers from the same suppression as for D s(2317)→D+s+π0.The experiment[4]has observed the decay channel D s(2457)→D s+γwith a small branch-ing ratio.In our picture,this process can happen via u¯u+d¯d annihilating into photons.One gluon or photon exchange between two constituent mesons is necessary to make this process possible.Therefore,the rate is small.There has also been a search for D s+ππchannel and a negative result was reported.In fact,there are several suppression eﬀects for the process. The decay of D s(2457)which is believed as1+,into D sππor D∗sππis a P-wave reaction and moreover,it is a three-bodyﬁnal state process,the P-wave suppression and theﬁnal state phase space integration greatly reduce the decay rate.3.D s(2632).The newly observed resonance raises a challenge to the theory.In our picture,it is the SU(3) partner of D s(2317)and its peculiar narrow total width can be naturally understood.D s(2632)is supposed to be dominated by the c¯s(s¯s)structure.Since s¯s can only reside in η,η′,φ,one might suspect if D s(2632)could be a D sηmolecule,however,it is unlikely because its mass is100MeV above the threshold of D s andη.There is no suitable molecular structure which canﬁt this quark composition(as long as the spin-parity of D s(2632)is0+(in analog to D s(2317)).Thus we suppose that D s(2632)is a pure four-quark state.An intuitive picture of this state could be that the c quark stays in the middle and the three s quarks with gluon clouds move around it.Its mass is3m s+m c+∆E B where∆E B is the binding energy and can only be evaluated by invoking concrete models.Since constituent mass of strange quark is about150MeV heavier than the constituent mass of u and d quarks,the state of c¯s(s¯s)should be roughly300MeV heavier than c¯s(u¯u+d¯d),and this simple evaluation is consistent with the measurement.Because of the special structure,one can expect the total width of D s(2632)to be small. The reason is the Pauli exclusive principle.As well known,the destructive interference between two anti-d-quarks in D+→Kπcan remarkably reduce the total width of D±and makes its lifetime to be2.5times longer than that of D0[18,19].The same mechanism may apply here. The two anti-strange quarks in the initial state would eventually join the other quarks(c and s)to constituteﬁnal mesons,and an interchange of the two anti-s-quarks can result in a minus sign or in other words,a destructive interference.The most natural decay mode is D s(2632)→D sηwith the s¯s pair forming anηand c¯s forming a D s meson.With the s¯s annihilates into u¯u+d¯d pair,it can also decay to D+K. This is a higher order process,so that is suppressed.Moreover,the color matching of the quarks (anti-quarks)in the mesons leads to an additional suppression.The experimental data show that the DK decay channel is6times smaller than the D sηchannel and it is indeed understandable in our picture.4.Predictions.In this four-quark state picture,one straightforward prediction is the existence of the vector partner of D s(2632).By a simple SU(3)f manipulation,we can guess the mass of a new1+resonance D∗sj.The following relationM Ds(2632)−M Ds(2317)=M D∗sj−M Ds(2457),(3)seems to hold and we can expect the mass of D∗sj to beD∗sj≈2770MeV/c2.(4) Since it is not a molecular state(similar to D s(2632)),one cannot expect to make a precise estimate on its mass,so that if a measured mass is50MeV deviated from our predicted value, it is not surprising.D∗sj is a1+meson whose total width is small,i.e.a narrow resonance and is of quantum number of c¯s(or¯c s).Its decay modes should be dominated by D∗sηand D∗K. This narrow resonance should be tested in the near future experiments.Another interesting prediction is that these states should have decay modes to e+e−pair with the light quark pair inside the hadron annihilates into two photons with relatively large branching ratios.Speciﬁcally,we would consider D s(2632)→D∗s e+e−,D s(2317)→D∗s e+e−and D s(2457)→D s e+e−.In the normal c¯s meson,the branching ratios of these decay modes are very tiny.The measurement on these decays will be a crucial test to the four quark state explanation.5.Conclusion.In this letter,we propose a simple unitive picture for the three new members of the D s family which are observed in recent experiments.We consider D s(2317)and D s(2457)to be of c¯s(u¯u+d¯d)quark composition and moreover motivated by the data,it is supposed that the quarks(antiquarks)are re-combined into molecular ly,D s(2317)is a molecular state of DK while D s(2457)is a D∗K molecular state.By constrast,the newly observed D s(2632)is of the c¯s(s¯s)structure and it is exactly the partner of D s(2317)of0+.In this picture,we have naturally explain the narrowness of the widths of all the three resonances and discussed some decay modes.We further predict another narrow resonance which is a1+meson and its mass is located at about2770MeV with a narrow width,we also predict that these states should have relatively large branching ratios for decaying into D(∗)s and a e+e−pair.The four-quark pictures for the three states accommodate the newly observed resonances D s(2317),D s(2457)and D s(2632)and everything seems to be consistent with all existing exper-imental data.Once those predictions are conﬁrmed by further experiments,the existence of the four quark structure can be established.Then it may open a new page in the hadronic physics.In this short note,we do not account for any concrete dynamics,but only apply the SU(3)ﬂa-vor symmetry to analyze the quark structure of all the states D s(2317),D s(2457)and D s(2632) and try to understand their qualitative behaviors,such as the very narrow total widths and some decay modes.Since the dynamics is not involved,some details would be missing,as we discussed in last section about our prediction.It is,of course,interesting and compelling to pursue this line by adding the dynamics in the calculations.Acknowledgment:One of the authors(YQC)wishes to thank C.Liu for bringing this subject to his attention. This work is partially supported by the National Natural Science Foundation of China.References[1]A.Evdokimov et al.The SLEX Collaboration,hep-ex/0406045.[2]B.Aubert etal.The BABAR collaboration,Phys.Rev.Lett.90(2003)242001;F.Porter,hep-ex/0312019.[3]P.Krokovny et al.The BELLE collaboration,hep-ex/0308019.[4]D.Besson et al.The CLEO Collaboration,Phys.Rev.D68(2003)032002.[5]A.Pompili,hep-ph/0311036.[6]F.Porter,hep-ex/0312019.[7]R.Cahn and D.Jackson,hep-ph/0305012.[8]M.DiPierro and E.Eichten,Phys.Rev.D64(2001)114004.[9]W.A.Bardeen,E.J.Eichten and C.T.Hill,Phys.Rev.D68,054024(2003)[arXiv:hep-ph/0305049];W.A.Bardeen and C.T.Hill,Phys.Rev.D49,409(1994)[arXiv:hep-ph/9304265].[10]M.A.Nowak,M.Rho and I.Zahed,Phys.Rev.D48,4370(1993)[arXiv:hep-ph/9209272];M.A.Nowak,M.Rho and I.Zahed,arXiv:hep-ph/0307102.[11]T.Barnes,F.Close and H.Lipkin,hep-ph/0305025;A.Szczepaniak,hep-ph/0305060.[12]E.Beveren and G.Rupp,hep-ph/0305035.[13]H.Cheng and W.Hou,hep-ph/0305038.[14]H.Georgi,Lie Algebra in Particle Physics,Benjamin/Cummings Pub.Co.1982,New York.[15]J.Weinstein and N.Isgur,Phys.Rev.D41(1990)2236.[16]A.Rujula,H.Georgi and S.Glashow,Phys.Rev.Lett.38(1977)317.[17]S.F.Tuan,Phys.Lett.bf B473(2000)136.[18]I.Bigi,B.Blok,M.Shifman,N.Uraltsev and A.Vainstein,in B decays,edited by S.Stone,World Scientiﬁc Pub.Co.1994.[19]W.Dai,X.He,X.Li and G.Zhao,Phys.Rev.D60(1999)034005.[20]N.Isgur and M.Wise,Phys.Lett.B232(1989)113;237(1980)527.。

Holder Inequalities and Bounds on the Masses of Light Quarks

2 2 ¯ = 4m q ( mu + md ) u ¯u + dd ¯q = −2fπ mπ
(10)
where fπ = 93 MeV. We will use recent determinations of the gluon condensate αG2 [10] αG2 = (0.07 ± 0.01) GeV4 (11)
with a ﬁnite correlation length [4]. Thus the nonperturbative contributions to Π5 consist of the QCD condensate eﬀects Πcond and instanton eﬀects Πinst . 5 5 To leading order in the quark mass, Πcond is given by [5, 6] 5 Πcond (Q2 ) = (mu + md ) 5
(3) (4) (5) (6)
Divergent polynomials in Q2 are ignored in (3) since they correspond to subtraction constants in dispersion relations which do not contribute to sum-rules. In the QCD sum-rule approach, nonperturbative eﬀects are parametrized by the QCD condensates representing inﬁnite correlation-length vacuum eﬀects [1]. In addition to the QCD condensate contributions, scalar and pseudoscalar correlation functions must also take into account the eﬀects of instantons which represent vacuum eﬀects 1

Three-body relativistic flux tube model from QCD Wilson-loop approach

2
Quark-antiquark ﬂux tube model
In QCD the quark-antiquark gauge invariant propagator can be written G(x1 , x2 , y1, y2 ) = 1 c c = 0|Tψ2 (x2 )U (x2 , x1 )ψ1 (x1 )ψ 1 (y1 )U (y1 , y2 )ψ2 (y2 )|0 = 3 1 F F = Tr U (x2 , x1 )S1 (x1 , y1 |A)U (y1 , y2 )C −1 S2 (y2 , x2 |A)C . 3
(1)
Here c denotes the charge-conjugate ﬁelds, C is the charge-conjugation matrix, U the path-ordered gauge string U (b, a) = P exp ig 1

b
a
dxµ Aµ (x) ,
CERN-TH/95-194 IFUM 511/FT
Three-body relativistic ﬂux tube model from QCD Wilson-loop approach
arXiv:hep-ph/9507300v1 13 Jul 1995
N. Brambilla, G.M. Prosperi
CERN-TH/95-194 IFUM 511/FT July 1995
Preprint submitted to Elsevier Preprint
1 February 2008
1
Introductiห้องสมุดไป่ตู้n
In a recent paper we have shown that, for the quark-antiquark system, neglecting spin-dependent terms, it is possible to derive rigorously the so-called relativistic ﬂux tube model from the Wilson area law in QCD. To achieve the result, the essential ingredients were the path-integral representation of the Pauli-type propagator we used in the derivation of the semirelativistic potential, the replacement of the 1/m2 expansion of the purely kinetic terms by their closed form and the explicit integration over the momenta of the functional integral. In this letter we want to show how a similar result can be worked out for the three-quark system starting from the corresponding representation for the three-quark propagator we have considered in ref. [1]. The resulting Lagrangian turns out to correspond to a star ﬂux tube conﬁguration. As in the quark-antiquark case, the Hamiltonian can be expressed only in the form of an expansion in 1/m2 or in the string tension σ . In the ﬁrst situation we obtain the Wilson loop semirelativistic potential, but for the spin-dependent part. In the second one we end up with a very complicated result which, however, reduces to the three-quark string model already considered in the literature for low angular momentum. In Section 2 we review the quark-antiquark case to clarify the procedure and to establish notation; in Sec. 3 we derive the new Lagrangian and in Sec. 4 we obtain the Hamiltonian of the resulting model.

The Quest for Light Sea Quarks Algorithms for the Future

a r X i v :h e p -l a t /0110033v 1 10 O c t 20011The Quest for Light Sea Quarks:Algorithms for the FutureW.Schroers a ∗,N.Eicker a ,M.D’Elia b ,Ph.de Forcrand c ,C.Gebert d ,Th.Lippert a ,I.Montvay d ,B.Orth a ,M.Pepe e ,K.Schilling aaDepartment of Physics,University of Wuppertal,D-42097Wuppertal,Germany b Dipartimento di Fisica dell’Universit`a di Genova and INFN,I-16146,Genova,ItalycInst.f¨u r Theoretische Physik,ETH H¨o nggerberg,CH-8093Z¨u rich,Switzerland d Theory Division,DESY,Notkestr.85,D-22603Hamburg,GermanyeLaboratoire de Physique Th´e orique Universit´e de Paris-Sud,Bˆa timent 210F-91405Orsay-CedexAs part of a systematic algorithm study,we present ﬁrst results on a performance comparison between a multibosonic algorithm and the hybrid Monte Carlo algorithm as employed by the SESAM collaboration.The standard Wilson fermion action is used on 32×163lattices at β=5.5.1.IntroductionThe past six years have witnessed a number of large simulations of full QCD with two degen-erate sea quarks and Wilson like actions.All of them were based on the hybrid Monte Carlo algo-rithm (HMC)which has been systematically im-proved over the years.These simulations,how-ever,albeit being milestones for lattice QCD in the pre-Teracomputing age [1],suﬀer from severe restrictions that prevent them to be truly realis-tic:we need to go to lighter quark masses and operate with three ﬂavors with masses eventually approaching the physical mass.This situation has been motivating research and development of alternative algorithms of the multibosonic variety (MBA)that promise in prin-ciple to overcome some of the restrictions of HMC mentioned.It has been shown that MBA can in-deed be geared to reach the eﬃciency of HMC [2]and encouraging results have been achieved with MBA for the treatment of ﬁnite density QCD [3],supersymmetric ﬁeld theories [4]as well as the case of three dynamical fermion ﬂavors [5].In this contribution we wish to describe our benchmarking of MBA,performed in the quest to achieve lighter quark masses than previouslyVolume:Ω=32×163,β=5.50.1590.1414.39(3)0.800(10)0.1600.1174.89(3)0.670(14)2two diﬀerent quark mass settings,as given in ta-ble1.Note that both quark masses are from the regime of current full QCD simulations.3.Deﬁning the MBA algorithmThe parameter space of MBA algorithms is large w.r.t.the HMC situation:MBA optimiza-tion proceeds(a)byﬁnding a not too high or-der polynomial approximating the inverse of the fermion matrix at the given working point,(b) by devising an eﬃcient“trajectory”in form of an updating sequence for the degrees of freedom and (c)by exploiting a method to correct for the re-maining systematic error.Today,all MBAs use a noisy correction step for(c).For a detailed dis-cussion see e.g.ref.[7,8].For the construction of appropriate polyno-mials we followed the prescription proposed in ref.[9,10]which employs the GMRES algorithm toﬁnd a non-hermitian approximation of the in-verse Wilson matrix with ultraviolet-ﬁltering on a small sample of thermalized gaugeﬁeld conﬁg-urations.This was done to minimize the polyno-mial order and hence the number of bosonﬁelds. Quadratically optimized polynomials[11]need about2−3times more bosonﬁelds for similar acceptance rates.The other key element for the optimization of an MBA is the choice of“trajectory”,T,i.e.,the basic updating pattern.For that matter,one has to devise an appropriate blend of gauge and boson ﬁeld updates.We have chosen the“trajectory”to beT=CO g(O b O g)5H b(O g O b)5O g,(2) with C being theﬁnal accept/reject correction step,O g(O b)a gauge(boson)ﬁeld overrelax-ation sweep and H b a bosonﬁeld global quasi-heat bath[12].The number of bosonﬁelds is tuned such as to ensure a good acceptance rate in the ball-park of60−70%.As a result we obtained the values quoted in table2.4.ResultsThe determination of accurate autocorrelation times,τint,needs very long Monte Carlo time 0.1592460.3%0.8760.1604265.9%1.763κObserv.HMC MBA 0.160Plaq.0.56077(6)0.56067(5)(amπ)0.3041(36)0.3184(68)(amρ)0.4542(77)0.4674(81)3 0.159HMC352111.0(0.4)712(26)103<1620103<1620103MBA621744.7(3.4)786(60)103528103704103Table4Lengths of MC runs,estimated plaquette autocorrelation times and total cost,E ind,for producing one conﬁguration decorrelated w.r.t.various observables,from HMC and MBA.From table4oneﬁnds that E ind(plaq)is about equal for MBA and HMC at the larger quark mass.The total eﬀort in the case of the HMC increases by a factor of8when stepping down in quark mass,while at the same time the cost for MBA‘only’rises by a factor of2.7.Conse-quently,the MBA has become almost a factor of three more eﬃcient than HMC.5.ConclusionsWe have demonstrated in a realistic setting that MBA-type sampling techniques are not at all inferior to the state-of-the-art HMC techniques and appear to show superior scaling behavior in quark masses.We have established here a better scaling behavior of MBA,leading to a gain factor of almost three in favor of the particular MBA variant at currently reachable quark masses.It remains to be seen to what extent this encouraging result will pertain in the regime of smaller quark masses.The ultimate aim is to push full QCD simulation with Wilson like fermions below the point mπ/mρ≤0.5. Another point to check is whether the MBA variant considered in this paper with its non-hermitian approximation will remain operational for practical simulations in the deeper chiral regime.Because the GMRES method suﬀers from numerical instabilities for increasing orders n,one might have to take recourse—even on 32×163lattices—to128-bit precision.A more extended MBA study in this direction is in prepa-ration[13].AcknowledgmentsN.E.is supported under DFG grant Li701/3-1,M.P.by the European Community’s Human potential program under HPRN-CT-2000-00145 Hadrons/Lattice QCD,and W.S.by the DFG Graduiertenkolleg“Feldtheoretische Methoden in der Elementarteilchentheorie und Statistischen Physik”.The numerical productions were run on the APE-100systems installed at NIC Zeuthen. REFERENCES1.Th.Lippert,these proceedings; awa,ibid;H.Wittig,ibid.2. C.Alexandrou et al.,Phys.Rev.D61(2000)074503.3.S.Hands et al.,Eur.Phys.J.C17(2000)285.4.I.Campos et al.[DESY-M¨u nster Coll.],Eur.Phys.J.C11(1999)507.5. C.Gebert, F.Farchioni,I.Montvay,andW.Schroers,these proceedings.6.Th.Lippert,Habilitationsschrift,Univ.ofWuppertal,2001.7.P.de Forcrand,Parallel Computing25(1999)1341.8.W.Schroers,PhD-Thesis,Univ.of Wupper-tal,2001.9. A.Borrelli et al.,Nucl.Phys.B477(1996)809.10.P.de Forcrand,Nucl.Phys.Proc.Suppl.73(1999)822.11.I.Montvay,mun.109(1998)144.12.P.de Forcrand,Phys.Rev.E59(1999)3698.13.W.Schroers,N.Eicker,M.D’Elia,Ph.deForcrand,C.Gebert,Th.Lippert,I.Mont-vay,B.Orth,M.Pepe,and K.Schilling,in preparation.。

Dyson-Schwinger calculations of meson form factors

a rXiv:n ucl-t h /848v124Aug2Dyson–Schwinger calculations of meson form factors Pieter Maris Dept.of Physics,Kent State University,Kent OH 442421The Dyson–Schwinger Equations Our aim is to obtain the hadron spectrum and observables such as coupling constants and form factors from the underlying theory,QCD 1.The set of Dyson–Schwinger equations [DSEs]form a useful tool for this purpose [1].In rainbow-ladder truncation,they have been successfully applied to calculate the masses and decay constants of light pseudoscalar and vector mesons [2,3].The dressed-quark propagator,as obtained from its DSE,together with the Bethe–Salpeter amplitude and the qqγvertex as obtained from the homogeneousand inhomogeneous Bethe–Salpeter equations [BSE]respectively,form the necessary elements for form factor calculations in impulse approximation [4,5].The DSE for the renormalized quark propagator in Euclidean space isS (p )−1=i Z 2/p +Z 4m (µ)+Z 1 d 4q2γµS (q )Γa ν(q,p ),(1)where D µν(k )is the dressed-gluon propagator,Γaν(q ;p )the dressed-quark-gluon vertex,and k =p −q .The most general solution of Eq.(1)has the form S (p )−1=i/p A (p 2)+B (p 2)and is renormalized at spacelike µ2according to A (µ2)=1and B (µ2)=m (µ)with m (µ)the current quark mass.The DSE for the qqγvertexΓµ(p+,p−)is the inhomogeneous BSE Γµ(p+,p−)=Z2γµ+ d4q2Q are the incoming and outgoing quark momenta,and simi-larly for q±.The kernel K is the renormalized,amputated¯q q scattering kernel that is irreducible with respect to a pair of¯q q lines.Solutions of the homoge-neous version of Eq.(2)deﬁne vector meson bound states at timelike photon momenta Q2=−m2v.It follows thatΓµ(p+,p−)has poles at those locations. Pseudoscalar solutionsΓps(p+,p−;Q)of the homogeneous BSE deﬁne bound states such as pions and kaons.Together with the canonical normalization con-dition for q¯q bound states,Eqs.(1)and(2)completely determine all elements needed for form factor calculations in impulse approximation.To solve the BSE,we use a ladder truncation,with an eﬀective quark-antiquark interaction that reduces to the perturbative running coupling at large mo-menta[3].In conjunction with the rainbow truncation for the quark DSE,the ladder truncation preserves both the vector Ward–Takahashi identity[WTI] for the quark-photon vertex and the axial-vector WTI.This ensures the exis-tence of massless pseudoscalar mesons connected with dynamical chiral sym-metry breaking[2].In combination with impulse approximation,it also guar-antees current conservation[5].Our model preserves the one-loop renormal-ization group behavior of QCD and reproduces perturbative results in the ul-traviolet region.As long as the interaction is suﬃciently strong in the infrared region,it leads to chiral symmetry breaking and conﬁnement.The model gives a good description of theπ,ρ,K,K⋆andφmasses and electroweak decay constants[3],with only two parameters in the eﬀective interactionﬁtted to fπand the condensate,and realistic current quark massesﬁtted to mπand m K. 2Results for meson electromagnetic form factorsMeson form factors in impulse approximation are described by two diagrams, with the photon coupled to the quark and to the antiquark respectively.We can deﬁne a form factor for each of these diagrams,e.g.2PνF a¯b¯b(Q2)=N c d4q2P,q±=k−12Q,P±=P±100.51 1.522.533.5Q 2 [GeV 2]00.10.20.30.40.5Q 2 F π(Q 2) [G e V 2]00.20.40.6Q 2F K (Q 2) [G e V 2]Fig.1.The electromagnetic form factors and radii of the light pseudoscalar mesons,adapted from [5].The pion data are from Ref.[6],the charge radii from [7].charged and neutral kaon form factors are given by F K +=23F u ¯s ¯s and F K 0=−13F d ¯s¯s respectively.Our results for Q 2F π,Q 2F K +,and Q 2F K 0are shown in Fig.1,together with the corresponding charge radii.The impulse approximation for the γ⋆πγvertex with γ⋆momentum Q isΛµν(P,Q )=iα3 d 4q2is a direct consequence of only gaugeinvariance and chiral symmetry;this value is reproduced by our calculations and corresponds well with the experimental width of 7.7eV.In Fig.2we show our results with realistic quark masses,normalized to the experimental g πγγ.All our form factor results are remarkably close to the data,without any read-justment of the parameters.Up to about Q 2=3GeV 2,they can be ﬁtted quite well by monopoles.Asymptotically,our calculated form factors behave like Q 2F (Q 2)→c up to logarithmic corrections.However,numerical limitations30.0 1.0 2.03.0Q 2 [GeV 2]0.00.20.40.60.81.0F γ∗πγ(Q 2)Fig.2.The γ⋆πγform factor,with data from CLEO and CELLO [8].prevent us from accurately determining these constants.Around Q 2=3GeV 2,our result for Q 2F πis well above the pQCD result 16πf 2παs (Q 2)[9],and clearlynot yet asymptotic;F γ⋆πγlies in between monopoles ﬁtted to the Brodsky–Lepage asymptotic limit,8π2f 2π[10],and to the DSE limit,16。

1、下载文档前请自行甄别文档内容的完整性，平台不提供额外的编辑、内容补充、找答案等附加服务。
2、"仅部分预览"的文档,不可在线预览部分如存在完整性等问题,可反馈申请退款(可完整预览的文档不适用该条件!)。
3、如文档侵犯您的权益，请联系客服反馈,我们会尽快为您处理(人工客服工作时间：9:00-18:30)。

classic physics to quantum physics depended mainly on new physical ideas rather than complex mathematics and extra dimensions of space. Today we face a similar situation. The standard model [2] “is in excellent accord with almost all current data.... It has been enormously successful in predicting a wide range of phenomena,” but it cannot deduce the mass spectra of quarks. So far, no theory has been able to successfully do so. Like black body spectrum, the quark mass spectrum may need a new theory outside the standard model. M. K. Gaillard, P. D. Grannis, and F. J Sciulli have already pointed out [2] that the standard model “is incomplete... We do not expect the standard model to be valid at arbitrarily short distances. However, its remarkable success strongly suggest that the standard model will remain an excellent approximation to nature at distance scales as small as 10−18 m... high degree of arbitrariness suggests that a more fundamental theory underlies the standard model.” The history of quantum physics shows that a new physics theory’s primary need is new physical ideas. This paper gives new physical ideas using phenomenological formulae. Using these formulae, we try to deduce the rest masses of quarks.
∗ Mp =Mn = 939 Mev, m∗ u =md and |EBind (p)| = |EBind (n)| .
(2)
|EBind (p)|= |EBind (n)|= |EBind | is a unknown complex function. As a phenomenological approximation, we assume that |EBind | = 3∆. ∆ is an unknown large constant (∆ >>
arXiv:hep-ph/0507117v2 3 Aug 2005
E- mail: jxu@
Abstract
Using phenomenological formulae, we can deduce the rest masses and intrinsic quantum numbers (I, S, C, B and Q) of quarks, baryons and mesons from only one unﬂavored elementary quark family ǫ. The deduced quantum numbers match experimental results exactly, and the deduced rest masses are 98.5% (or 97%) consistent with experimental results for baryons (or mesons). This paper predicts some quarks [dS (773), dS (1933) and uC (6073) ], baryons [Λc (6599), Λb (9959)] and mesons [D(6231), B(9502)]. PACS: 12.39.-x; 14.65.-q; 14.20.-c. Keywords: phenomenological, beyond the standard model.
Deducing Quark Rest Masses with Phenomenological Formulae
Jiao Lin Xu
The Center for Simulational Physics, The Department of Physics and Astronomy University of Georgia, Athens, GA 30602, USA
Mp and neutron mass Mn are:
Mp = mu +mu +md − |EBind (p)|, Mn = mu +md +md − |EBind (n)|,
(1)
where |EBind (p)| and |EBind (n)| are the strong binding energy of the three quarks inside p and n. Omitting electromagnetic masses, we have
2
2 , rest mass m∗ state of the elementary quark ǫu is the u-quark with Q = 3 u, I = s =
1 2
1 1 and IZ = 2 . The excited state of the elementary quark ǫd is the d-quark with Q = - 3 ,
3
III
Phe
1). In order to deduce the short-lived quarks, we assume a phenomenological energy band formula. There are energy band excited states of the elementary quark ǫ whose energies are given by the following formula: → → → → E(− κ ,− n ) = 313 + ∆ + 360 E(− κ ,− n) − → − → 2 2 E( κ , n ) = [(n1 -ξ ) +(n2 -η ) +(n3 -ζ )2 ], (4)
I
Introduction
One hundred years ago, classic physics had already been fully developed. Most phys-
ical phenomena could be explained with this physics. Black body spectrum, however, could not be explained by the physics of that time, leading Planck to propose a quantization postulate to solve this problem [1]. The Planck postulate eventually led to quantum mechanics. Physicists already clearly knew that the black body spectrum was a new phenomenon outside the applicable area of classic physics. The development from 1
II
The Elementary Quarks and Their Free Excited States
1). We assume that there is only one elementary quark family ǫ with s = I =
1 2
and
two isospin states (ǫu has IZ =
1 . Iz = 1 , Q =1, S = C = B = 0) and n(939) (with I = 1 . Iz = p(939) (with I = 2 2 2 −1 , 2
Q
= S = C = B = 0) and mesons π (with I =1, Q =+1, 0, -1, S = C = B = 0).
1 2
2 1 and Q = + 3 , ǫd has IZ = - 1 and Q = - 3 ). For ǫu 2
(or ǫd ), there are three colored (red, yellow or blue) quarks. Thus, there are six Fermi elementary quarks in the ǫ family with S = C = B = 0 in the vacuum. The elementary quarks ǫu and ǫd have the SU(2) symmetries. 2). As a colored elementary quark ǫu (or ǫd ) is excited from the vacuum, its color, electric charge, rest mass and spin do not change, but it will get energy. The excited