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Meson Mass Splittings in the Nonrelativistic Model ∗

arXiv:hep-ph/9209242v1 15 Sep 1992September 11, 1992LBL-32872UCB-PTH-92/33Meson Mass Splittings in the Nonrelativistic Model ∗Richard F. LebedDepartment of PhysicsUniversity of CaliforniaandTheoretical Physics GroupPhysics DivisionLawrence Berkeley Laboratory1 Cyclotron RoadBerkeley, California 94720AbstractMass splittings between isodoublet meson pairs and between 0−and1−mesons of the same valence quark content are computed in a detailednonrelativistic model. The field theoretic expressions for such splittingsare shown to reduce to kinematic and Breit-Fermi terms in the nonrela-tivistic limit.

Algebraic results thus obtained are applied to the specificcase of the linear-plus-Coulomb potential, with resultant numbers com-pared to experiment.∗This work was supported by the Director, Office of Energy Research, Office of High Energyand Nuclear Physics, Division of High Energy Physics of the U.S. Department of Energy underContract DE-AC03-76SF00098.

DisclaimerThis document was prepared as an account of work sponsored by the United States Gov-ernment.Neither the United States Government nor any agency thereof, nor The Regentsof the University of California, nor any of their employees, makes any warranty, express orimplied, or assumes any legal liability or responsibility for the accuracy, completeness, or use-fulness of any information, apparatus, product, or process disclosed, or represents that its usewould not infringe privately owned rights. Reference herein to any specific commercial prod-ucts process, or service by its trade name, trademark, manufacturer, or otherwise, does notnecessarily constitute or imply its endorsement, recommendation, or favoring by the UnitedStates Government or any agency thereof, or The Regents of the University of California.

Theviews and opinions of authors expressed herein do not necessarily state or reflect those of theUnited States Government or any agency thereof of The Regents of the University of Californiaand shall not be used for advertising or product endorsement purposes.Lawrence Berkeley Laboratory is an equal opportunity employer.ii

1IntroductionThe splitting of the masses of mesons in an isospin doublet, sometimes calledelectromagnetic splitting, has traditionally been attributed primarily to explicitisospin breaking (i.e. mu ̸= md) and differences between the charges of thevalence quark-antiquark pairs (Qu ̸= Qd), with hyperfine, spin-orbit, and othereffects neglected in comparison.

Such a model serves to explain the observedsplittings K0 −K+ = 4.024 ± 0.032 MeV and D+ −D0 = 4.77 ± 0.27 MeV, buthas failed in light of the surprisingly small B0 −B+ = 0.1 ± 0.8 MeV.It is precisely this mass difference which has led to the proposal of a varietyof models. Some of these [1, 2, 3, 4] are based on the nonrelativistic model ofhadron masses put forth by De R´ujula, Georgi, and Glashow [5] soon after thedevelopment of QCD.

Such models have the unfortunate tendency to predictnumbers no smaller than B0 −B+ ≃2 MeV, well outside the current exper-imental limits. Using more phenomenological models [6, 7], one can obtain asmaller splitting in closer agreement with experiment.

Nevertheless, it may seemodd that the usual nonrelativistic model, which works well for the D- and eventhe K - mesons, should fail in the case of the B, which boasts an even heavierquark.The primary conclusion of this work is that it is possible to explain the masssplittings of heavy mesons (D and B, but not K) in an ordinary nonrelativisticmodel, as long as we take into account all corrections to consistent orders ofmagnitude, that expectation values of the mesonic wavefunctions in generalhave mass dependence, and that the running of the strong coupling constant isnot negligible.In this spirit, the paper is organized as follows: in the second section weconsider the problem of computing mesonic mass contributions in field theory.Then, in the third section, we demonstrate that the nonrelativistic limit of thefield-theoretic result leads to kinematic terms and the Breit-Fermi interaction,exactly as stated in De R´ujula et al. This is followed in Section 4 by an exhibi-tion of the full mass splitting relations for isodoublet 0−and 1−meson pairs, aswell as (0−, 1−) pairs with the same valence quarks.

Section 5 discusses the ap-plication of quantum-mechanical theorems, including a very useful generalizedvirial theorem, to the problem of reducing the number of independent expecta-1

tion values in the splitting formulae. These theorems are applied to the popularchoice of a linear-plus-Coulomb potential in Section 6, with numerical resultspresented in Section 7.2Mass Computation in Field TheoryTypically, the computation of mesonic mass splittings in a nonrelativistic modelis accomplished by starting with the Breit-Fermi interaction [8, Secs.

38-42]HBF =Xi>j(αQiQj + kαs)( 1|⃗rij| −12mimj |⃗rij|(⃗pi · ⃗pj + ˆrij · (ˆrij · ⃗pi)⃗pj)−π2 δ3(⃗rij)" 1m2i+ 1m2j+4mimj43⃗si · ⃗sj +34 + ⃗si · ⃗sjδqi ¯qj#−12 |⃗rij|3" 1m2i⃗rij × ⃗pi · ⃗si −1m2j⃗rij × ⃗pj · ⃗sj+2mimj(⃗rij × ⃗pi · ⃗sj −⃗rij × ⃗pj · ⃗si + 3(⃗si · ˆrij)(⃗sj · ˆrij) −⃗si · ⃗sj)#), (1)where ⃗ri, ⃗pi, mi, ⃗si, and Qi denote the coordinate, momentum, (constituent)mass, spin, and charge (in units of the protonic charge) of the ith quark, re-spectively; ⃗rij ≡⃗ri −⃗rj; α and αs are the (running) QED and QCD couplingconstants; and k = −43 (−23) is a color binding factor for mesons (baryons). Thisexpression includes an annihilation term if qi = ¯qj are in a relative j = 1 state.From this, one chooses the terms that are considered significant and then calcu-lates the appropriate quantum mechanical expectation values.

We will pursuethis course of action in the next section; however, this author feels that it wouldbe worthwhile to first consider the derivation of this interaction for the mesonicsystem from the more fundamental field theories of QED and QCD, since thisapproach entails greater generality and may provide impetus for work beyondthe scope of this paper.We first consider the question of the mass of a composite system from thepoint of view of the S-matrix and interaction-picture perturbation theory. The2

mass of a system, defined as the expectation value of the total Hamiltonian inthe center-of-momentum frame of the constituents, receives contributions fromboth the noninteracting and interacting pieces of the Hamiltonian; the formergives rise to the masses and kinetic energies of the constituents, and the latterproduces the interaction energy. Technically, the matrix element of the non-interacting piece in the interaction picture produces terms which contribute tointeractions between renormalized constituents.

Thus one may think of interac-tions between “dressed” constituents, a topic to which we return momentarily.Let us follow the method of Gupta [9] to derive the interaction potentialfrom the field-theoretical interaction Hamiltonian.We begin by writing theS-matrix in the Cayley formS = 1 −12iK1 + 12iK ,(2)and expand the Hermitian operator K:K =XnKn. (3)The purpose of this expansion, rather than expanding S directly, is to preserveunitarity in each partial sum of S. The physical effect of this parametrization isto eliminate diagrams with real intermediate states from the S-matrix expansion.Computing the terms Kn, one findsK1 =Z +∞−∞dt HIint(t),(4)where I indicates the interaction picture.

Now observe that we may invent aneffective Hamiltonian, HIeff, such that its first-order contribution is equivalentto the contribution from HIint to all orders. Thus,K =Z +∞−∞dt HIeff(t).

(5)The interaction energy is then∆E =Df I HIeff(0)iIE⟨f I | iI⟩,(6)withiIEandf IEactually the same state since the system is stable.3

In our case, in which HIeff is composed of the interaction terms of QEDand QCD, the lowest-order contribution is K2, corresponding to two interactionvertices: the exchange of one vector boson. It is easily shown thatK2 = iS2 = (2π)4δ4(Pf −Pi)M(2)fi =Df IZdt HI (2)eff (t)iIE,(7)where the superscript (2) indicates second-order in the coupling constant, and Mis the usual invariant amplitude for the process.

Eliminating the delta functionsthat arise in the right-most expressions we find∆E(2) =Df I HI (2)eff (0)iIE⟨f I | iI⟩= M(2)fi . (8)Beyond second-order the relation between interaction energy and the invariantamplitude becomes less trivial, but nevertheless Gupta has shown that it can bedone.

However, we do not continue to fourth-order in this work, and henceforthsuppress the (2) in the following.In general, Mfi at any given order is represented by diagrams of the formindicated in Figure 1. The composite state is formed by superposition of theconstituent particle wavefunctions in such a way that the desired overall quan-tum numbers for the composite state are obtained.

For the mesonic system,Mfi is represented by the diagram in Figure 2, where the lowest-order interac-tion is the exchange of a single gauge boson. This class of diagrams allows foronly the valence quark and antiquark (no sea qq pairs or glue), and thus wouldbe a poor model if we chose these to be current quarks.

Instead, the quarks inour diagrams will be constituent quarks, and the gauge couplings will assumetheir running values. In this way we can model the hadronic cloud, as well asrenormalizations of the lines and vertices of our diagram, so that its particles are“dressed” in two senses.

There is also an annihilation diagram if the quark andantiquark are of the same flavor. In this work we consider only the exchangediagram, since the mesons of greatest interest to us are those with one heavyand one light quark.The next step is to obtain the amplitude Mfi, in which the constituent legsare bound in the composite system, from the Feynman amplitude M [Figure 3]for the same interaction with free external constituent legs.

To do this, we needonly constrain the free external legs in a way which reflects the wavefunction4

and rotational properties of the meson state. In general, if the variables zn arethe degrees of freedom of the meson state |Φ⟩, then we may write|Φ⟩=ZXdzn φ(zn) O(zn) |0⟩.

(9)The function φ is an amplitude in the variables zn, i.e. a wavefunction; and Ois a collection of Fock space operators which specifies the rotational propertiesof |Φ⟩.

The integral-sum symbol indicates summation over both continuous anddiscrete zn. In this notation, we obtain the result∆E =ZXdzf dzi φ∗(zf)φ(zi)f(zi, zf)M(zi, zf),(10)where f(zi, zf) ≡⟨0| O†(zf)O(zi) |0⟩is a constraint function.

We have writtenthe energy contribution in this very general way in order to demonstrate thepower of the technique.Now we apply this prescription to the usual case of Feynman rules. Thenzn are quark momenta, φ is the mesonic momentum-space wavefunction, and fspecifies the spin of the meson, as we shall see below.

The energy contribution isevaluated in the quark center of momentum frame (i.e. the meson rest frame), inwhich the relative momenta of the quark-antiquark pair, intitially and finally, aredenoted by ⃗p and ⃗p ′, respectively.

Fourier transformation of the wavefunctionsfrom momentum-space to position-space yields∆ECM=Zd3⃗xfZd3⃗xi ψ∗(⃗xf)K(⃗xf, ⃗xi)ψ(⃗xi),whereK(⃗xf, ⃗xi)=Zd3⃗p ′Zd3⃗p exp[i(⃗p ′ · ⃗xf −⃗p · ⃗xi)]Xspinsf(spins)M(⃗p ′, ⃗p, spins),andZd3⃗x ψ∗(⃗x)ψ(⃗x) = 1. (11)As a technical point of fact, it is necessary to keep track of the normalizationconventions used for wavefunctions, Fourier transforms, and Feynman rules inorder to obtain the true convention-independent ∆E.

As it stands, Eq. 11 locksus into a particular set of Feynman rule normalizations, which should be madeclear in the following expression.

The kinematic conventions are established inFigure 4. Then the Feynman amplitude for free external quark legs and a virtual5

photon isM = i"1(2π)3/2#4 sMEfsMEismεfsmεih¯vHi(⃗Pi)(−iQeγµ)vHf( ⃗Pf)i −igµνk2 h¯uhf(⃗pf)(−iqeγµ)uhi(⃗pi)i,(12)with Qqe2 replaced by g2s for the gluon-mediated diagram.Note the use ofhelicity rather than spin eigenstate spinors, which is done in order to implementa relativistic description of the mesons. In a nonrelativistic picture in whichmeson spin originates solely from the spin of the quarks (s-waves), spin-0(1)mesons have spin-space wavefunctions described by the usual singlet and tripletquark wavefunction ¯Qq combinations:¯Q↑q↓± ¯Q↓q↑√2,↑, ↓spins.

(13)The above expression remains true in a relativistic picture if we take the initialand final spin-quantization axes to coincide with the axes of relative momenta ⃗pand ⃗p ′, respectively, and then take ↑, ↓as helicity eigenstates. This is nothingmore than the simplest nontrivial case of the Jacob-Wick formalism [10].

It isthen a simple matter to write the constraint function for singlet (triplet) mesons:f(helicities) = 1√2(δhi↑δHi↓± δhi↓δHi↑) 1√2(δhf ↑δHf ↓± δhf↓δHf ↑),(14)and so the object of interest is the constrained matrix element Msing or Mtrip,which is the Feynman amplitude multiplied by the constraint function andsummed over spins (or helicities). This is the object that is Fourier transformedin Eq.

11.In summary, mass contributions due to a binding interaction in a systemof particles may be computed by writing down the Feynman amplitude inducedby the interaction Hamiltonian, constraining the component particles to satisfythe symmetry properties of the system, and convolving with the appropriatesystem wavefunction. The specific implementation of this technique to spin-0and spin-1 mesons with constituent quarks in a relative ℓ= 0 state is describedby Eqs.

11, 12, and 14.6

3The Nonrelativistic LimitWith the method for computing mass contributions in hand, we find ourselveswith two possible courses of action. The first is to compute Msing or Mtrip in afully relativistic manner, and then Fourier transform the result to obtain ∆ECM.The second is to immediately reduce the spinor bilinears via Pauli approximants,thus producing a nonrelativistic expansion.

Let us explore both directions forthe pseudoscalar case; the vector case is not much different.The relativistic result is noncovariant, because the energy contribution isevaluated specifically in the CM frame of the quarks. We see this reflected in thecomputation of the matrix element.

For example, it is convenient to eliminatespinors from the calculation by means of relations likeXhuh(⃗pA)¯uh(⃗pB) =(mA + p/A)q2mA(EA + mA)(1 + γ0)2(mB + p/B)q2mB(EB + mB),(15)and the explicit γ0 is a signal of the noncovariance. Once the spinor reductionsand the resultant trace are performed, we find the expressionMsing = −(Qqe2 + g2s)N T 1k2,(16)where N results from the normalization factors, and T is the gamma-matrixtrace.

They are given byN=1(2π)6125 [Ei(Ei + M)Ef(Ef + M)εi(εi + m)εf(εf + m)]−1/2andT=8 {(pi · Pi) [2εfEf + 3(mEf + Mεf + mM)]+(pf · Pf) [2εiEi + 3(mEi + Mεi + mM)] + (pi · Pi)(pf · Pf)−(pi · pf) [2EiEf + M(Ei + Ef + M)]−(Pi · Pf) [2εiεf + m(εi + εf + m)] + (pi · pf)(Pi · Pf)−(pi · Pf) [mEi + Mεf + mM]−(Pi · pf) [mEf + Mεi + mM] −(pi · Pf)(Pi · pf)+ [−2mM(Ei −Ef)(εi −εf)7

+mM (m(Ei + Ef) + M(εi + εf) + mM)+2m2EiEf + 2M2εiεfio. (17)Also,k2 = (pi −pf)2 = (εi −εf)2 −(⃗p −⃗p ′)2.

(18)It is, in principle, possible to Fourier transform the product Msing of theseunwieldy functions to obtain the full relativistic result for ∆ECM; this has notyet been performed. We can also perform the expansion of the energy factorsin powers ofpm, where all such momentum-over-mass quotients that occur aretaken to be of the same order.However, this is unnecessary work, for if we require only a nonrelativisticexpansion, there is a much faster way, namely expansion of the spinor bilinearsvia the Pauli approximants¯u(⃗p ′)⃗γu(⃗p)=⟨χ′| (⃗p + ⃗p ′)2m+ i⃗σ × (⃗p ′ −⃗p)2m|χ⟩+ o" pm3#¯u(⃗p ′)γ0u(⃗p)=⟨χ′| 1 + (⃗p + ⃗p ′)28m2+ i⃗σ · (⃗p ′ × ⃗p)4m2|χ⟩+ o" pm4#.

(19)Using these expansions in Eq. 12 and taking |χ⟩, |χ′⟩in helicity basis, we quicklyfindMsingNR⇒Qqe2 + g2s(2π)6(⃗p −⃗p ′)2(1 + (⃗p + ⃗p ′)24mM−(⃗p −⃗p ′)28 1m2 −4mM + 1M2+o" pm4#).

(20)The gluon diagram has the additional physical constraint that the initialand final q¯q pairs are combined into a color singlet; this introduces an additionalfactor of −43. Then Fourier transformation of this result produces∆ECM,sing=αQq −43αs1r+12mM1r(⃗p 2 + ˆr · (ˆr · ⃗p )⃗p )−π2 1m2 −4mM +1M2 Dδ3(⃗r )E+ · · ·(21)8

In comparison, the energy contribution from the Breit-Fermi interaction (Eq. 1)for a quark-antiquark pair of masses m, M in the CM reduces to⟨HBF⟩=αQq −43αs1r+12mM1r(⃗p 2 + ˆr · (ˆr · ⃗p )⃗p )−π2Dδ3(⃗r )E 1m2 + 1M2 +4mM (G + δS,1δqflavors)−12 1r3 *⃗L ·⃗sqm2 + ⃗s ¯QM2 + 2⃗SmM+S122mM+(22)where G ≡43D⃗sq · ⃗s ¯QE, which is −113for S = 0(1).

Also, ⃗S ≡⃗sq +⃗s ¯Q, and S12is the ∆L = 2 tensor operatorS12 ≡3(⃗σ1 · ˆr)(⃗σ2 · ˆr) −⃗σ1 · ⃗σ2. (23)For mesons with differently-flavored quarks in a relative ℓ= 0 state, many ofthe terms drop out.

Let us defineB≡1r,C≡1r(⃗p 2 + ˆr · (ˆr · ⃗p )⃗p ),D≡Dδ3(⃗r )E.(24)Then Eq. 22 becomes⟨HBF⟩=αQq −43αs B +12mM C −π2 1m2 +1M2 + 4GmMD,(25)and this is exactly Eq.

21 where G = −1.We have been up to now considering only the contributions to the massoriginating from the binding interaction due to one-gluon and one-photon ex-changes; there are, of course, also contributions from the kinetic energy of thequarks. Were we calculating these quantities in a relativistic theory, we wouldsimply compute KE =D√m2 + ⃗p 2E.

The square root may be formally expandedin norelativistic quantum mechanics as well, resulting in an alternating series9

in ⟨⃗p 2n⟩. However, for large enough n in NRQM, these expectation values tendto diverge.

For example, in the hydrogen atom, divergence occurs for s-wavesat n = 3. Furthermore, if the system is not highly nonrelativistic, the inclusionof the ⟨⃗p 4⟩may cause us to grossly underestimate the true value of the kineticenergy.

The problem is that there is no positive ⟨⃗p 6⟩term to balance the largenegative ⟨⃗p 4⟩term. For these reasons, we incorporate the alternating nature ofthe series in a computationally simple way by making the ansatzKE =qm2 + ⟨⃗p 2⟩.

(26)In order to evaluate the expectation values in the above equations, we willneed to choose a potential. In the meantime, let us simply denote it with U(r).Then at last we have the mass formula:Mmeson =qM2 + ⟨⃗p 2⟩+qm2 + ⟨⃗p 2⟩+ ⟨U(r)⟩+ ⟨HBF⟩.

(27)The static potential U(r) takes the place of L, the universal quark bindingfunction, in Eq. 1 of Ref [5].4Mass Splitting FormulaeThe static potential in which the quarks interact determines the form of theNRQM wavefunction.

The strong Coulombic term gives the largest energy con-tribution of terms within the Breit-Fermi interaction, and therefore would also beexpected to substantially alter the wavefunction in perturbation theory. There-fore, we include the strong Coulombic term in the static potential:V (r) ≡U(r) −43αsr .

(28)Then the mass formula Eq. 27 becomes, using Eq.

25,Mmeson=qM2 + ⟨⃗p 2⟩+qm2 + ⟨⃗p 2⟩+ ⟨V (r)⟩+ αQqB+αQq −43αs 12mM C −π2 1m2 +1M2 + 4GmMD. (29)Now at last we are in a position to write explicit formulae for the masssplittings of interest.Denoting the mass of a meson of spin S and valence10

quarks ¯Q,q as MS( ¯Qq), we define:∆0Q≡M0( ¯Qu) −M0( ¯Qd)∆1Q≡M1( ¯Qu) −M1( ¯Qd)∆∗Qu≡M1( ¯Qu) −M0( ¯Qu)∆∗Qd≡M1( ¯Qd) −M0( ¯Qd),(30)where u and d, the up and down constituent quarks, are nearly degenerate inmass: defining ∆m ≡mu −md and m ≡mu+md2, we have ∆mm ≪1. Therefore,the differences in Eq.

30 are expanded in Taylor series in ∆mm about m. It is alsoconvenient to defineA≡D⃗p 2E,β≡11 + m/M ,µ≡usual reduced mass,¯µ≡mβ,Dαs≡β µαs∂αs∂µ!µ=¯µ,DX≡β µ∂X∂µ!µ=¯µ,X = A, B, C, D, ⟨V (r)⟩. (31)Then the expressions for mass splitting are∆0,1Q="2m2 + DA√m2 + A +DA√M2 + A# ∆m2m + D⟨V ⟩∆mm−43αs∆m12m2M (DC −C + CDαs)−π2m3" 1 + 4G mM + m2M2!

(DD + DDαs) −21 + 2G mMD#)+αQ"B +12mM C −π2m2 1 + 4G mM + m2M2!D#+o"∆mm3#+ oα∆mm. (32)11

Note that no derivatives appear in the αEM terms because we take both αEMand ∆mm(but not αs) as expansion parameters. Furthermore, the running ofαs(µ) is explicitly taken into account.For vector-pseudoscalar splittings, we have∆∗Qq=8π3mM43αs −αQqD ± 43αs∆m2m (DD + DDαs −D)+o"∆mm2#+ oα∆mm,with ± for q = u(d).

(33)Let us remind ourselves of the physical significance of the terms in the previoustwo equations. Terms containing A signify kinetic energy contributions, includ-ing intrinsic quark masses.

The potential term is identified, of course, by V ; B,C, and D denote static Coulomb, Darwin, and hyperfine terms, respectively.5Quantum-mechanical TheoremsIn order to apply the foregoing results, we will need to evaluate the expectationvalues A,B,C,D, and ⟨V (r)⟩for our potential V (r). There are two quantum-mechanical theorems which make the evaluation of these expectation values andtheir mass derivatives simpler [11].

The first is theTheorem 1 (Feynman-Hellmann Theorem) For normalized eigenstates ofa Hamiltonian depending on a parameter λ,∂E∂λ =*∂H(λ)∂λ+. (34)In the particular case that λ = µ,∂E∂µ = −1µ (E −⟨V (r)⟩) +*∂V∂µ+.

(35)The other result may be less familiar. For reasons that will become clear,let us call it theTheorem 2 (Generalized Virial Theorem) Consider bound eigenstatesuℓ(r) in a spherically symmetric potential V(r) such thatlimr→0 r2V (r) = 0.12

Then, writing the Schr¨odinger equation asu′′ℓ(r) + 2µ¯h2"E −V (r) −¯h2ℓ(ℓ+ 1)2µr2#uℓ(r) = 0,and defining aℓbylimr→0uℓ(r)rl+1 ≡aℓ,theni) aℓis a nonzero constant;ii) for q ≥−2ℓ,(2ℓ+ 1)2a2ℓδq,−2l=−2µ¯h2*rq−1 2q(E −V (r)) −rdVdr!++(q −1)2ℓ(ℓ+ 1) −12q(q −2) Drq−3E. (36)Clearly this theorem will prove most useful for potentials easily expressedas a sum of terms which are powers in r. But in fact there are some interestinggeneral results included.

For example, the q = ℓ= 0 case generates the well-known result for s-waves,|Ψ(0)|2 =µ2π¯h2*dVdr+,(37)whereas the q = 1 case producesE −⟨V (r)⟩= 12*rdVdr+,(38)the quantum-mechanical virial theorem.Using partial integration, the Schr¨odinger equation, and the GVT, it ispossible to show the following (¯h = 1):A=2µ (E −⟨V (r)⟩) ,C=4µ"E1r−*V (r)r+−14*dVdr+(1 + δℓ,0)#,D=µ2π*dVdr+δℓ,0,Z ∞0 duℓ(r)dr!2dr=A −ℓ(ℓ+ 1) 1r2. (39)13

In addition, we must also uncover what we can about the µ-dependence ofexpectation values. For a general potential this is actually an unsolved problem.However, unless the potential has very special µ-dependence, it can be shownthat only in the case V (r) = V0rν is it possible to scale away the dimensionfulparameters V0 and µ in the Schr¨odinger equation.

In that case, the µ-dependencewill be entirely contained in the scaling factors, and computing DX will be trivial.Unfortunately, in the potential we consider in the next section, we will see thatthis is not the case, and we must resort to subterfuge to obtain the requiredinformation.6Example: V (r) = ra2 −κrThe potential V (r) =ra2 −κr , where κ =43αs, is interesting because it phe-nomenologically includes quark confinement via the linear term. This potentialwas considered in greatest detail by Eichten et al.

[12] to describe the masssplitting structure of the charmonium system (and was later applied to bot-tomonium).The Schr¨odinger equation was solved numerically; currently, noanalytic solution is known. However, it is possible to extract a great deal ofinformation from their tabulated results, as we shall see below.This is possible because of the GVT.

If we rescale the Schr¨odinger equationwith the linear-plus-Coulomb potential to d2dρ2 −ℓ(ℓ+ 1)ρ2+ λρ + ζ −ρ!wℓ(ρ) = 0,(40)whereρ ≡2µa21/3r,λ ≡κ(2µa)2/3,ζ ≡(2µa4)1/3E,wℓ(ρ) ≡uℓ(r) a22µ!1/6,(41)then the GVT gives(q = 0)a20 δ0,ℓ=2µa2 "1 + λ* 1ρ2+−2ℓ(ℓ+ 1)* 1ρ3+#,(q = 1)0=3 ⟨ρ⟩−2ζ −λ*1ρ+. (42)14

Also, definingDv2E≡Z ∞0 dwℓ(ρ)dρ!2dρ,(43)we findDv2E= −⟨ρ⟩+ ζ + λ*1ρ+−ℓ(ℓ+ 1)* 1ρ2+. (44)It is a happy accident of this potential that all of the quantities in theexpectation values we need, for any ℓ, may be expressed in terms of the threequantities ζ,D1ρ2E, and ⟨v2⟩, which are exactly those values tabulated for the1s-state, as functions of λ, in Eichten et al.

Table I. Defining σ ≡2µa21/3 andtaking ℓ= 0 (as per our mesonic model), we findA=σ2 Dv2E,B=σ2λh3Dv2E−ζi,C=σ2"2Bζ + σ −3 + λ* 1ρ2+!#,D=σ34π"λ* 1ρ2++ 1#. (45)So now we can compute all of the necessary expectation values numerically.

Thesuperficial singularity in B(λ = 0) is false; B(0) is computed by extrapolationof the computed values of B for nonzero λ and is found to be finite.The mass derivatives must be handled in a different fashion. We begin bydefining˜Dζ ≡µ∂ζ∂µ,˜Dv ≡µ∂⟨v2⟩∂µ ,˜Dρ ≡µ∂D1ρ2E∂µ,and˜Dαs ≡µαs∂αs∂µ .

(46)From the Feynman-Hellmann theorem (Eq. 35) we may show˜Dζ = ζ3 −Dv2E!

1 + 32˜Dαs. (47)15

As mentioned in the previous section, scaling of the Schr¨odinger equation canbe accomplished for µ-independent potentials that are monomials. In the caseλ = 0 (a purely linear potential), the scaling would be perfect, and ζ,D1ρ2E,and ⟨v2⟩would be µ-independent.

In the λ ̸= 0 case, the derivatives must befound numerically. Again, fortunately, we have a table of numerical values ofthe desired expectation values, as a function of λ(µ).

We fit the expectationvalues Y (=D1ρ2E, ⟨v2⟩) to the functional formY (λ) = Y0 + KλnY . (48)Then, using Eq.

41, we find˜DY =23 + ˜DαsnY (Y −Y0) . (49)Finally, define˜DX ≡µ∂X∂µ for X = A, B, C, D,(50)so thatDX = β ˜DXµ=¯µ .

(51)Then we find˜DA=23A + σ2 ˜Dv,˜DB=3σ2λ˜Dv −12B ˜Dαs,(λ ̸= 0),˜DC=53C + 2σ2(−ζ + ˜DζB + ζ ˜DB + σ"λ2 ˜Dρ + ˜Dαs* 1ρ2+!+ 1#),˜DD=σ34π(λ"53 + ˜Dαs * 1ρ2++ ˜Dρ#+ 1). (52)In the exceptional case of ˜DB, we simply note that, for λ = 0, we have perfectscaling of the wave equation, and we can quickly show that ˜DBλ=0 = 13Bλ=0.This provides us with everything we need to produce numerical results.Before leaving the topic, let us mention that many complications of µ-derivatives of expectation values vanish if the potential itself has the appropriateµ-dependence, for then scaling of the wave equation is possible.

For example,one can scale the Schr¨odinger equation for the potentialV (r) = cµ2r −κr ,(53)16

where c is a pure number.7Numerical ResultsThe method of obtaining results from the theory requires us to choose severalnumerical inputs, most of which are believed known to within a few percent.Let us choose the following inputs to the model:m = 340 MeV,Ms = 540 MeV,Mc = 1850 MeV,Mb = 5200 MeV,a = 1.95 GeV−1. (54)The light quark constituent mass is arrived at by assuming that nucleons consistof quarks with negligible anomalous magnetic moments, which can be addednonrelativistically to provide the full nucleonic magnetic moment.Likewise,the strange quark mass issues from the same considerations applied to strangebaryons [5].

The c and b quark masses are simply found by dividing the thresholdenergy value for charm and bottom mesons by two (however, smaller masses havebeen predicted using semileptonic decay results in addition to meson masses[13]). The confinement constant is inferred from charmonium levels [12].One important input not yet mentioned is ∆m, the up-down quark massdifference.

Traditionally, this assumes a value of ≈−3 to −8 MeV, in order toaccount for the electromagnetic mass splittings of the lighter hadrons. In thismodel, with the inputs listed in Eq.

54, we find that the experimental splittingsfor the D- and B-mesons (both vector and pseudoscalar) can be satisfied withinone standard deviation of experimental error for values of ∆m in the narrowrange of −4.05 to −4.10 MeV. In contrast, it is found that for no choice of ∆mcan one simultaneously fit D- and K-meson data simultaneously, as was donein the earlier models.Before exhibiting the quantitative results, let us describe the method bywhich they are obtained.

Once particular inputs for the above variables arechosen, one can compute the various mass splittings for the values of λ ∝αs thatoccur in Table I of Ref. [12], and in-between values may be interpolated.

We thenfit vector-pseudoscalar splittings (computed via Eq. 33) to the correspondingexperimental data (since these numbers have the smallest relative errors of the17

splittings we consider) and thus obtain a value of αs. For the three systems K,D, and B, we use the three values of αs to estimate graphically (and admittedlyrather crudely) its mass derivative.

Applying the values of the strong couplingconstant and its derivative to the splittings in Eq. 32, we generate all of theother values.

If the resultant numbers do not fall within the experimental errorbars for such splittings, we vary the input parameters (most importantly, ∆m)until a simultaneous fit is achieved.Table I displays the various contributions to mass splittings derived in thisfashion for B- and D-mesons. Although the kinetic term (which includes theexplicit difference ∆m) and the static Coulomb term are unsurprisingly large, asignificant contribution to the mass splitting arises in the strong hyperfine term.That strong contributions to the so-called electromagnetic mass splittings couldbe important was observed by Chan [2], and was exploited in the subsequentliterature.

It is exactly this term which is most significant in driving the Bsplittings toward zero. Note also the decrease in the derived value of αs as thereduced mass of the system increases when we move from the D system to the Bsystem, consistent with asymptotic freedom in QCD.

It was this running whichmotivated the inclusion of mass derivatives of the strong coupling constant inthis model. If they are not included, one actually obtains a value of ∆m > 0, incontrast with all estimates from both nonrelativistic and chiral models.The net result is that one can satisfactorily fit the data for the D and Bsystems simultaneously in the most natural nonrelativistic model with a physi-cally reasonable potential.

The comparison of the results of this calculation for∆m = −4.10 MeV to experimental data is presented in Table II.However, the table also exhibits very bad agreement for the K system(despite the fact that the fit to vector-pseudoscalar splittings yields the valueαs = 0.424, which runs in the correct direction). One may view this as a failure ofthe nonrelativistic assumptions of the model in a variety of ways.

Most obviousare the ansatz Eq. 26, which is certainly not an airtight assumption in even thebest of circumstances, and the crudeness of the estimate of ∂αs∂µ .

Other possibleproblems include the assumption that the quarks occur only in a relative ℓ= 0state (relevant for K∗-mesons), and the assumption that the strong effects aredominated by a confining potential and one-gluon exchange, since at the lowerenergies associated with the K system, o(α2s) terms and more complex models of18

confinement may be required. The failure of these assumptions can drasticallyalter the strong hyperfine interaction, which determines the size of αs, and hencethe other mass splittings.Some may find the small size of αs somewhat puzzling.

This is primarily theresult of the confining term of the model potential: it causes the wavefunctionto be large at the origin, and thus a small αs is required to give the sameexperimentally measured vector-pseudoscalar splitting (see Eq. 33).

Such smallvalues for the strong coupling constant might lead to excessively small values ofΛQCD and large values for mesonic decay constants f ¯Qq. Indeed, given the naiveexpressions for these quantities:αs(µ) =12π(33 −2nf) log µ2Λ2,(55)and, assuming the relative momenta of the quarks is small,f 2¯Qq =12M ¯Q + mq|Ψ(0)|2 ,(56)let us consider, for example, the D system.

Then αs = 0.363 and µ = 287 MeV,and with three flavors of quark, we calculate ΛQCD = 42 MeV and fB = 342MeV. However, one may state the following objections: first, ΛQCD is computedfrom the full theory of QCD, but the nonrelativistic potential approach includesthe confinement in an ad hoc fashion, by including a confinement constant a,which is independent of αs.

Furthermore, choosing ΛQCD as the renormalizationpoint forces an artificial singularity at µ = ΛQCD; the problem is that little isknown about the low-energy behavior of strong interactions. At low energies thecomputation and interpretation of ΛQCD requires a more careful considerationof confinement.

With respect to the decay constant, the assumption that thequarks are relatively at rest leads to the evaluation of the wavefunction at zeroseparation. Inclusion of nonzero relative momentum will presumably result inthe necessity of considering separations of up to a Compton wavelength r ≈1µ,for which the wavefunction is smaller in the 1s-state.

Thus decay constants maybe smaller than computed in the naive model.There is one further qualitative success of this model, a partial explanationof the experimental facts that D∗s −Ds = 141.5 ± 1.9 MeV ≈D∗−D, andB∗s −Bs = 47.0 ± 2.6 MeV ≈B∗−B, namely, the approximate independence of19

vector-pseudoscalar splitting on the light quark mass. In our model, the leadingterm of the splitting is, using Eqs.

33 and 45,∆∗Qq ≈169Ma2αsβ"λ* 1ρ2++ 1#. (57)Inasmuch as β, λD1ρ2E, and αs are slowly varying in the light quark mass m, thefull expression reflects this insensitivity, in accord with experiment.

In fact, wemay fit the experimental values above to obtain more running values of αs:∆∗cs = 141.5 MeVforαs = 0.351,∆∗bs = 47.0 MeVforαs = 0.295,(58)and again these decrease as the mass scale increases. Note, however, one kinkin this interpretation: the heavy-strange mesons all have larger reduced massesthan their unflavored counterparts, yet the corresponding values of αs are nearlythe same.8ConclusionsIn this paper, we have seen how mass contributions to a bound system of par-ticles are derived from an interaction Hamiltonian in field theory, and how thiscalculation is then reduced to a problem in nonrelativistic quantum mechanics.For the system of a quark and antiquark bound in a meson, the exchange of onemediating vector boson reduces to the Breit-Fermi interaction in the nonrela-tivistic limit.

It is also important to consider contributions to the total energyfrom the kinetic energy and the long-range potential of the system; in fact, thehigher-order momentum expectation values can be so large that it is necessaryto impose an ansatz (Eq. 26) in order to estimate their combined effect.

Futurework may suggest better estimates.It is found in the case of a linear-plus-Coulomb potential that the largestcontributions to electromagnetic mass splittings originate in the kinetic energy,static Coulomb, and strong hyperfine terms. However, it is likely that simi-lar results hold for other ans¨atze and potentials.

As in other models, vector-pseudoscalar mass differences are determined by strong hyperfine terms.20

With typical values for quark masses, the confinement constant, and theup-down quark mass difference, we can obtain agreement for the mass split-tings of the D- and B-mesons. The failure of the model for K mass splittingsis attributed to the collapse of the nonrelativistic assumptions in that case.The model also qualitatively explains the similarity of heavy-strange to heavy-unflavored vector-pseudoscalar splittings, although additional work is needed toexplain why these numbers are nearly equal, despite the expected inequality ofαs at the two different energy scales.Another interesting problem is the running of αs itself at low energies.

Asmentioned in the results section, this running cannot be neglected if we are toobtain sensible results, and yet our approximation of this running is based oncrude assumptions. The size of αs also enters into another possible development,namely, whether terms of o(α2s) are important, particularly for the K system.More reliable estimates are required.In addition to the explicit formulae derived in this paper, the techniquesemployed here may be applied to later efforts: in particular, the explicit consid-eration of the mass-dependence of expectation values and the use of quantum-mechanical theorems to relate various expectation values for certain potentials.The methods and formulae in this work may prove to be a starting point forsubsequent research.AcknowledgmentsI would like to thank J. D. Jackson, L. J.

Hall, and M. Suzuki for invaluablediscussions during the preparation of this work.References[1] C. P. Singh, A. Sharma, and M. P. Khanna, Phys. Rev.

D 24, 788 (1981). [2] L.-H. Chan, Phys.

Rev. Lett.

51, 253 (1983). [3] D. Y. Kim and S. N. Sinha, Ann.

Phys. 42, 47 (1985).

[4] D. Flamm, F. Sch¨oberl, and H. Uematsu, Nuovo Cimento 98A, 559 (1987).21

[5] A. De R´ujula, H. Georgi, and S. L. Glashow, Phys.

Rev. D 12, 147 (1975).

[6] K. P. Tiwari, C. P. Singh, and M. P. Khanna, Phys. Rev.

D 31, 642 (1985). [7] J. L. Goity and W.-S. Hou, Phys.

Lett. 282B, 243 (1992).

[8] H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One- and Two-Electron Atoms (Plenum, New York, 1977). [9] S. N. Gupta, Quantum Electrodynamics (Gordon and Breach, New York,1977), pp.

191-206. [10] M. Jacob and G. C. Wick, Ann.

Phys. 7, 404 (1959).

[11] C. Quigg and J. L. Rosner, Phys. Rep. 56C, 167 (1979).

[12] E. Eichten, K. Gottfried, T. Kinoshita, K. D. Lane, and T.-M. Yan, Phys.Rev. D 17, 3090 (1978).

[13] M. G. Olsson, University of Wisconsin, Madison Report No. MAD/PH/656,June, 1991 (unpublished).22

Table I: Contributions to mass splittings of heavy mesonsD mesonsB mesonsαs0.3630.312SourceMeVMeVIsospin pairsKinetic energy-4.109-3.523Potential energy1.057-1.645Strong Darwin-0.834-0.635EM Darwin-0.7690.147Static Coulomb-2.4421.252∆0QStrong hyperfine2.1484.075EM hyperfine0.424-0.561Total-4.525-0.889∆1QStrong hyperfine3.6835.244EM hyperfine1.817-0.825Total-1.5960.0171−−0−Strong hyperfine(leading)141.3046.04(subleading)± 0.77± 0.58∆∗QuEM hyperfine0.93-0.18Total143.0046.45∆∗QdEM hyperfine-0.460.09Total140.0745.5423

Table II: Meson mass splittings compared to experimentMass splittingNotationPredicted (MeV)Expt. (MeV)K+ −K0∆0s-0.98-4.024 ± 0.032K∗+ −K∗0∆1s-0.15-4.51 ± 0.37 aK∗+ −K+∆∗su398.6397.94 ± 0.24 aK∗0 −K0∆∗sd397.8398.43 ± 0.28 aD0 −D+∆0c-4.53-4.77 ± 0.27D∗0 −D∗+∆1c-1.60-2.9 ± 1.3D∗0 −D0∆∗cu143.0142.5 ± 1.3D∗+ −D+∆∗cd140.1140.6 ± 1.9 aB+ −B0∆0b-0.89-0.1 ± 0.8B∗+ −B∗0∆1b0.02NAB∗+ −B+∆∗bu46.546.0 ± 0.6 bB∗0 −B0∆∗bd45.546.0 ± 0.6 ba obtained as a difference of world averagesb average of charged and neutral states24

Figure CaptionsFIG. 1.

Diagrammatical representation of Mfi.FIG. 2.

Diagram for Mfi in the mesonic system.FIG. 3.

Free quark Feynman amplitude M.FIG. 4.

Notation and conventions for the mesonic system.25

✓✒✏✑interaction✓✒✏✑✻✻composite systemsuperposition✻✻constituentsFig. 1✓✒✏✑✲✲✓✒✓✒✓✒✏✑✏✑✓✒✏✑g,γ¯Qq✻meson✻meson wavefunctionFig.

2✲✲✓✒✓✒✓✒✏✑✏✑g,γ¯QqFig. 326

✓✒✏✑✲✲✓✒✓✒✓✒✏✑✏✑✓✒✏✑P →P →↑kpiPipfPfq, helicity h¯Q, helicity H⃗pi =⃗P2 + ⃗p⃗Pi =⃗P2 −⃗p⃗pf =⃗P2 + ⃗p ′⃗Pf =⃗P2 −⃗p ′p0i ≡εiP 0i ≡Eip0f ≡εfP 0f ≡Efp2i = p2f ≡m2P 2i = P 2f ≡M2Fig. 427

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