Accuracy of Auxiliary Field Approach for Baryons
aa r X i v : . [ h e p - ph ] D ec EPJ manuscript No. (will be inserted by the editor)
Accuracy of Auxiliary Field Approach for Baryons
I.M.Narodetskii , C.Semay , and A.I.Veselov Institute of Theoretical and Experimental Physics, 117218 Moscow, Russia Groupe de Physique Nucl´eaire Th´eorique, Universit´e de Mons-Hainaut, Place du Parc 20, BE-7000 Mons, Belgium.Received: date / Revised version: date
Abstract.
We provide a check of the accuracy of the auxiliary field formalism used to derive the EffectiveHamiltonian for baryons in the Field Correlator Method. To this end we compare the solutions for theEffective Hamiltonian with those obtained from the solution of the spinless Salpeter equation. Comparingthese results gives a first estimate of the systematic uncertainty due to the use of the auxiliary fieldformalism for baryons.
PACS.
The advent of new ideas concerning quark-quark forcesin QCD has led to a revival of interest in baryon spec-troscopy. Various versions of the constituent quark model[1] reproduce the octet and decuplet ground states buthave very different and even contradictory predictions onthe spectrum of excited states. It is therefore very im-portant to develop model independent methods that aredirectly connected to the QCD Lagrangian and can helpin alternatively understanding baryon spectroscopy.One of such approaches is based on the Field Correla-tor Method (FCM) in QCD [2]. FCM provides a promis-ing formulation of the nonperturbative QCD that givesadditional support for the quark model assumptions. Theapplication of this method for light mesons, heavy quarko-nia, heavy-light mesons and light and heavy baryons canbe found in Refs. [3]. The key ingredient of the FCM isthe use of the auxiliary fields (AF) initially introduced inorder to get rid of the square roots appearing in the rel-ativistic Hamiltonian . Using the AF formalism allowsone to write a simple local form of the Effective Hamilto-nian (EH) for the three quark system [6], which comprisesboth confinement and relativistic effects and contains onlyuniversal parameters: the string tension σ , the strong cou-pling constant α s , and the bare (current) quark masses m i . Send offprint requests to : I.M.Narodetskii a This work was supported by RFBR grants 06-02-17120,08-02-00657, and 08-02-00677. C. Semay thanks the F.R.S.-FNRS for financial support. Historically the AF formalism was first introduced in [4] totreat the kinematics of the relativistic spinless particles. Fora brief review of the AF formalism relevant to the problemconsidered in this paper see Sec. II of [5].
The EH has the form H = X i =1 (cid:18) m i µ i + µ i (cid:19) + H + V. (1)In Eq. (1), H is the non-relativistic kinetic energy oper-ator for masses µ i , V is the sum of the string potential V Y ( r , r , r ) and a Coulomb interaction term V Coulomb arising from the one-gluon exchange. The string potentialis V Y ( r , r , r ) = σ r min , (2)where r min is the minimal string length corresponding tothe Y-shaped configuration. Finally the µ i are the oper-ator AF that have to be determined from the variationalprinciple.Note that the the sum of the mass term and H in Eq.(1) can be conveniently written as X i =1 (cid:18) m i µ i + µ l (cid:19) + H = X i =1 (cid:18) p i + m i µ i + µ i (cid:19) . (3)After taking the extremum of this expression in µ i oneends with the standard relativistic kinetic energy operator P i =1 p p i + m i .In this paper we use an approximate approach to con-sider the AF formalism first suggested in [7]. The AF aretreated as c-number variational parameters. In this ap-proach one replaces the operators µ i ( τ ) depending on timeparameter τ by the c-numbers µ i independent of τ . Theeigenvalue problem is solved for each set of µ i ; then onehas to minimize h H i with respect to µ i . Such an approachallows for a very transparent interpretation of AF: start-ing from bare quark masses m i , we naturally arrive at thedynamical masses µ i that appear due to the interaction I.M.Narodetskii et al.: Accuracy of Auxiliary Field Approach for Baryons and can be treated as the dynamical masses of constituentquarks.An obvious disadvantage of the AF approach is that,as a variational method, it provides only an upper boundto the mass spectrum. So far the accuracy of this approx-imate solution for relativistic systems has been checkednumerically only for mesons [5,8]. The principle objectiveof this work is to test the AF method for baryons. Weimplement the AF method to calculate the baryon massesand then perform similar calculations using the relativisticHamiltonian H = X i =1 q p i + m i + V. (4)Although being formally simpler the Hamiltonian (1) isequivalent to (4) up to the elimination of the AF (see e.g.Ref. [9]). We refer to an eigenvalue equation with Hamilto-nian (4) as the spinless Salpeter equation (SSE). In QCD,it arises from the Bethe-Salpeter equation replacing theinteraction by the instantaneous potential V and consid-ering a limited Fock space containing qqq states only.In this paper, we study the confinement plus Coulombenergies for the ground S -wave and orbitally excited P -wave states of nnn , nns and ssn baryons and disregardthe spin dependent forces, which are not relevant for ourconsideration.The baryon masses in the AF approach are calculatedusing the hyperspherical method, while those in the SSEare calculated variationally. The numerical algorithm tosolve the three-body problem variationally is based on anexpansion of the wave function in terms of harmonic oscil-lator functions with different sizes [10]. The details of tech-nical aspects can be found elsewhere [11]. It was provedto give results of good accuracy if the expansion is pushedsufficiently far (let say up to 16-20 quanta). Moreover itcan deal easily either with a non-relativistic or relativisticexpression for the kinetic energy operator.We find an accuracy of the AF method for hyperons tobe about 6 % at worst, which is quite reasonable to justifyapplication of the AF formalism.The paper is organized as follows. In Sec. 2, we brieflyreview the EH method. The application of this methodfor the baryons was described in detail elsewhere [12,13].Here we give only a brief summary important for our par-ticular calculation. In Sec. 3, we discuss the hypersphericalapproach, which is a very effective numerical tool to solvethis Hamiltonian. In Sec.4, we provide a few numericalexamples illustrating the accuracy of the hypersphericalsolutions. In Sec. 5, predictions of the AF method arecompared with those obtained from the solution of thespinless Salpeter equation (SSE). Section 6 contains ourconclusions. Here and below the symbol n stands for the light quarks u or d . The baryon mass in the FCM is given by M AFB = M AF + C AF , (5) M AF = X i =1 (cid:18) m i µ i + µ i (cid:19) + E ( µ i ) (6)where E ( µ i ) is an eigenvalue of the Shr¨odinger operator H + V , the constant AF µ i are defined from the minimumcondition ∂ M AF ( m i , µ i ) ∂ µ i = 0 , (7)and C AF is the quark self-energy correction which is cre-ated by the color magnetic moment of a quark propagatingthrough the vacuum background field [14]. This correc-tion, which can be added perturbatively, adds an overallnegative constant to the hadron masses: C AF = − σπ X i η ( t i ) µ i , t i = m i /T g , (8)where 1 /T g is the gluonic correlation length. In what fol-lows we use T g = 1 GeV.The function η ( t ) is defined as η ( t ) = t Z ∞ z K ( tz ) e − z dz, (9)where K is the McDonald function. A straightforwardcalculation yields [14] η ( t ) = 1 + 2 t (1 − t ) − t (1 − t ) / ln 1 + √ − t t , t < , = 1 + 2 t (1 − t ) − t ( t − / arctan ( p t − , t > η (0) = 1 and η ( t ) ∼ /t as t → ∞ .The baryon mass in the SSE approach is given by M SSEB = M SSE + C SSE , (11)where M SSE is an eigenvalue of the relativistic Hamilto-nian (4) and the C SSE are given by (8) with the obvioussubstitution µ i → ω i , where ω i = h q p i + m i i (12)are the average kinetic energies of the current quarks.We will not perform a systematic study in order todetermine the best set of parameters to fit the baryonspectra. Instead, in what follows we employ some typicalvalues of the string tension σ and the strong coupling con-stant α s , which have been used for the description of theground state baryons [12]: σ = 0.15 GeV and α s = 0.39.In our calculations we use the values of the current light .M.Narodetskii et al.: Accuracy of Auxiliary Field Approach for Baryons 3 quark masses, m u = m d = 9 MeV, and m s = 175 MeV.As in Ref. [12] we neglect the spin dependent potentialsresponsible for the fine and hyperfine splittings of baryonstates.Our aim is to compare the baryon masses given by Eqs.(5) and (11). To this end we first solve the non-relativisticSchr¨odinger equation with the confining and Coulomb in-teractions to determine the constituent quark masses µ i and the baryon masses M AFB . Efficient methods to dealwith the Y-shape interaction rely either on Monte-Carloalgorithms [15,16] or the hyperspherical method [17]. Weuse the latter approach.
In this section, we briefly review the hyperspherical method,which we use to calculate the masses of the ground andexcited hyperon states.The baryon wave function depends on the three-bodyJacobi coordinates ρ ij = r µ ij µ ( r i − r j ) , λ ij = r µ ij, k µ (cid:18) µ i r i + µ j r j µ i + µ j − r k (cid:19) , (13)( i, j, k cyclic), where µ ij and µ ij,k are the appropriate re-duced masses: µ ij = µ i µ j µ i + µ j , µ ij, k = ( µ i + µ j ) µ k µ i + µ j + µ k , (14)and µ is an arbitrary parameter with the dimension ofmass, which drops out in the final expressions. There arethree equivalent ways of introducing the Jacobi coordi-nates, which are related to each other by linear transfor-mations with the Jacobian equal to unity. In what followswe omit the indices i and j .In terms of the Jacobi coordinates the kinetic energyoperator H in (1) is written as H = − µ (cid:18) ∂ ∂ ρ + ∂ ∂ λ (cid:19) == − µ (cid:18) ∂ ∂R + 5 R ∂∂R + L ( Ω ) R (cid:19) , (15)where R is the six-dimensional hyperradius that is invari-ant under quark permutations, R = ρ + λ ,ρ = R sin θ, λ = R cos θ, ≤ θ ≤ π/ , (16) Ω denotes five residuary angular coordinates, and L ( Ω )is an angular operator L = ∂ ∂θ + 4 cot θ ∂∂θ − l ρ sin θ − l λ cos θ , (17) whose eigenfunctions (the hyperspherical harmonics) sat-isfy L ( Ω ) Y [ K ] ( θ, n ρ , n λ ) = − K ( K + 4) Y [ K ] ( θ, n ρ , n λ ) , (18)with K being the grand orbital momentum.The wave function ψ ( ρ , λ ) is written in a symbolicalshorthand as ψ ( ρ , λ ) = X [ K ] ψ [ K ] ( R ) Y [ K ] ( Ω ) , (19)where the set [ K ] is defined by the orbital momentum ofthe state and the symmetry properties.We truncate this set using the approximation K = K min . We comment on the accuracy of this approximationlatter on. Our task is then extremely simple in principle:we have to choose a zero-order wave function correspond-ing to the minimal K for a given L ( K min = 0 for L = 0and K min = 1 for L = 1). The corresponding hyper-spherical harmonics are Y = r π , K = 0 , Y ρ = r π ρ R , Y λ = r π λ R , K = 1 . (20)For nns baryons we use the basis in which the strangequark is singled out as quark 3 but in which the non-strange quarks are still antisymmetrized. In the same way,for the ssn baryon we use the basis in which the nonstrange quark is singled out as quark 3. The nns basisstates diagonalize the confinement problem with eigen-functions that correspond to separate excitations of thenon-strange and strange quarks ( ρ - and λ excitations,respectively). In particular, excitation of the λ variableunlike excitation in ρ involves the excitation of the “odd”quark ( s for nns or n for ssn ). The nonsymmetrized uds and ssq bases usually provide a much simplified pictureof the states. The physical P-wave states are neither pureSU(3) states nor pure ρ or λ excitations but linear combi-nations of all states with a given J . Most physical statesare, however, closer to pure ρ or λ states than to pureSU(3) states [18]. Note that for the nnn baryons, the ρ and λ excitation energies are degenerate.Introducing the reduced function u γ ( R ) Ψ γ ( R, Ω ) = u γ ( R ) R / · Y ν ( Ω ) , (21)where γ = 0 for L = 0, γ = ρ, λ for L = 1 , the newvariable x = √ µ R = X i µ µ M r + µ µ M r + µ µ M r ! / , (22) In what follows, for ease of notation we will drop the mag-netic quantum numbers of the vector spherical harmonics. I.M.Narodetskii et al.: Accuracy of Auxiliary Field Approach for Baryons and averaging the interaction V = V Y + V C over the six-dimensional sphere Ω with the weight | Y γ | , one obtainsthe one-dimensional Schr¨odinger equation for u γ ( x ) d u γ ( x ) dx +2 (cid:18) E − ( K + )( K + )2 x − V γ ( x ) (cid:19) u γ ( x ) = 0 , (23)where V γ ( x ) = V γ Y ( x ) + V γ Coulomb ( x ) ,V γ Y ( x ) = Z | Y γ ( θ, χ ) | V Y ( r , r , r ) dΩ = σ b ν x √ µ , (24)and V γ Coulomb ( x ) = − α s Z | Y γ ( θ, χ ) | X i < j r ij dΩ = − α s a γ x √ µ . (25)In what follows we denote µ = µ = µ, µ = κ µ. (26)Then the straightforward analytical calculation of the in-tegrals in (25) yields a √ µ = 163 π (cid:18) √ r κ κ (cid:19) √ µ, (27) a ρ √ µ = 3215 π (cid:18) √ r κ κ κ + 61 + κ (cid:19) √ µ, (28) a λ √ µ = 3215 π (cid:18) √ r κ κ κ κ (cid:19) √ µ. (29)For κ = 1 (the nnn system) a ρ = a λ . The correspond-ing expressions for b γ are more complicated (see, e.g., theappendix of Ref. [13]). A few words concerning the accuracy of the approxima-tion K = K min are in order. An illustration of the accuracyof the hyperspherical approximation K = K min is givenby the results presented in Table 1. This Table comparesthe eigenvalues E in Eq. (6) for the nnn , nns and ssn systems obtained using the variational method and thosecalculated from Eq. (23) with K = K min 4 . In all casesthe dynamical masses µ i are the same as were found fromthe minimum condition (7) for the Y-shaped string poten-tial [13]. For technical reasons the variational calculations Recall that, as was stated in Sec. 3, the ρ and λ excitationenergies for the nnn baryon are degenerate. have been performed not for the genuine string junctionpotential but for its approximation by a sum of the one-and two-body confining potentials [19] V M = 12 ( V ∆ + V CM ) , (30)where V ∆ with the sum of the two-body confining poten-tials is V ∆ = σ X i 20 MeV or 1 − E for all states (compare columns 6and 9). Let us note that Hamiltonian 1 with potential V M gives eigenvalues wich are, to some MeV, the arithmeticmean of the eigenvalues with potential V ∆ and V CM . Sothe contributions of V ∆ and V CM to V M are nearly evenlydistributed.The last column 10 contains the eigenvalues E M ar calculated using the variational method briefly describedin Sect. 1. Comparing the column 9 and 10 of Table 1 weconclude that the hyperspherical and variational resultsare close enough to validate the approximation K = K min . Table 2 compares the baryon masses computed using theAF and SSE formalisms. In this Table we list the massesof the nnn , nns and ssn states with L = 0,1. The en-tries labeled AF have been calculated from Eq. (23) with K = K min , while the entries labeled SSE have been cal-culated using the variational method for the relativisticHamiltonian (4). In both cases, we approximate the Yshaped string potential by the expression (30). As wasmentioned in the Introduction the comparison of the AF .M.Narodetskii et al.: Accuracy of Auxiliary Field Approach for Baryons 5 results with those evaluated from the solution of SSE hasbeen performed only for the qq mesons with the conclusionthat the variational AF method gives a systematic over-estimation of order 5-7 % [5,8]. Our calculations show thesimilar results: the relative deviation ε = M AFB − M SSEB M SSEB (33)is positive and for most considered states does not ex-ceed 6% . The accuracy of the AF approach does notseem to be very sensitive to the bare light-quark masses.The quantum numbers of states have a stronger influ-ence on the accuracy. In particular, ε for the L = 1states are uniformly smaller than those for the L = 0states. Curiously, the self-energy corrections C MF C and M SSE agree even with better accuracy (typically within5% or even better) in spite of the fact that the differ-ence µ i and ω i in some cases (e.g. for the λ excitation inthe ssn ) comprises 30%. As for the excitation energies, ∆ = M B ( L = 1) − M B ( L = 0) evaluated using theAF and SSE methods, they practically coincide for the ssn baryons and differ no more than ∼ 30 MeV for the nns baryons. Taking into consideration that we neglectthe spin interactions the baryon energies calculated us-ing SSE agree reasonably with the data [20]. For instance,for L = 0 we get ( N + ∆ ) theory = 1062 MeV versus ( N + ∆ ) exp = 1085 MeV and ( Λ + Σ + 2 Σ ∗ ) theory =1220 MeV versus ( Λ + Σ + 2 Σ ∗ ) exp = 1267 MeV. Asimilar correspondence exists for the other states consid-ered in this work. In this paper we have tested the quality of our previousstudy of the masses of the S- and P- baryon states ob-tained in the FCM with the use of the AF formalism. Tothis end we have compared the AF results with those ob-tained from the solution of the SSE with the same interac-tion. The main purpose was to check whether the resultsobtained within these two methods are similar. We havefound that they agree within ∼ 100 MeV for the absolutevalues of masses and with much better accuracy for theexcitation energies. Thereby our study supports the AFbasic assumptions by the compatibility of its mass pre-dictions with the masses derived from the SSE. Moreover,this comparative study gives better insight into the quarkmodel results, where the constituent masses encode theQCD dynamics. References 1. S. Capstick and N. Isgur, Phys. Rev. D 4, (1986) 2809:L. Ya. Glozman and D. O. Riska, Phys. Rep. 68, (1996)263. An obvious exception is the nnn state with L = 0 forwhich ε reaches 14%. 2. H. G. 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Baryon L Excitation µ µ E Y E C M E ∆ E M0 E M0 var nnn ρ, λ 457 457 1638 1697 1532 1615 1612 nns ρ 482 459 1611 1670 1506 1589 15871 λ 441 534 1614 1676 1508 1593 1591 ssn ρ 520 424 1592 1653 1487 1571 15691 λ 483 506 1588 1646 1485 1567 1566 Table 2. Comparison of baryon masses calculated using the AF approach and SSE. The symbol ν i denotes either the constituentquark masses µ i or the average kinetic energies of the current quarks ω i . Shown are the masses M AF and M SSE without theself-energy corrections, the self-energy corrections C AF and C SSE , M = M + C (all in units of MeV), and the relative error ε defined by Eq. (33).Baryon L Excitation Method ν = ν ν M C M ε (%) nnn − 702 1209 13.8SSE 394 394 1788 − 726 1062 nns − 648 1298 6.4SSE 396 484 1877 − 657 1220 ssn − 598 1384 6.3SSE 404 465 1904 − 602 1302 nnn ρ, λ AF 457 457 2301 -627 1674 9.1 ρ, λ SSE 440 440 2186 -651 1534 nns ρ AF 482 459 2356 − 581 1751 6.0 ρ SSE 464 465 2245 − 594 1652 λ AF 441 534 2330 − 592 1738 1.5 λ SSE 415 592 2315 − 603 1712 ssn ρ AF 520 424 2362 − 552 1810 4.1 ρ SSE 478 530 2302 − 564 1738 λ AF 483 506 2367 − 540 1827 4.4 λ SSE 503 391 2295 −−