The quark Dyson-Schwinger equation shows that there are distinct differences
between light and heavy quarks. The dynamical mass function of the light quarks
is characterised by a sharp increase below 1 GeV, whereas the mass function of
the heavy quarks is approximately constant in this infrared region. As a
consequence, the heavy-meson masses increase linearly with the current-quark
masses, whereas the light pseudoscalar meson masses are proportional to the
square root of the current-quark masses.Comment: 4 pages, 3 figures, Contribution to the IVth International Workshop
  on Progress in Heavy Quark Physics, 20-22 Sept. 1997, Rostoc

Maris, P.

Roberts, C. D.

English

arXiv

The quark Dyson-Schwinger equation shows that there are distinct differences between light and heavy quarks. The dynamical mass function of the light quarks is characterized by a sharp increase below 1 GeV, whereas the mass function of the heavy quarks is approximately constant in this infrared region. As a consequence, the heavy meson masses increase linearly with the current quark masses, whereas the light pseudoscalar meson masses are proportional to the square root of the current quark masses

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.iw-==j4fl+y/c.f- WFNo \Differences between heavy and light quarks ~E’CEIV~P. Maris and C. D. Roberts JUL26WPhysics Division, Argonne National Laboratory, Argonne IL 60439-4843 0 ~~}1AbstractThe quark Dyson–Schwinger equation shows that there are distinct differences betweenlight and heavy quarks. The dynamical mass function of the light quarks is characterisedby a sharp increase below 1 GeV, whereas the mass function of the heavy quarks isapproximately constant in this infrared region. As a consequence, the heavy meson massesincrease linearly with the current quark masses, whereas the light pseudoscalar mesonmasses are proportional to the square root of the current quark masses.Quark propagatorThe Dyson–Schwinger equations [DSES] form an excellent tool to study nonperturbativeaspects of quark propagators and bound states 1. Various properties of the light mesonscan be described very well in this framework but this method is not limited to the lightquark sector. Nonperturbative effects, such as chiral symmetry breaking and confinement,are reflected in the dressed quark propagator S(p)’1 = iv” pA(p2) + B(p2). In Euclideanspace, the DSE for the quark propagator iswhere DPV(k) is the dressed gluon propagator, I’~(q; p) is the dressed quark-gluon vertexand m(p) is the renormalised current-quark mass. The quark-gluon-vertex, quark wave-function, and quark mass renormalisation constants, 21, 22, and 24 respectively, dependon the renormalisation point ~, as does the current-quark mass.The set of DSES form an infinite hierarchy of integral equations, and for actual calculationswe need to truncate this set of equations. Following Ref. 1, we use a truncation based onthe so-called ladder approximation, namely(2)Qualitatively reliable studies of the DSE for the dressed gluon propagator indicate thatthe effective interaction is significantly enhanced in the infrared such that on this do-main it is well represented by an integrable singularity. Combining this observationwith asymptotic freedom in the ultraviolet, motivates an Ansatz2 for G((p – q)’) with ab–function to represent the integrable infrared singularity, plus a tail term, regular in theinfrared region, to incorporate asymptotic freedom. The renormalisation constants arefixed by applying the renormalisation condition A(p’) = 1 and B(p2) = m(~).Now that we have fully specified the truncation of the quark DSE, we can solve numer-ically the resulting coupled integral equations for A(p2) and B (p2). In Fig. 1, we showthe dynamical quark mass function M’(p2) = 1?(p2)/A(p2) obtained for different quarkflavours. With a nonzero current quark mass, the dynamical mass function behaves as(3)where y~ = 1233-2Nfis the anomalous mass dimension, and m the renormalisation-point-independent current-quark mass. Dynamical chiral symmetry breaking is manifest when,DISCLAIMERThis report was prepared as an account of work sponsoredbyanagency of the United States Government. Neither theUnited States Government nor any agency thereof, nor anyof their employees, make any warranty, express or implied,or assumes any legal liability or responsibility for theaccuracy, completeness, or usefulness of any information,apparatus, product, or process disclosed, or represents thatits use would not infringe privately owned rights. Referenceherein to any specific commercial product, process, orservice by trade name, trademark, manufacturer, orotherwise does not necessarily constitute or imply itsendorsement, recommendation, or favoring by the UnitedStates Government or any agency thereof. The views andopinions of authors expressed herein do not necessarilystate or reflect those of the United States Government orany agency thereof.DISCLAIMERPortions of this document may be illegiblein electronic image products. images areproduced from the best available originaldocument...# ,.10’10°10-’sG-z- 10‘=z10-’10-’1041(.?--------------- ------------. ------- -------- .--q----------------- ----<---- WUark----- a-quark ~— u/d-quark .0 chiral limit ~p’ (Ge@320------- --- ,/’---- *,Lw*-/ --- ---/ ---I// --- b-quark,...’ ---- *quark,/’,,. ----- s-quark 1---- /“” — uf&quark---*-------------------------0 1 2 3 4 5p (GeV)Fig. 1. The dynamica3mass function of the quark propagator in the chiral limit (circles), andfor up/down (solid), strange (short dssh), charm (dash-dot), and bottom (long dash) quarks.for m = O, one obtains itl(p2) # O in solving Eq. (1), which is impossible at any tilteorder in perturbation theory. Analysing our chiral limit solution we find27r27mM(p2) = y - (’w)”~2(Mdr’m‘ ‘2>AQCD’(4)with the renormalisation-point-independent chiral condensate — (Qq)O= (0.227 GeV)3.The large-p2 momentum dependence of the quark mass function, both Eq. (3) and 13q. (4),is characteristic of the QCD renormalisation group at one-loop4.In the infrared region, we see that dynamical chiral symmetry breaking implies more thanjust (~q}~ # O. For small momenta, p < 1 GeV, there is a qualitative and distinct dif-ference between light and heavy quarks. The light-quark mass function is very similarto the chid limit mass function, and is characterised by a significant infrared enhance-ment, a direct result of that in the effective coupling, ~ (k2). Introducing the Euclideanconstituent-quark mass, ME, asthe solution of p2 = M2 (p2), the ratio: lkf~jm(p), is a si.n-gle, indicative and quantitative measure of the nonperturbative effects of gluon-dressing onthe quark propagator. For current-quark mass values of %/d = 3.7 MeV, m, = 82 MeV,m= = 0.59 GeV and m~ = 2.0 GeV at the renormalisation point p = 20 GeV, we findflavour u~d s c b tfViE/m(p) 150102.2 1.4 -+1which clearly indicates the magnitude of this effect for light quarks. It provides a qualita-tive understanding of the nature of the “constituent-quark” mass. The magnitude of thisratio is a signal of the effect of the infrared enhancement of the quark-quark interactionon quark propagation characteristics.For heavy quarks, the ratio ME/m(p) takes a value of O(1) because the current-quarkmass is much larger than the mass-scale characterizing the infrared enhancement inthe effective coupling, AQCD. This means that in the spacelike region the momentum-dependence of the heavy-quark mass function is dominated by perturbative effects andsuggests, in phenomenological applications, that one can replace the heavy-quark massfunction by a constant mass5.2?. .. .. .Note that the both heavy and light quarks, and also the gluons, are explicitly confined;the essence of confinement is that the propagator of a confined particle does not havea Lehmann representation 6. Therefore, quark confinement entails that there is no pole-mass, which would be the solution of p2 + &f2(p2) = O.Meson bound statesThe renormalised Bethe-Salpeter Equation [BSE] for meson bound states is[rH(p; P)l,u = /~[s(~+)rH(,;p)s(,-)]sr~;;(,,p;p),(5)where: H specifies the flavour structure of the meson; q+ = q + qp P, q- = q —(1 —IIP)P, ,with P the total momentum of the bound state and qp the momentum partitioning; and‘r,.. . ,Urepresent colour-, Dirac- and flavour-matrix indices. Solutions of Eq. (5) exist onlyfor particular, separated, timelike values of P2; and the eigenvector associated with eacheigenvalue, the Bethe-Salpeter amplitude [BSA]: I’H(p; P), is the one-particle-irreducible,fully-amputated quark-meson vertex. Once we know this BSA, we can use it to calculatemeson observable such as the decay constant and various form factors and other physicalobservable. For pseudoscalar mesons, the solution of Eq. (5) has the general form2In order to calculate the BSA, we must have an explicit form for the kernel K~~(q, p; P)in Eq. (5). Requiring that the axial-vector Ward indentity” be preserved constrains thiskernel. The ladder approximation fulfllls this constraint’d4q— G((P – !7)2)Df&(P– &Yps(g+)rH(% P)s(q-)’%. (7)rH(p;’) =‘~ j (2@4Numerical solutions of Eq. (7) for increasing values of the current quark mass show thatthe meson mass grows with the quark mass. In Fig. 2 we present the meson mass mH asfunction of the renormalisation-point-independent current quark mass ti, for both (u~)(~*, K, D, B) and (fictitious) (q?) (To) pseudoscalar mesons. In both cases, the mesonmass mH is for light quarks, proportional to the square root of the current quark mass,whereas for heavy quarks, the meson mass grows linearly with the current quark mass:(8)The pseudoscalar decay constant increases with quark masses, at least up to quark massesof about 0.5 GeV x 3 ti.. The reversal of this trend, that is dictated by the heavy quarklimit ~H@ = constant,5 is reached at larger current quark masses.Although the BSA does depend on the momentum partitioningVP in Eq. (5), the physicalobservable, such as the meson mass and decay constant, do not, provided one takes intoaccount the nonleading Dirac amplitudes FH and G’H in Eq. (6). These amplitudes arealso essential in order to satisfy the axial-vector Ward identity, and are quantitativelyimportant for physical observables2.*In the chiral limit, this Ward identity relates the meson B&4 to the quark propagation the pions arethe Go\dstone bosons associated with dynamical chiral symmetry breaking.31.21.0~ 0“89= 0.6E0.4 “0.20.0Fig. 2. The pseudoscalar mesonmass for (UT) (plusses) and equalquark (qij) (diamonds) boundstates, ss a function of the current-quark mass rnq.0.00 0.10 0.20 0.30 0.40 0.50mq (GeV)The ultraviolet asymptotic behaviour of all amplitudes l?(p; ), F(p; P), p2 G(p; F’) andp2 H(p; P), is fully determined by the chiral-limit quark mass function: all behave likeEq. (4), both for light and heavy pseudoscalar mesons. A difference between the BSA ofheavy mesons and light mesons is that heavy mesons are broader in momentum space:they approach the asymptotic behavior at larger momenta, and are thus narrower incoordinate space than light mesons.ConclusionThe dynamical mass function is a useful measure of nonperturbative effects. In the in-frared region, the mass functions of the light flavours, in particular the u- and d-quarks,are very similar to the chid limit mass function IWO@2). They are all characterised bya sharp increase below 1 GeV: the constituent quark mass is much larger than the cur-rent quark mass. The mass function of the heavy quarks does not show such behaviour:nonperturbative effects are much less important for the heavy quarks. For the b-quarks,and to a lesser extent the c-quarks, the mass function can be replaced by a constant: theconstituent mass is approximately the same as the current quark mass. One of the con-sequences of th~ d~fference is reflected in the mass spectrum of the pseudoscalar mesons:the heavy meson masses increase linearly with the current quark masses, whereas the lightmeson masses are proportional to the square root of the current quark masses.AcknowledgmentsThis work was supported by the US Department of Energy, Nuclear Physics Division, un-der contract number W-31-109-ENG-38 and benefited from the resources of the NationalEnergy Research Scientific Computing Center.1. C. D. Roberts and A. G. WHliams, Frog. Part. Nucl. Phys. 33 (1994) 447.2. P. Maris and C. D. Roberts, nucl-th/9708029, to appear in Phys. Rev. C.3. N. Brown and M. R. Pennington, Phys. Rev. D 39, 2723 (1989).4. K. Lane, Phys. Rev. 10, 2605 (1976); H. D. Politzer, Nucl. Phys. B 117, 397 (1976).5. M. A. Ivanov ‘et at., nucl-th/9704039, to appear in Phys. Lett. B; C. D. Roberts et al.,nucl-th/yymmn~, these proceedings.6. C. D. Roberts, A. G. Williams and G. Krein, Int. J. Mod. Phys. A 4, 1681 (1992); P. Maris,Phys. Rev. D 52, 6087 (1995).7. A. Bender, C. D. Roberts and L. v. Smekal,Phys. Lett. B 380, 7 (1996).4

Differences between heavy and light quarks.

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Differences between heavy and light quarks

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