---

Derek B. Leinweber∗and Thomas D. Cohen

arXiv:hep-ph/9307261v2 9 Sep 1993Unquenching the ρ mesonDerek B. Leinweber∗and Thomas D. CohenDepartment of Physics and Center for Theoretical PhysicsUniversity of Maryland, College Park, MD 20742(July 9, 1993)AbstractTwo-pion induced self-energy contributions to the ρ-meson mass are exam-ined in relation to the quenched approximation of QCD, where the physicsassociated with two-pion intermediate states has been excluded from vector-isovector correlation functions. Corrections to quenched QCD calculations ofthe ρ-meson mass are estimated to be small at the order of a few percent ofthe ρ-meson mass.

The two-pion contributions display nonanalytic behavioras a function of the pion mass as the two-pion cut is encountered. The impli-cations of this nonanalytic behavior in extrapolations of full QCD calculationsare also discussed.

We note that for full QCD, the error made in making alinear extrapolation of the ρ mass, neglecting nonanalytic behavior, increasesas one approaches the two-pion cut.12.38.-t, 12.38.Gc, 14.40.-n, 14.40.CsTypeset using REVTEX1

I. INTRODUCTIONThe lattice regularized approach to quantum field theory provides the best forum for theexamination of the fundamental nonperturbative aspects of QCD. In the low momentumtransfer regime, it is the only approach which in the foreseeable future holds a reasonablepromise of confirming or rejecting the validity of QCD as the underlying theory of the stronginteractions.Recently, it has become possible to perform quenched QCD calculations in which allsystematic uncertainties are quantitatively estimated.

Thus if the effects of quenching canbe understood, the validity of QCD may be tested in the nonperturbative regime.Ofparticular note is the recent determination of the QCD coupling constant, αMS, from the1S −1P mass splitting of charmonium [1].In this case one believes that the effects ofquenching may be estimated with minimal model dependence. Corrections to the utilizationof the quenched approximation of QCD have been estimated and are currently the dominantsource of uncertainty in the final predictions [2].In this paper we will continue efforts along this line through the examination of systematicuncertainties in hadron mass spectrum calculations.

In particular, the importance of thetwo-pion induced self-energy contribution to the ρ-meson mass is evaluated in relation tothe quenched approximation of QCD and to full QCD. In the quenched approximation, thephysics associated with two-pion intermediate states has been excluded in the numericalsimulations.This investigation is motivated by recent results from the GF11 group [3] for the low-lyinghadron mass spectrum in the quenched approximation of QCD.

Their analysis is the firstto systematically extrapolate QCD calculations to physical quark mass, zero lattice spacingand infinite volume. Their predictions display an impressive agreement with experiment.Of eight hadron mass ratios, six agree within one standard deviation and the remaining tworatios agree within 1.6σ.

However, since the quenched approximation leaves out so muchimportant physics, one might question whether these results are actually too good [4].2

In the perturbative regime, many of the effects of not including disconnected quark loopswhen preparing an ensemble of gauge configurations may be accounted for in a simple renor-malization of the strong coupling constant. However, one also anticipates nonperturbativeeffects in making the quenched approximation.

Unlike a global renormalization of the cou-pling constant, these effects are expected to be channel specific. For example, the quenchedapproximation of QCD leaves out the physics associated with the decay of the ρ meson totwo pions.

This physics must be accounted for and added to the quenched results prior tocomparing with experimental data. Moreover, the calculated hadron masses are extrapo-lated as a function of the pion mass squared to the point at which the pion mass vanishesusing linear extrapolation functions.

Such an approach neglects nonlinear and indeed, non-analytic behavior in the continuum extrapolation function. For example, in the case of theρ-meson mass, one expects nonanalyticity associated with the onset of the two-pion cut.A priori one does not know the relative importance of two-pion intermediate states ofthe ρ meson in describing the ρ mass.

The substantial width of the ρ meson at 151.5 ± 1.2MeV indicates its coupling to pions is not small and correspondingly these dynamics mayhave significant influence on the ρ-meson mass.Geiger and Isgur were the first to study the possible importance of the two-pion inducedself-energy of the ρ meson in relation to lattice QCD calculations [5]. Their results are basedon a string breaking quark model and predict large corrections to smooth extrapolations ofthe ρ-meson mass at approximately 70 MeV.

These authors were motivated by the longstanding problem of QCD predictions for the N/ρ-mass ratio being too large. Their hopewas the two-pion induced self-energy correction would sufficiently raise the ρ mass to solvethis problem.

However, we now understand that both the finite lattice spacing and the finitevolume of the lattice act together to push up this mass ratio. Current estimates [3] for thisratio corrected to the infinite volume, continuum limit are 1.22±0.11 in excellent agreementwith the experimental value of 1.22.Geiger and Isgur [5] advocate using the nonlinearities in the ρ mass as a function of thequark mass to correct for the linear extrapolation of lattice results.

We wish to stress that3

such a procedure is only sensible for the extrapolation of full QCD calculations. As we shallargue, the entire two-pion induced self-energy is absent in the quenched approximation andshould be added on to quenched QCD results prior to comparison with experiment.

Wenote that for the model of Ref. [5], quenched lattice calculations of the ρ-meson mass wouldbe reduced by 160 MeV instead of increased by 70 MeV.

This would further exacerbate the“mN/mρ problem” discussed in their paper, rather that curing it.In the quenched approximation, the ρ meson cannot decay to two light pseudoscalars.As discussed in Ref. [6] it is not possible to generate intermediate states of the ρ meson inthe quenched approximation in which one has two isovector pseudoscalars.

One might worryabout the presence of the light isoscalar pseudoscalar η′ which fails to obtain its heavy massin the quenched approximation [6–9]. However, decay of the ρ meson to a π η′ is forbiddenby G-parity and decay to two η′-mesons is of course forbidden by the isospin invariance ofthe strong interactions.

Hence the physics associated with two-pion intermediate states ofthe ρ should simply be added onto the results extracted from calculations of quenched QCD,provided the lattice spacing is defined by physical observables which do not have a similardependence on pion decays.Current calculations of full QCD typically employ quark masses which place 2mπ >mρ. As a result, the functional form of the extrapolation function should account for thenonanalytic behavior in the ρ-meson mass as the two-pion cut is encountered.

On the lattice,the spectral density does not have a cut but rather a series of poles at the points satisfying√s = 2m2π + p2n1/2 ,(1)where pn are the discrete momenta allowed on the lattice. Obviously, to fully account forthe two-pion induced self-energy of the ρ, one must first extrapolate the lattice results tozero lattice spacing and infinite volume prior to extrapolating the quark masses to physicalvalues.In evaluating the integrals describing the coupling of pions to the ρ one must take intoaccount the q2 dependence of the ρ to two π coupling constant gρππ reflecting the internal4

structure of these mesons. While this q2 dependence was extracted from a string breakingquark model in Ref.

[5], we elect to take a more agnostic approach and consider differentmethods of cutting offthe integral. In particular we investigate a sharp θ-function cutoffanda dipole cutoff.

The underlying reason for our agnosticism is the belief that models of thesort underlying Ref. [5] are not likely to correctly describe the structure of pseudo-Goldstonebosons such as the pion, as they do not incorporate chiral symmetry.

While there are modelsof the ρππ vertex which incorporate chiral symmetry and chiral symmetry breaking [10–12],these models are based on particular dynamical assumptions. Accordingly, it is difficult toassess the reliability of such models.

Instead, we consider a range of possibilities for thevertex and to simplify this task, we consider convenient phenomenological forms.One other paper addressing this issue [13] sidesteps the problems surrounding the q2 de-pendence of gρππ by fixing gρππ to a constant and making two subtractions of the divergentintegral at q2 = 0. These subtractions are absorbed into a mass and wave function renormal-ization.

However, this approach excludes any analysis of ρ-meson mass extrapolations as thesubtraction terms themselves have an unknown mπ dependence which has been lost in therenormalization procedure. Moreover, contributions from virtual two-pion states have beenabsorbed into the bare lattice parameters which is inconsistent with the dynamics containedin the quenched approximation.The outline of this paper is as follows.

In Section II the model used in examining thetwo-pion induced self-energy is outlined. Two methods for regulating the divergent self-energy are explored.

In section III the relevance of the self-energy corrections to quenchedQCD simulations is discussed. Section IV addresses the quark mass extrapolation of fullQCD calculations and the importance of nonlinear behavior in the ρ-meson mass.

Finally,the implications of this investigation are summarized in Section V.5

II. THE SELF-ENERGYIn modeling the two-pion induced self-energy of the ρ meson, Σρππ, the standard ρππinteraction motivated by low-energy current algebra is used.

The effective Lagrange inter-action has the form [14]Lint = −i gρππ ρµπ↔∂µ π+ g2ρππ π2 ρ2 . (2)The pions are further assumed to interact exclusively through the ρ channel as summarizedin the following Schwinger-Dyson equation for the ρ propagatorGµν = G0µν + G0µσ Σστ Gτν ,(3)whereG0µν =−iq2 −M20 + iǫ gµν −qµqνq2!,(4)in Landau gauge, and M0 is the bare ρ-meson mass.

The self-energy Σρππ is defined throughthe solutionGµν =−iq2 −M20 −Σρππ + iǫ gµν −qµqνq2!,(5)whereΣστ ≡Σρππ gστ −qσqτq2!. (6)Σστ is given by the standard one loop integrals−i Σστ =Zd4k(2 π)4 g2ρππ(qσ −2 kσ) (qτ −2 kτ)h(q −k)2 −m2π + iǫihk2 −m2π + iǫi −2gµνk2 −m2π + iǫ.

(7)Physically, the integral is convergent due to the momentum dependence of gρππ. However,this momentum dependence is unknown.

In light of this uncertainty, it is reasonable toparameterize the momentum dependence in terms of some given functional form with anadjustable parameter controlling how the function falls offas a function of momentum trans-fer. We consider two regulation prescriptions.

In one, we assume a monopole form for gρππ,and for comparison, we also consider a sharp θ-function cutoff.6

The simplest fashion for introducing a covariant cutofffunction is through the use of adispersion relation. The second term of (7) is q independent and serves only to subtract thequadratic divergence of the first term in maintaining current conservation.

As a result, wewrite a dispersion relation for Σ(q2) with one subtraction at q2 = 0,Σ(q2) ≡1πZ ∞0ds q2sIm Σ(s)s −q2 . (8)Of course, the imaginary part of Σρππ may be easily determined using any number of tech-niques for rendering the integral of (7) finite.

The imaginary part isIm Σρππ(q2) = g2ρππ48π q2 1 −4 m2πq2!3/2θq2 −4 m2π. (9)The value of gρππ at q2 = m2ρ is fixed by equating the imaginary parts ofM20 + Σρππ ≡mρ + i Γ22,(10)at q2 = m2ρ.

The physical values [15] mρ = 768.1 MeV and Γ = 151.5 MeV fix gρππ at ∼6.0.A. θ-Function CutoffTo illustrate the physics associated with the real part of the self-energy we first considerthe integral of (8) cut offcovariantly by a sharp θ-function at s = Λ2.

The functional formisRe Σρππ = g2ρππ48 π2 q2(ln1 −σΛ1 + σΛ+ 8 m2πq2σΛ −σ3q ln σΛ −σqσΛ + σq! ),(11)whereσq = 1 −4 m2πq2!1/2, and σΛ = 1 −4 m2πΛ2!1/2.

(12)Figure 1 illustrates the real part of Σρππ evaluated at q2 = m2ρ for a variety of cutoffsranging from slightly above m2ρ to 4 GeV2. For small cutoffs, most of the strength in theintegral lies below the ρ mass and consequently the ρ mass is pushed up due to the mixingwith pion states.

Of course, this behavior is completely consistent with that anticipated bysimple quantum mechanical arguments. For larger cutoffs the strength above the ρ massacts to reduce the ρ mass.7

B. Dipole CutoffWhile the θ-function is useful as an illustrative tool, it suffers from being physicallyartificial and the results can be very sensitive to the value of the cutoff, as illustrated infigure 1. In an attempt to better represent the q2 dependence of gρππ a monopole form foreach vertex is introduced and the dispersion relation of (8) is evaluated withg2ρππ →g2ρππ q2 + Λ2s + Λ2!2.

(13)This form maintains the normalization of gρππ defined at the physical ρ mass and rendersthe integral of (8) finite.Of course, this approach is not without a few unphysical side effects. The most obviousproblem is the introduction of spurious poles in the space like region for the s dependenceof gρππ.

However, the dispersion integral only samples the time like region and the presenceof these unphysical poles should not affect the results. One could consider other functionalforms.

However, the effective physical value for the regulator mass, Λ, is itself unknown.Our aim is to estimate the importance of the two-pion induced self-energy relative to the ρmass, as opposed to attempting to evaluate the actual correction. For this reason we viewa consideration of the dipole regulator to be adequate.Evaluation of the dispersion relation of (8) with (13) leads to the following functionalform for the real part of ΣρππRe Σρππ = g2ρππ48 π2 q2( 1 + 8 m2πq2+ 12 m2πΛ2!

1 + q2Λ2!+ 1 + 10 m2πΛ2+ 6 m2π q2Λ4!βΛ ln βΛ −1βΛ + 1! (14)−σ3q ln 1 −σq1 + σq!

),whereσq = 1 −4 m2πq2!1/2, and βΛ = 1 + 4 m2πΛ2!1/2. (15)The imaginary part is recovered as in (9).8

Figure 2 illustrates the real part of the self-energy and its derivative with respect to m2πat Λ2 = 2 GeV2. The derivative clearly displays the nonanalytic behavior encountered atmρ = 2mπ.

The second derivative is discontinuous at mρ = 2mπ and is infinite from above.The imaginary part of the self-energy is also illustrated in figure 2.A comparison with figure 1 indicates that at Λ2 = 2 GeV2 the results are not too sensitiveto the manner in which the integral is regulated. Figure 3 illustrates the real part of theself-energy for the same values of Λ2 used in figure 1.

The sensitivity of the results to thevalue of Λ is greatly reduced and all curves display the same qualitative behavior.In the limit of Λ →∞both (11) and (14) reduce toRe Σρππ = g2ρππ48 π2 q2(1 + 8 m2πq2+ ln m2πq2!−σ3q ln 1 −σq1 + σq!−ln Λ2q2! ),(16)displaying the logarithmic divergence as Λ2 →∞.III.

APPLICATION TO QUENCHED QCDThe effects of quenching QCD may be categorized as perturbative or global effects andnonperturbative or channel specific effects. As discussed in the introduction, the effects in theperturbative regime may be accounted for through a simple renormalization of the couplingconstant.

However, the effects in the nonperturbative regime will, of course, be channelspecific.The ρ-meson channel might be particularly vulnerable to the nonperturbativeeffects of quenching QCD due to the fact that the ρ is unstable in full QCD and stablein the quenched approximation.As we shall see however, our estimated “unquenching”corrections turn out to be rather small.Figure 4 displays ρ and squared pion masses for the three lightest quark masses usedin the quenched QCD analysis of the GF11 group [3] for the 32 × 30 × 32 × 40 lattice atβ = 6.17 for 219 configurations. The dashed line illustrates the linear relationship assumedin extrapolating the hadron masses to the critical point where the pion mass vanishes.9

Provided the lattice spacing is defined by physical observables which do not have a similardependence on pion decays, the physics associated with two-pion intermediate states of theρ should simply be added onto the results extracted from calculations of quenched QCD.The correction is particular to the ρ meson and is unlikely to be accounted for if the latticespacing is fixed by the nucleon mass for example.The solid line of figure 4 illustrates the addition of the self-energy correction to the linearextrapolation where the regulator mass Λ2 = 1 GeV2 has been selected [16]. This choice ofΛ is physically motivated and indicates the correction to the ρ mass at the physical point isnegligible.

The dot-dashed curve illustrates the correction when Λ2 = 2 GeV2.Figure 3 indicates that Σρππ is less than 6% of the squared ρ mass for Λ2 < 2 GeV2. Thiscorresponds to a less than 3% correction to the ρ mass itself.

The conclusion that may bedrawn from this analysis, which differs from previous considerations of these issues, is thatthe predictions of the GF11 group [3] are not “too good to be true”. The channel specificnonperturbative correction to the ρ-meson mass may actually be rather small.The magnitude of the corrections estimated here is much smaller than that anticipatedin the analysis of Ref.

[5] for any reasonable choice of Λ. Geiger and Isgur predict an un-quenching correction to the ρ mass of approximately −160 MeV in contrast to our predictionof a 0 to 25 MeV reduction of the ρ mass.IV. FULL QCD CALCULATIONSIn full QCD simulations, the two-pion induced self-energy corrections are, of course,already included.

However, the continuum predictions derived here will differ from thoseanticipated on the lattice [13] largely due to the discretization of the momenta on the latticeand the lattice regularization itself. This renders the divergent integral of (7) to a finite sumover a few two-pion states.

In fact, if one hopes to recover the continuum physics, it will benecessary to first correct the hadron masses determined at unphysical quark masses to thecontinuum, infinite volume limit prior to extrapolating to physical quark masses.10

Figure 4 suggests that it may be extremely difficult to see the effects of virtual two-pion intermediate states in full QCD. Since a great deal of the integral strength is lost forcurrent lattice regularization parameters, the correction curve in figure 4 is most likely anoptimistically large deviation from the linear relation.

Even when full QCD calculationsreach the current state of quenched calculations, it is questionable whether one will becapable of discerning the effects of virtual two-pion states of the ρ while the ρ is stable. Ofcourse, once the ρ becomes unstable and decays to two pions, the lower lying two-pion stateswill need to be subtracted from the correlation function prior to extracting the ρ mass.

Thisrenders an examination of the ρ-meson mass above the two-pion cut nearly impossible. Itshould be mentioned that these arguments support the recent findings of Ref.

[17] where anattempt to observe the effects of virtual pion states in ρ correlation functions failed.Ultimately, full QCD calculations will reach the point where linearly extrapolating the ρmass to physical quark masses ignoring nonanalytic effects will introduce a relevant system-atic error. To this end we present figure 5 which illustrates the relative amount which shouldbe added onto the ρ mass extracted from a linear extrapolation of full QCD data correctedto the continuum, infinite volume limit [18].

We remind the reader that these effects arebeing calculated using the dipole form for the cutoffwhich we have more or less arbitrarilychosen. Thus the curves are illustrative only.

The x-axis indicates the point at which thederivative is determined for the linear extrapolation. Values for Λ2 are as in figures 1 and 3.For example, if the effective physical value for Λ2 is 1 GeV2 and the derivative is determinedat m2π ≃0.25 GeV2, the ρ mass extracted from a linear extrapolation should be augmentedby approximately 25 MeV prior to comparing with experiment.The size of the corrections to linear extrapolations illustrated in figure 5 are generallysmaller than the 70 MeV addition predicted by Geiger and Isgur.

However, it is possibleto recover a correction to the ρ-meson mass extrapolation similar in magnitude to that ofGeiger and Isgur’s analysis, provided one linearly extrapolates from the onset of the cut asin their investigation. Of course, it is not practical to extrapolate from the onset of the cutin actual lattice calculations.

In fact, figure 5 indicates that this is the worst possible place11

to attempt an extrapolation to physical quark masses. The derivative is more likely to beaveraged from a number quark masses corresponding to squared pion masses of 0.2 to 0.6GeV2.

Hence the corrections are expected to be the order of 10 to 20 MeV and possiblynegligible if the effective physical value for Λ2 is 2 GeV2 or more.An important point to mention here is that a simple calculation of the ρ-meson mass infull QCD will not circumvent the problems associated with its decay to pions. In fact, asone works hard to drive down the quark mass, linear extrapolations to the physical pointwill increasingly underestimate the ρ mass.

Determination of the last few percent of thephysical ρ-meson mass requires additional information describing the q2 dependence of ρππinteractions such that a suitable extrapolation function may be identified. Fortunately, thesecorrections are small and may be neglected until systematic uncertainties associated withthe finite volume and finite spacings of the lattice are understood and eliminated.V.

SUMMARYWe have calculated the two-pion induced self-energy correction to the ρ-meson mass ina manner that allows an estimation of the correction to quenched QCD calculations and ananalysis of extrapolations of full QCD results.The analysis of full QCD extrapolations indicates linear extrapolations of typical ρ-mesonmasses extracted from lattice correlation functions will underestimate the physical ρ massby 10 to 20 MeV. An important point to draw from the analysis is that as the quark massesbecome lighter, linear extrapolations to the physical point will increasingly underestimatethe ρ-meson mass.We estimate the corrections to quenched calculations to be the order of a few percentand quite possibly negligible (0 to −25 MeV).

These results lend credence to the success ofquenched QCD in describing the physical low-lying hadron mass spectrum.12

ACKNOWLEDGMENTSWe thank Don Weingarten and Hong Chen for providing us with the lattice data il-lustrated in figure 4. We also thank Manoj Banerjee for his objective input, and MichaelHerrmann for helpful conversations.

This work is supported in part by the U.S. Departmentof Energy under grant DE-FG02-93ER-40762.T.D.C. acknowledges additional financialsupport from the National Science Foundation though grant PHY-9058487.13

REFERENCES∗Present Address: Department of Physics, The Ohio State University, 174 West 18thAvenue, Columbus, OH 43210-1106. [1] A. El-Khadra, G. Hockney, A. Kronfeld, and P. Mackenzie, Phys.

Rev. Lett.

69, 729(1992). [2] Ultimately, these uncertainties will be eliminated by directly calculating the observablesin full QCD calculations.

However, explorations of the parameter space for full QCDare in the early stages. A thorough understanding and elimination of lattice systematicsin full QCD remains a focus of current investigations.

[3] F. Butler et al., Phys. Rev.

Lett. 70, 2849 (1993).

[4] S. R. Sharpe, Recent Progress in Lattice QCD. U. Washington report no.

UW/PT-92-22. [5] P. Geiger and N. Isgur, Phys.

Rev. D 41, 1595 (1990).

[6] T. D. Cohen and D. B. Leinweber, Comments Nucl. Part.

Phys. 21, 137 (1993).

[7] C. Bernard and M. Golterman, Phys. Rev.

D 46, 853 (1992). [8] S. R. Sharpe, Phys.

Rev. D 46, 3146 (1992).

[9] Only the first two terms of the geometric series of pseudoscalar-meson and glueball states(which can generate the η′ mass when geometrically summed) survive in the quenchedapproximation. [10] J. Praschifka, C. Roberts, and R. Cahill, Int.

J. Mod. Phys.

A 2, 1797 (1987). [11] C. Roberts, R. Cahill, and J. Praschifka, Int.

J. Mod. Phys.

A 4, 719 (1989). [12] L. Hollenburg, C. Roberts, and B. McKellar, Phys.

Rev. C 46, 2057 (1992).

[13] T. A. DeGrand, Phys. Rev.

D 43, 2296 (1991).14

[14] M. Herrmann, B. Friman, and W. Norenberg, Z. Phys. A 343, 119 (1992), GSI PreprintGSI-93-12, accepted Nucl.

Phys. A.

[15] Particle Data Group, K. Hikasa, et al. , Phys.

Rev. D 45, S1 (1990).

[16] Ideally, the lattice results should be corrected to the infinite volume, continuum limitprior to adding the self-energy correction. [17] C. Bernard et al., Finite-size and quark mass effects on the QCD spectrum with twoflavors, Report no.

IUHET-232. [18] In preparing figure 5 we have accounted for the quark mass (or mπ) dependence of theρ-meson mass.

For illustrative purposes we set mρ = m0 + 0.5 (GeV−1) m2π and fixed m0to reproduce the physical ρ mass at the physical pion mass.15

FIGURESFIG. 1.The real part of Σρππ from (11) evaluated at q2 = m2ρ for a variety of cutoffs, ats = Λ2.

In this and the following figures, the finely dashed vertical line marks the position of thephysical point. For small cutoffs, most of the strength in the integral lies below the ρ mass and asa result the ρ mass is pushed up.FIG.

2.The two-pion induced self-energy (a) and its derivative with respect to m2π (b) fora regulator mass of Λ2 = 2 GeV2. Both real (solid line) and imaginary (dashed line) parts areillustrated.

The derivative clearly displays the nonanalytic behavior encountered at mρ = 2mπ.FIG. 3.The real part of the self-energy for dipole regulator masses Λ taking the same valuesused in figure 1.

The sensitivity of the results to the value of Λ is greatly reduced.FIG. 4.ρ and squared pion masses for the three lightest quark masses used in the quenchedQCD analysis of the GF11 group [3] on their largest lattice.

The dashed line illustrates the linearrelationship assumed in extrapolating the hadron masses to the critical point. The solid curvedisplays the self-energy correction for a regulator mass of Λ2 = 1 GeV2 where the correction tothe ρ mass at the physical point is negligible.

The dot-dash curve corresponds to Λ2 = 2 GeV2. Alattice scale parameter of 2.73 GeV has been applied to the otherwise dimensionless lattice results.FIG.

5.The amount to be added to the ρ mass extracted from a linear extrapolation of fullQCD data. The x-axis indicates the point at which the derivative is determined for the linearextrapolation.

Values for Λ2 are as in figures 1 and 3.16

REFERENCESPresentAddress:DepartmentofPhysics,TheOhioStateUniversity,WestthAvenue,Columbus,OH0-0.[]A.El-Khadra,G.Hockney,A.Kronfeld,andP.Mackenzie,Phys.Rev.Lett.,().[]Ultimately,theseuncertaintieswillbeeliminatedbydirectlycalculatingtheobservablesinfullQCDcalculations.However,explorationsoftheparameterspaceforfullQCDareintheearlystages.AthoroughunderstandingandeliminationoflatticesystematicsinfullQCDremainsafocusofcurrentinvestigations.[]F.Butleretal.,Phys.Rev.Lett.0,().[]S.R.Sharpe,RecentProgressinLatticeQCD.U.Washingtonreportno.UW/PT--.[]P.GeigerandN.Isgur,Phys.Rev.D,(0).[]T.D.CohenandD.B.Leinweber,CommentsNucl.Part.Phys.,().[]C.BernardandM.Golterman,Phys.Rev.D,().[]S.R.Sharpe,Phys.Rev.D,().[]Onlythersttwotermsofthegeometricseriesofpseudoscalar-mesonandglueballstates(whichcangeneratethe0masswhengeometricallysummed)surviveinthequenchedapproximation.[0]J.Praschifka,C.Roberts,andR.Cahill,Int.J.Mod.Phys.A,().[]C.Roberts,R.Cahill,andJ.Praschifka,Int.J.Mod.Phys.A,().[]L.Hollenburg,C.Roberts,andB.McKellar,Phys.Rev.C,0().[]T.A.DeGrand,Phys.Rev.D,().[]M.Herrmann,B.Friman,andW.Norenberg,Z.Phys.A,(),GSIPreprintGSI--,acceptedNucl.Phys.A.[]ParticleDataGroup,K.Hikasa,etal.,Phys.Rev.D,S(0).[]Ideally,thelatticeresultsshouldbecorrectedtotheinnitevolume,continuumlimitpriortoaddingtheself-energycorrection.[]C.Bernardetal.,Finite-sizeandquarkmasseectsontheQCDspectrumwithtwoavors,Reportno.IUHET-. []Inpreparinggurewehaveaccountedforthequarkmass(orm)dependenceofthe-mesonmass.Forillustrativepurposeswesetm=m0+0:(GeV)mandxedm0toreproducethephysicalmassatthephysicalpionmass.

atm'0:GeV,themassextractedfromalinearextrapolationshouldbeaugmentedbyapproximatelyMeVpriortocomparingwithexperiment.Thesizeofthecorrectionstolinearextrapolationsillustratedingurearegenerallysmallerthanthe0MeVadditionpredictedbyGeigerandIsgur.However,itispossibletorecoveracorrectiontothe-mesonmassextrapolationsimilarinmagnitudetothatofGeigerandIsgur'sanalysis,providedonelinearlyextrapolatesfromtheonsetofthecutasintheirinvestigation.Ofcourse,itisnotpracticaltoextrapolatefromtheonsetofthecutinactuallatticecalculations.Infact,gureindicatesthatthisistheworstpossibleplacetoattemptanextrapolationtophysicalquarkmasses.Thederivativeismorelikelytobeaveragedfromanumberquarkmassescorrespondingtosquaredpionmassesof0.to0.GeV.Hencethecorrectionsareexpectedtobetheorderof0to0MeVandpossiblynegligibleiftheeectivephysicalvalueforisGeVormore.Animportantpointtomentionhereisthatasimplecalculationofthe-mesonmassinfullQCDwillnotcircumventtheproblemsassociatedwithitsdecaytopions.Infact,asoneworkshardtodrivedownthequarkmass,linearextrapolationstothephysicalpointwillincreasinglyunderestimatethemass.Determinationofthelastfewpercentofthephysical-mesonmassrequiresadditionalinformationdescribingtheqdependenceofinteractionssuchthatasuitableextrapolationfunctionmaybeidentied.Fortunately,thesecorrectionsaresmallandmaybeneglecteduntilsystematicuncertaintiesassociatedwiththenitevolumeandnitespacingsofthelatticeareunderstoodandeliminated.V.SUMMARYWehavecalculatedthetwo-pioninducedself-energycorrectiontothe-mesonmassinamannerthatallowsanestimationofthecorrectiontoquenchedQCDcalculationsandananalysisofextrapolationsoffullQCDresults.TheanalysisoffullQCDextrapolationsindicateslinearextrapolationsoftypical-mesonmassesextractedfromlatticecorrelationfunctionswillunderestimatethephysicalmassby0to0MeV.Animportantpointtodrawfromtheanalysisisthatasthequarkmassesbecomelighter,linearextrapolationstothephysicalpointwillincreasinglyunderestimatethe-mesonmass.Weestimatethecorrectionstoquenchedcalculationstobetheorderofafewpercentandquitepossiblynegligible(0toMeV).TheseresultslendcredencetothesuccessofquenchedQCDindescribingthephysicallow-lyinghadronmassspectrum.ACKNOWLEDGMENTSWethankDonWeingartenandHongChenforprovidinguswiththelatticedatail-lustratedingure.WealsothankManojBanerjeeforhisobjectiveinput,andMichaelHerrmannforhelpfulconversations.ThisworkissupportedinpartbytheU.S.DepartmentofEnergyundergrantDE-FG0-ER-0.T.D.C.acknowledgesadditionalnancialsupportfromtheNationalScienceFoundationthoughgrantPHY-0.

continuum,innitevolumelimitpriortoextrapolatingtophysicalquarkmasses.Figuresuggeststhatitmaybeextremelydiculttoseetheeectsofvirtualtwo-pionintermediatestatesinfullQCD.Sinceagreatdealoftheintegralstrengthislostforcurrentlatticeregularizationparameters,thecorrectioncurveingureismostlikelyanoptimisticallylargedeviationfromthelinearrelation.EvenwhenfullQCDcalculationsreachthecurrentstateofquenchedcalculations,itisquestionablewhetheronewillbecapableofdiscerningtheeectsofvirtualtwo-pionstatesofthewhiletheisstable.Ofcourse,oncethebecomesunstableanddecaystotwopions,thelowerlyingtwo-pionstateswillneedtobesubtractedfromthecorrelationfunctionpriortoextractingthemass.Thisrendersanexaminationofthe-mesonmassabovethetwo-pioncutnearlyimpossible.ItshouldbementionedthattheseargumentssupporttherecentndingsofRef. []whereanattempttoobservetheeectsofvirtualpionstatesincorrelationfunctionsfailed.Ultimately,fullQCDcalculationswillreachthepointwherelinearlyextrapolatingthemasstophysicalquarkmassesignoringnonanalyticeectswillintroducearelevantsystem-aticerror.TothisendwepresentgurewhichillustratestherelativeamountwhichshouldbeaddedontothemassextractedfromalinearextrapolationoffullQCDdatacorrectedtothecontinuum,innitevolumelimit[].Weremindthereaderthattheseeectsarebeingcalculatedusingthedipoleformforthecutowhichwehavemoreorlessarbitrarilychosen.Thusthecurvesareillustrativeonly.Thex-axisindicatesthepointatwhichthederivativeisdeterminedforthelinearextrapolation.Valuesforareasinguresand.Forexample,iftheeectivephysicalvalueforisGeVandthederivativeisdeterminedFIG..TheamounttobeaddedtothemassextractedfromalinearextrapolationoffullQCDdata.Thex-axisindicatesthepointatwhichthederivativeisdeterminedforthelinearextrapolation.Valuesforareasinguresand.

FIG..andsquaredpionmassesforthethreelightestquarkmassesusedinthequenchedQCDanalysisoftheGFgroup[]ontheirlargestlattice.Thedashedlineillustratesthelinearrelationshipassumedinextrapolatingthehadronmassestothecriticalpoint.Thesolidcurvedisplaystheself-energycorrectionforaregulatormassof=GeVwherethecorrectiontothemassatthephysicalpointisnegligible.Thedot-dashcurvecorrespondsto=GeV.Alatticescaleparameterof.GeVhasbeenappliedtotheotherwisedimensionlesslatticeresults.Figureindicatesthatislessthan%ofthesquaredmassfor

FIG..Therealpartoftheself-energyfordipoleregulatormassestakingthesamevaluesusedingure.Thesensitivityoftheresultstothevalueofisgreatlyreduced.III.APPLICATIONTOQUENCHEDQCDTheeectsofquenchingQCDmaybecategorizedasperturbativeorglobaleectsandnonperturbativeorchannelspeciceects.Asdiscussedintheintroduction,theeectsintheperturbativeregimemaybeaccountedforthroughasimplerenormalizationofthecouplingconstant.However,theeectsinthenonperturbativeregimewill,ofcourse,bechannelspecic.The-mesonchannelmightbeparticularlyvulnerabletothenonperturbativeeectsofquenchingQCDduetothefactthattheisunstableinfullQCDandstableinthequenchedapproximation.Asweshallseehowever,ourestimated\unquenching"correctionsturnouttoberathersmall.FiguredisplaysandsquaredpionmassesforthethreelightestquarkmassesusedinthequenchedQCDanalysisoftheGFgroup[]forthe00latticeat=:forcongurations.Thedashedlineillustratesthelinearrelationshipassumedinextrapolatingthehadronmassestothecriticalpointwherethepionmassvanishes.Providedthelatticespacingisdenedbyphysicalobservableswhichdonothaveasimilardependenceonpiondecays,thephysicsassociatedwithtwo-pionintermediatestatesoftheshouldsimplybeaddedontotheresultsextractedfromcalculationsofquenchedQCD.Thecorrectionisparticulartothemesonandisunlikelytobeaccountedforifthelatticespacingisxedbythenucleonmassforexample.Thesolidlineofgureillustratestheadditionoftheself-energycorrectiontothelinearextrapolationwheretheregulatormass=GeVhasbeenselected[].Thischoiceofisphysicallymotivatedandindicatesthecorrectiontothemassatthephysicalpointisnegligible.Thedot-dashedcurveillustratesthecorrectionwhen=GeV.

FIG..Thetwo-pioninducedself-energy(a)anditsderivativewithrespecttom(b)foraregulatormassof=GeV.Bothreal(solidline)andimaginary(dashedline)partsareillustrated.Thederivativeclearlydisplaysthenonanalyticbehaviorencounteredatm=m.

Thisformmaintainsthenormalizationofgdenedatthephysicalmassandrenderstheintegralof()nite.Ofcourse,thisapproachisnotwithoutafewunphysicalsideeects.Themostobviousproblemistheintroductionofspuriouspolesinthespacelikeregionforthesdependenceofg.However,thedispersionintegralonlysamplesthetimelikeregionandthepresenceoftheseunphysicalpolesshouldnotaecttheresults.Onecouldconsiderotherfunctionalforms.However,theeectivephysicalvaluefortheregulatormass,,isitselfunknown.Ouraimistoestimatetheimportanceofthetwo-pioninducedself-energyrelativetothemass,asopposedtoattemptingtoevaluatetheactualcorrection.Forthisreasonweviewaconsiderationofthedipoleregulatortobeadequate.Evaluationofthedispersionrelationof()with()leadstothefollowingfunctionalformfortherealpartofRe=gq( +mq+m! +q!+ +0m+mq!ln +!

()qln q+q! );whereq= mq!=;and= +m!=:()Theimaginarypartisrecoveredasin().Figureillustratestherealpartoftheself-energyanditsderivativewithrespecttomat=GeV.Thederivativeclearlydisplaysthenonanalyticbehaviorencounteredatm=m.Thesecondderivativeisdiscontinuousatm=mandisinnitefromabove.Theimaginarypartoftheself-energyisalsoillustratedingure.Acomparisonwithgureindicatesthatat=GeVtheresultsarenottoosensitivetothemannerinwhichtheintegralisregulated.Figureillustratestherealpartoftheself-energyforthesamevaluesofusedingure.Thesensitivityoftheresultstothevalueofisgreatlyreducedandallcurvesdisplaythesamequalitativebehavior.Inthelimitof!both()and()reducetoRe=gq(+mq+ln mq!qln q+q!ln q!

);()displayingthelogarithmicdivergenceas!.

FIG..Therealpartoffrom()evaluatedatq=mforavarietyofcutos,ats=.Inthisandthefollowinggures,thenelydashedverticallinemarksthepositionofthephysicalpoint.Forsmallcutos,mostofthestrengthintheintegralliesbelowthemassandasaresultthemassispushedup.q= mq!=;and= m!=:()Figureillustratestherealpartofevaluatedatq=mforavarietyofcutosrangingfromslightlyabovemtoGeV.Forsmallcutos,mostofthestrengthintheintegralliesbelowthemassandconsequentlythemassispushedupduetothemixingwithpionstates.Ofcourse,thisbehavioriscompletelyconsistentwiththatanticipatedbysimplequantummechanicalarguments.Forlargercutosthestrengthabovethemassactstoreducethemass.B.DipoleCutoWhilethe-functionisusefulasanillustrativetool,itsuersfrombeingphysicallyarticialandtheresultscanbeverysensitivetothevalueofthecuto,asillustratedingure.Inanattempttobetterrepresenttheqdependenceofgamonopoleformforeachvertexisintroducedandthedispersionrelationof()isevaluatedwithg!g q+s+! :()

gqqq! :()isgivenbythestandardoneloopintegralsi=Zdk()g<:(qk)(qk)h(qk)m+iihkm+iigkm+i=;:()Physically,theintegralisconvergentduetothemomentumdependenceofg.However,thismomentumdependenceisunknown.Inlightofthisuncertainty,itisreasonabletoparameterizethemomentumdependenceintermsofsomegivenfunctionalformwithanadjustableparametercontrollinghowthefunctionfallsoasafunctionofmomentumtrans-fer.Weconsidertworegulationprescriptions.Inone,weassumeamonopoleformforg,andforcomparison,wealsoconsiderasharp-functioncuto.Thesimplestfashionforintroducingacovariantcutofunctionisthroughtheuseofadispersionrelation.Thesecondtermof()isqindependentandservesonlytosubtractthequadraticdivergenceoftherstterminmaintainingcurrentconservation.Asaresult,wewriteadispersionrelationfor(q)withonesubtractionatq=0,(q)Z0dsqsIm(s)sq:()Ofcourse,theimaginarypartofmaybeeasilydeterminedusinganynumberoftech-niquesforrenderingtheintegralof()nite.TheimaginarypartisIm(q)=gq mq!=qm:()Thevalueofgatq=misxedbyequatingtheimaginarypartsofM0+m+i;(0)atq=m.Thephysicalvalues[]m=:MeVand=:MeVxgat:0.A.-FunctionCutoToillustratethephysicsassociatedwiththerealpartoftheself-energywerstconsidertheintegralof()cutocovariantlybyasharp-functionats=.ThefunctionalformisRe=gq(ln++mqqln q+q!

);()where

bosonssuchasthepion,astheydonotincorporatechiralsymmetry.Whiletherearemodelsofthevertexwhichincorporatechiralsymmetryandchiralsymmetrybreaking[0{],thesemodelsarebasedonparticulardynamicalassumptions.Accordingly,itisdiculttoassessthereliabilityofsuchmodels.Instead,weconsiderarangeofpossibilitiesforthevertexandtosimplifythistask,weconsiderconvenientphenomenologicalforms.Oneotherpaperaddressingthisissue[]sidestepstheproblemssurroundingtheqde-pendenceofgbyxinggtoaconstantandmakingtwosubtractionsofthedivergentintegralatq=0.Thesesubtractionsareabsorbedintoamassandwavefunctionrenormal-ization.However,thisapproachexcludesanyanalysisof-mesonmassextrapolationsasthesubtractiontermsthemselveshaveanunknownmdependencewhichhasbeenlostintherenormalizationprocedure.Moreover,contributionsfromvirtualtwo-pionstateshavebeenabsorbedintothebarelatticeparameterswhichisinconsistentwiththedynamicscontainedinthequenchedapproximation.Theoutlineofthispaperisasfollows.InSectionIIthemodelusedinexaminingthetwo-pioninducedself-energyisoutlined.Twomethodsforregulatingthedivergentself-energyareexplored.InsectionIIItherelevanceoftheself-energycorrectionstoquenchedQCDsimulationsisdiscussed.SectionIVaddressesthequarkmassextrapolationoffullQCDcalculationsandtheimportanceofnonlinearbehaviorinthe-mesonmass.Finally,theimplicationsofthisinvestigationaresummarizedinSectionV.II.THESELF-ENERGYInmodelingthetwo-pioninducedself-energyofthemeson,,thestandardinteractionmotivatedbylow-energycurrentalgebraisused.TheeectiveLagrangeinter-actionhastheform[]Lint=ig$@+g:()ThepionsarefurtherassumedtointeractexclusivelythroughthechannelassummarizedinthefollowingSchwinger-DysonequationforthepropagatorG=G0+G0G;()whereG0=iqM0+i gqqq! ;()inLandaugauge,andM0isthebare-mesonmass.Theself-energyisdenedthroughthesolutionG=iqM0+i gqqq!

;()where

self-energyofthemesoninrelationtolatticeQCDcalculations[].Theirresultsarebasedonastringbreakingquarkmodelandpredictlargecorrectionstosmoothextrapolationsofthe-mesonmassatapproximately0MeV.TheseauthorsweremotivatedbythelongstandingproblemofQCDpredictionsfortheN=-massratiobeingtoolarge.Theirhopewasthetwo-pioninducedself-energycorrectionwouldsucientlyraisethemasstosolvethisproblem.However,wenowunderstandthatboththenitelatticespacingandthenitevolumeofthelatticeacttogethertopushupthismassratio.Currentestimates[]forthisratiocorrectedtotheinnitevolume,continuumlimitare:0:inexcellentagreementwiththeexperimentalvalueof..GeigerandIsgur[]advocateusingthenonlinearitiesinthemassasafunctionofthequarkmasstocorrectforthelinearextrapolationoflatticeresults.WewishtostressthatsuchaprocedureisonlysensiblefortheextrapolationoffullQCDcalculations.Asweshallargue,theentiretwo-pioninducedself-energyisabsentinthequenchedapproximationandshouldbeaddedontoquenchedQCDresultspriortocomparisonwithexperiment.WenotethatforthemodelofRef.[],quenchedlatticecalculationsofthe-mesonmasswouldbereducedby0MeVinsteadofincreasedby0MeV.Thiswouldfurtherexacerbatethe\mN=mproblem"discussedintheirpaper,ratherthatcuringit.Inthequenchedapproximation,themesoncannotdecaytotwolightpseudoscalars.AsdiscussedinRef.[]itisnotpossibletogenerateintermediatestatesofthemesoninthequenchedapproximationinwhichonehastwoisovectorpseudoscalars.Onemightworryaboutthepresenceofthelightisoscalarpseudoscalar0whichfailstoobtainitsheavymassinthequenchedapproximation[{].However,decayofthemesontoa0isforbiddenbyG-parityanddecaytotwo0-mesonsisofcourseforbiddenbytheisospininvarianceofthestronginteractions.Hencethephysicsassociatedwithtwo-pionintermediatestatesoftheshouldsimplybeaddedontotheresultsextractedfromcalculationsofquenchedQCD,providedthelatticespacingisdenedbyphysicalobservableswhichdonothaveasimilardependenceonpiondecays.CurrentcalculationsoffullQCDtypicallyemployquarkmasseswhichplacem>m.Asaresult,thefunctionalformoftheextrapolationfunctionshouldaccountforthenonanalyticbehaviorinthe-mesonmassasthetwo-pioncutisencountered.Onthelattice,thespectraldensitydoesnothaveacutbutratheraseriesofpolesatthepointssatisfyingps=m+pn=;()wherepnarethediscretemomentaallowedonthelattice.Obviously,tofullyaccountforthetwo-pioninducedself-energyofthe,onemustrstextrapolatethelatticeresultstozerolatticespacingandinnitevolumepriortoextrapolatingthequarkmassestophysicalvalues.Inevaluatingtheintegralsdescribingthecouplingofpionstotheonemusttakeintoaccounttheqdependenceofthetotwocouplingconstantgreectingtheinternalstructureofthesemesons.WhilethisqdependencewasextractedfromastringbreakingquarkmodelinRef.[],weelecttotakeamoreagnosticapproachandconsiderdierentmethodsofcuttingotheintegral.Inparticularweinvestigateasharp-functioncutoandadipolecuto.TheunderlyingreasonforouragnosticismisthebeliefthatmodelsofthesortunderlyingRef. []arenotlikelytocorrectlydescribethestructureofpseudo-Goldstone

I.INTRODUCTIONThelatticeregularizedapproachtoquantumeldtheoryprovidesthebestforumfortheexaminationofthefundamentalnonperturbativeaspectsofQCD.Inthelowmomentumtransferregime,itistheonlyapproachwhichintheforeseeablefutureholdsareasonablepromiseofconrmingorrejectingthevalidityofQCDastheunderlyingtheoryofthestronginteractions.Recently,ithasbecomepossibletoperformquenchedQCDcalculationsinwhichallsystematicuncertaintiesarequantitativelyestimated.Thusiftheeectsofquenchingcanbeunderstood,thevalidityofQCDmaybetestedinthenonperturbativeregime.OfparticularnoteistherecentdeterminationoftheQCDcouplingconstant,MS,fromtheSPmasssplittingofcharmonium[].Inthiscaseonebelievesthattheeectsofquenchingmaybeestimatedwithminimalmodeldependence.CorrectionstotheutilizationofthequenchedapproximationofQCDhavebeenestimatedandarecurrentlythedominantsourceofuncertaintyinthenalpredictions[].Inthispaperwewillcontinueeortsalongthislinethroughtheexaminationofsystematicuncertaintiesinhadronmassspectrumcalculations.Inparticular,theimportanceofthetwo-pioninducedself-energycontributiontothe-mesonmassisevaluatedinrelationtothequenchedapproximationofQCDandtofullQCD.Inthequenchedapproximation,thephysicsassociatedwithtwo-pionintermediatestateshasbeenexcludedinthenumericalsimulations.ThisinvestigationismotivatedbyrecentresultsfromtheGFgroup[]forthelow-lyinghadronmassspectruminthequenchedapproximationofQCD.TheiranalysisisthersttosystematicallyextrapolateQCDcalculationstophysicalquarkmass,zerolatticespacingandinnitevolume.Theirpredictionsdisplayanimpressiveagreementwithexperiment.Ofeighthadronmassratios,sixagreewithinonestandarddeviationandtheremainingtworatiosagreewithin:.However,sincethequenchedapproximationleavesoutsomuchimportantphysics,onemightquestionwhethertheseresultsareactuallytoogood[].Intheperturbativeregime,manyoftheeectsofnotincludingdisconnectedquarkloopswhenpreparinganensembleofgaugecongurationsmaybeaccountedforinasimplerenor-malizationofthestrongcouplingconstant.However,onealsoanticipatesnonperturbativeeectsinmakingthequenchedapproximation.Unlikeaglobalrenormalizationofthecou-plingconstant,theseeectsareexpectedtobechannelspecic.Forexample,thequenchedapproximationofQCDleavesoutthephysicsassociatedwiththedecayofthemesontotwopions.Thisphysicsmustbeaccountedforandaddedtothequenchedresultspriortocomparingwithexperimentaldata.Moreover,thecalculatedhadronmassesareextrapo-latedasafunctionofthepionmasssquaredtothepointatwhichthepionmassvanishesusinglinearextrapolationfunctions.Suchanapproachneglectsnonlinearandindeed,non-analyticbehaviorinthecontinuumextrapolationfunction.Forexample,inthecaseofthe-mesonmass,oneexpectsnonanalyticityassociatedwiththeonsetofthetwo-pioncut.Apriorionedoesnotknowtherelativeimportanceoftwo-pionintermediatestatesofthemesonindescribingthemass.Thesubstantialwidthofthemesonat::MeVindicatesitscouplingtopionsisnotsmallandcorrespondinglythesedynamicsmayhavesignicantinuenceonthe-mesonmass.GeigerandIsgurwerethersttostudythepossibleimportanceofthetwo-pioninduced

Submittedto:DOE/ER/0-0Phys.Rev.DU.MDPP#-00UnquenchingthemesonDerekB.LeinweberandThomasD.CohenDepartmentofPhysicsandCenterforTheoreticalPhysicsUniversityofMaryland,CollegePark,MD0July,AbstractTwo-pioninducedself-energycontributionstothe-mesonmassareexam-inedinrelationtothequenchedapproximationofQCD,wherethephysicsassociatedwithtwo-pionintermediatestateshasbeenexcludedfromvector-isovectorcorrelationfunctions.CorrectionstoquenchedQCDcalculationsofthe-mesonmassareestimatedtobesmallattheorderofafewpercentofthe-mesonmass.Thetwo-pioncontributionsdisplaynonanalyticbehaviorasafunctionofthepionmassasthetwo-pioncutisencountered.Theimpli-cationsofthisnonanalyticbehaviorinextrapolationsoffullQCDcalculationsarealsodiscussed.WenotethatforfullQCD,theerrormadeinmakingalinearextrapolationofthemass,neglectingnonanalyticbehavior,increasesasoneapproachesthetwo-pioncut.

---

출처: arXiv:9307.261 • 원문 보기