Baryon spectra with instanton induced forces

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PHYSICAL REVIEW C, VOLUME 64, 055202

Baryon spectra with instanton induced forces

C. Semay and F. BrauUniversitede Mons-Hainaut, Place du Parc, 20, B-7000 Mons, Belgium

B. Silvestre-BracInstitut des Sciences Nucle´aires, IN2P3, CNRS, Universite´ Joseph Fourier, Avenue des Martyrs, 53, F-38026 Grenoble-Cedex, Fran

~Received 8 June 2001; published 5 October 2001!

Except for the vibrational excitations ofK andK* mesons, the main features of spectra of mesons composedof quarksu, d, and s can be quite well described by a semirelativistic potential model including instantoninduced forces. The spectra of baryons composed of the same quarks is studied using the same model. Theresults and the limitations of this approach are described. Some possible improvements are suggested.

DOI: 10.1103/PhysRevC.64.055202 PACS number~s!: 12.39.Pn, 14.20.2c

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I. INTRODUCTION

The QCD based semirelativistic potential model is a scessful approach to describe both meson and baryon speIn most of these works, it is assumed that the quark intetion is dominated by a linear confinement potential, and ta residual interaction stems from the one-gluon exchamechanism. In particular, the spin-spin interaction implby this process is responsible of the nondegeneracy ofp andr mesons. The results obtained with such models are geally in good agreement with experimental data, but the msonsh andh8 cannot be described without adding an apppriate flavor mixing interaction.

Another QCD based candidate exists for the residualteraction: The effective forces computed by ’t Hooft froinstanton effects@1#. It is a pairing force that presents thpeculiarities to act only on quark-antiquark states with zspin and zero angular momentum in the nonrelativistic limIt also generates constituent masses for the light quarkflavor mixing appears naturally with this interaction that halready been used in various models to study light mesNonrelativistic potential model@2#, instantaneous BetheSalpeter formalism@3,4#, flux-tube model@5#, and semirela-tivistic potential approach@6,7#. In all these cases, quite gooresults are obtained.

A first attempt to test this instanton induced forcesbaryons is performed in Ref.@2#. With the same set of parameters, a description of all light mesons and baryonobtained~only the constant potential is changed from mesoto baryons!. The ground states of spectra are quite wellproduced but some meson and baryon excitations aretained too high: Vibrational excitations ofK andK* mesons,h8, f, vibrational excitations ofN, L, etc. Moreover, thismodel presents two serious drawbacks. First, it is a nonrtivistic model. As the velocity of a light quark inside a haron is not small compared with the speed of light, the intpretation of the parameters of the model is questionaSecond, the constituent masses and the coupling constanthe instanton induced forces have been considered hasparameters fitted to reproduce at best meson spectra. Aally, these quantities can be calculated from instanton the

More recently, an instantaneous Bethe-Salpeter thbody formalism has been applied to the study of barywith instanton induced forces@8–10#. In these works, the

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three-body generalization of the Hamiltonian developedRef. @4# is used, but with parameters fitted for the baryothat is to say, different from those found for the mesons. Tconstituent masses and the coupling constants of the inston induced forces are also considered has free parameThe results of these works are compared with our resultthe following.

In a previous work@11#, we have developed a semirelativistic model for mesons including the instanton inducforces, but with parameters calculated, as far as posswith the underlying theory. Very good results have beentained with the condition that the quarks are consideredeffective degrees of freedom with a finite size. In particulall ground states of vector and pseudoscalar mesons arereproduced, generally better than in Ref.@2#. The main flawstems from the usual problem of the vibrational excitatioof K and K* mesons. In this paper, we use this modeldescribe baryons composed ofu, d, ands quarks, in order totest the relevance of the instanton induced forces inframework of a semirelativistic potential model.

Note that for some authors@12# the pion should be treateas a pseudo-Goldstone boson and not as a quark-antiqstate. Nevertheless, in Refs.@2–7,11# devoted to the study omesons with instanton induced forces, the pion can alwbe obtained with a correct mass. We think that this is notchance since, with the resulting pion wave function, itpossible to compute the correct pion charge form factor@13#,reasonable values for the electromagnetic splittings@11#, and~more convincing! correct hadronic decay widths in whicpions are produced@14#. We believe that the instanton induced interaction can simulate processes giving to the pits very low mass, and that it is relevant to fix the parametof the Hamiltonian in order to obtain all pseudoscalar msons.

The main characteristics of the model are recalled in SII. The numerical technique used to compute the barymasses and the fitting procedures are briefly describeSec. III, where various baryon spectra obtained are alsocussed. Some concluding remarks are given in Sec. IV.

II. MODEL

The model we use is the natural generalization to barof the Hamiltonian built for mesons in Ref.@11#. It is worth

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C. SEMAY, F. BRAU, AND B. SILVESTRE-BRAC PHYSICAL REVIEW C64 055202

noting that this Hamiltonian is defined at the lowest ordthat is to say, that no relativistic correction is included in tpotential. Details can be found in Ref.@11#.

The three-quark Hamiltonian is written

H5(i 51

3

ApW i21mi

21 (i , j 51

3

Vi j , ~1!

with pW i the momentum of quarki (( i 513 pW i50W ), mi its con-

stituent mass, andVi j the interaction between quarksi and j.The interaction contains the Cornell potential and the instton interaction. The Cornell potential, which depends onlydistancer between two quarks, is given by

VC~r !51

2 F2k

r1ar1CG , ~2!

where the 1/2 factor takes into account the color contrition. The confining part of this potential represents a goapproximation of theY-shape string configuration.

The instanton induced interaction provides a suitablemalism to reproduce well the spectrum of the pseudoscmesons and to explain the masses of theh andh8 mesons. Inthe nonrelativistic limit, this interaction between two quarin a baryon is written@2,3#

VI~r !524~gP[nn]1g8P[ns] !PS50d~rW !, ~3!

whereg, g8 are two dimensioned constants,PS50 is the pro-jector on spin 0, andP[qq8] is the projector on antisymmetrical flavor stateqq8 (n for u or d is a nonstrange quark, ans is the strange quark!. The operatorP[nn] is simply a pro-jector on isosinglet states, but the operatorP[ns] is not soeasy to implement. Indeed, the instanton interaction istained under the hypothesis of a perfect SU~3! flavor sym-metry@2,3#. So, the baryon wave function is assumed to haa definite spin-flavor symmetry, as in the simple modelRef. @15# used to calculate baryon mass splitting. Withinmore realistic model, the strange quark is much heavier ta n quark, and the wave function cannot have a particuflavor symmetry other that an isospin symmetry for tn quarks. Consequently, the flavor matrix eleme^nsuP[ns] uns& have values in our model that are half the vues in Ref.@15#. To be compatible with this reference, whave placed a supplementary factor 2 in front of the operP[ns] in the computation code. The procedure to handlesame problem in Ref.@2# is not described.

The instanton induced forces also give a contributDmq to the current quark massmq

0 . As this interaction is notnecessarily the only source for the constituent mass, anomenological termdq is also added to the current ma@11#. Finally, the constituent masses in our models are giby

mn5mn01Dmn1dn , ~4!

ms5ms01Dms1ds . ~5!

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In the instanton theory, the quantitiesg, g8, Dmn , Dms aregiven by integrals over the instanton sizer, up to a cutoffvaluerc @see, for instance, Ref.@3#, formulas~5!–~9!#. Theseintegrals can be rewritten in a more interesting form for nmerical calculations by defining a dimensionless instansizex5rL whereL is the QCD scale parameter@11#

g5dp2

2

1

L3 Fms0a11~xc!2

cs

L2a13~xc!G , ~6!

g85dp2

2

1

L3 Fmn0a11~xc!2

cn

L2a13~xc!G , ~7!

Dmn5d1

L Fmn0ms

0a9~xc!2~cnms

01csmn0!

L2a11~xc!

1cncs

L4a13~xc!G , ~8!

Dms5d1

L F ~mn0!2a9~xc!22

cnmn0

L2a11~xc!1

~cn!2

L4a13~xc!G ,

~9!

with

an~xc!5E0

xcdxF9 lnS 1

xD132

9lnH lnS 1

xD J G6

xnF lnS 1

xD G232/9

.

~10!

In these equations, d53.63 102334p2/3 and ci

5(2/3)p2^qiqi& where^qiqi& is the quark condensate for thflavor i. Except the quantityxc5rcL, all parameters in-volved in Eqs.~6!–~9! have expected values from theoreticand/or experimental considerations. The integration in~10! must be carried out until the ratio of the ln term on tlnln term into the integral stays small@2,11#. This ratio in-creases withx from zero atx5x151/e to very large values.At x5x2'0.683 105, the value of this ratio is 1. This lavalue corresponds to the minimum of the instanton den~see Ref.@11#, Fig. 1!. Thus we define the parametere by

rc5xc /L with xc5x11e~x22x1! and eP@0,1#.~11!

In this work e is a pure phenomenological parameter whovalue must be comprized between 0 and 1. With this produre, the value of the cutoff instanton size in our modelcomprized between 0.3 and 0.5 fm, which is a reasonarange for this parameter@2,10#.

The quark masses used in our model are the constitmasses and not the current ones. It is then natural to supthat a quark is not a pure pointlike particle, but an effectdegree of freedom that is dressed by the gluon and quantiquark pair clouds. The form that we retain for the proability density of a quark is a Gaussian function

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BARYON SPECTRA WITH INSTANTON INDUCED FORCES PHYSICAL REVIEW C64 055202

r i~rW !51

~g iAp!3/2exp~2r 2/g i

2!. ~12!

It is generally assumed that the quark sizeg i depends on theflavor. So, we consider two size parametersgn andgs for nand s quarks, respectively. It is assumed that the dresexpressionOi j (rW) of a bare operatorOi j (rW), which dependsonly on the relative distancerW5rW i2rW j between the quarksqiandqj , is given by

Oi j ~rW !5E drW8Oi j ~rW8!r i j ~rW2rW8!, ~13!

wherer i j is also a Gaussian function of type~12! with thesize parameterg i j given by

g i j 5Ag i21g j

2. ~14!

This formula is chosen because the convolution of tGaussian functions, with size parametersg i andg j , respec-tively, is also a Gaussian function with a size paramegiven by Eq.~14!.

After convolution with the quark density, the Cornedressed potential has the following form:

VC~r !52kerf~r /g i j !

r1arFg i j exp~2r 2/g i j

2 !

Apr

1S 11g i j

2

2r 2D erf~r /g i j !G1C, ~15!

while the Dirac distribution inVI(r ) is transformed into aGaussian function

d~rW !51

~g i jAp!3exp~2r 2/g i j

2 !. ~16!

Despite this convolution, we consider, for simplicity, that tinstanton induced forces act always only onL50 states.Note that the strange size quark can be vanishing provthe nonstrange quark size is nonzero. Indeed,gs50 withgnÞ0 yields gnnÞ0, gnsÞ0; only gss50. This last valuecould pose a problem only in expression~16!. But this situ-ation never happens since the instanton interactionVI is van-ishing for asspair @see Eq.~3!#. For mesons, the situation ia little bit more complicated and it is discussed in Ref.@11#.

III. NUMERICAL RESULTS

A. Numerical technique

The eigenvalue equation is solved by developing the wfunctions in trial states built with harmonic oscillator statunlm&. In such a basis, the two-body matrix elements ofpotential are expressed in terms of the following quantiti

^n8lmuV~r !unlm&5 (p5 l

l 1n1n8

B~n8,l ,n,l ,p!I p , ~17!

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with

I p52

G~p13/2!E

0

`

dx x2p12 exp~2x2!V~bx!. ~18!

The quantitiesI p are the Talmi’s integrals, which depend oa nonlinear parameterb, the oscillator length. The coefficientsB(n8,l ,n,l ,p) are geometric factors@16# that can becalculated once for all. To accelerate the convergence,use two oscillator lengthsb, b8 in our basis. These two quantities are the scale parameters of the two internal radialtances that can be defined in a baryon. This method, whhas originally been developed in Ref.@17# for nonrelativistickinematics, works very well for relativistic kinematics, asis shown in Ref.@18#. The details of the technique usedcalculate the matrix elements of the relativistic kinetic eergy operator can be also found in Ref.@18#.

B. Fitting procedure

The purpose of this work is to extend to the baryonsresults obtained for the meson spectra in Ref.@11#, and thustry to obtain a satisfactory description of baryon spectra wa quite simple model. Our approach is indeed very simsince we use only a spinless Salpeter equation supplemeby a pure central potential and the nonrelativistic limit ofinstanton induced interaction, that is to say, that the potenis completely defined at the zero order of quark speed. Tis sufficient to describe the bulk properties of mesons abaryons. Note that the instanton induced interaction is estial to describe pseudoscalar mesons and the baryon grostate properties, such as theN-D splitting.

We need 13 parameters to obtain a satisfactory spectfor the mesons, and to be consistent, we keep the same sparameters for the baryons. This number could appear labut some are strongly constrained by theoretical or expmental considerations, while other are unavoidable~see dis-cussion in Ref.@11#!. The instanton interaction is defined bsix parameters: the current quark masses, the quark consates for the flavorsn ands, the QCD scale parameterL, andthe maximum sizerc (xc /L) of the instanton. Four otheparameters are introduced: the effective sizes of the quarnand s, and two termsdn and ds , which contribute to theconstituent quarks masses. It is worth mentioning thamong the above parameters,mn

0 , ms0 , L, cn , cs , dn , ds ,

and e are intermediate quantities used to compute the fparametersmn , ms , g, and g8 that enter directly into theHamiltonian. Three unavoidable parameters are also usedthe central part of the potential: The slope of the confinema—for which reliable estimations exist—the strengthk forthe Coulomb-like part, and the constantC that renormalizesthe masses. Consequently, we can say that only six quanare really free parameters~see Table I!.

To find the value of the parameters, we have minimizex2 function based on the masses of 11 well-known bary~see Table II!

x25(i

FMith2Mi

exp

DMiexp G 2

, ~19!

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TABLE I. List of parameters of the model. The column ‘‘Baryon’’ contains the values for the barspectra presented in Figs. 1–5. The column ‘‘Meson’’ contains the values for the meson model I of Re@11#.When available, the expected value of a parameter is also given in the column ‘‘Exp.’’ The valuesquantitiesmn , ms , g, andg8 computed with these parameters are also indicated.

Parameters Unit Baryon Meson Exp.

mn0 GeV 0.001 0.015 0.001–0.009@24#

ms0 GeV 0.103 0.215 0.075–0.170@24#

L GeV 0.238 0.245 0.20820.02310.025 @24#

^nn& GeV3 2~0.247!3 2~0.243!3 (20.22560.025)3 @25#

^ss&/^nn& 0.631 0.706 0.860.1 @25#

e 0.061 0.031 0–1@11#

a GeV2 0.168 0.212 0.2060.03 @26#

k 0.798 0.440C GeV 20.967 20.666gn GeV21 0.681 0.736gs GeV21 0.005 0.515dn GeV 0.327 0.120ds GeV 0.490 0.173mn GeV 0.378 0.192ms GeV 0.638 0.420g GeV22 2.498 2.743g8 GeV22 2.234 1.571

lf.n

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etersare

andn-

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rtain-

where the quantityDMiexp is the error on the experimenta

masses~it is fixed at the minimum value of 10 MeV, see Re@11#!. To perform the minimization, we use the most receversion of theMINUIT code from the CERN library@19#.

C. Baryon spectra

We first compute baryon spectra with the parametfound in our previous paper for meson spectra@11#, but theresults obtained are not very good. For instance, the roresonance is found 576 MeV above the nucleon—whichnot so bad—but theN-D mass difference computed is 21MeV, which is much too small. This mass difference is geerally considered as a minimum requirement to be repduced for a baryon model.

TABLE II. Quantum numbers and masses~the minimal uncer-tainty is fixed at 10 MeV, see Ref.@11#! of the baryons used in theminimization procedure to find the parameters listed in Table I.

Baryon I JP Masses~GeV!

N 12

12

1 0.93960.010N(1440) 1

212

1 1.45060.020D 3

232

1 1.23260.010N(1535) 1

212

2 1.53760.018L 0 1

21 1.11660.010

S 1 12

1 1.19360.010S* 1 3

21 1.38560.010

J 12

12

1 1.31560.010J* 1

232

1 1.53060.010V 0 3

21 1.67260.010

D(1600) 32

32

1 1.62560.075

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In a second step, we have searched for a set of paramto describe both baryon and meson spectra. All sets foundvery similar and present more or less the same qualitiesthe same flaws~best results are obtained with the supplemetary factor 2 in front of the operatorP[ns] ). For instance, inone of the best sets of parameters found, a goodN-D massdifference is obtained~280 MeV!, but the Roper resonancmass is then calculated around 150 MeV above its expmental value. Moreover, if the meson spectra obtained dodiffer significantly from the ones found in our previous pap@11#, two states are then very badly described with respecthe others: Theh8-meson is found 36 MeV above its exper

FIG. 1. Energy levels of the nucleon states~status!!!! and!!!) as a function of total angular momentum and parityJP. Theshaded boxes represent the experimental values with the unceties.

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mental value and the3DJ states are computed 43 MeV blow their experimental value. Despite a great numberminimizations, we never succeeded to find an ‘‘acceptabset of parameters to describe satisfactorily both mesonthe baryon sectors.

In order to test the relevance of our model for baryons,have then searched for parameters to describe baryonsOne of the best set of baryon spectra that we have foungiven in Figs. 1–5. The spectra present some characterithat can be found in several other works, in particular R@20#; only few states are not so well reproduced in our woFor example, the mass of the Roper resonance is arounMeV above its experimental value . The nucleon states wnegative parity have masses that are slightly too small.Roper of theD has a too high mass. TheL(1405) cannot bedescribed, as this is often the case. Even if our spectraclearly less good than the spectra found in Ref.@20#, theypresent many similar qualitative characteristics. But in geral, the agreement between calculated masses and exmental data is less good in our model. It is worth noting tall the baryon ground states can be well reproduced. Wthese parameters fitted to the baryons, the mesons massvery poorly obtained: For instance, if the computed pi

FIG. 2. Same as Fig. 1 but for theD states.

FIG. 3. Same as Fig. 1 but for theL states.

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mass is good~138 MeV!, the mass of ther meson is found260 MeV above its experimental value. The impossibilityobtain good meson and baryon spectra with the same paeters is also a characteristic of the model of Ref.@20#.

It is also interesting to compare our spectra with thoobtained in Refs.@8–10#. The model developed in thesworks and our model are similar in the sense that the inston induced interaction is the only spin-isospin-dependpart of the Hamiltonian, but the model of Refs.@8–10# dif-fers from ours by two main points:~i! the use of a spinlessSalpeter equation in our model instead of an instantaneBethe-Salpeter equation,~ii ! the presence of a Coulomb-likinteraction in our model. Below 2 GeV, spectra of both moels are very similar. They share more or less the same quties and the same flaws. The ground states are well reduced, but the Roper resonance and the firstJP51/22 stateare inverted. Again theL(1405) cannot be described. Wcan just note a slight improvement for otherL baryons. Thebaryon Regge trajectories are nevertheless better describthe model of Refs.@8–10#. This is an indication of the betterelevance of an instantaneous Bethe-Salpeter equation osimpler spinless Salpeter equation. Note that our modecharacterized by smaller values of parametersg andg8. This

FIG. 5. Same as Fig. 1 but for theJ andV states.

FIG. 4. Same as Fig. 1 but for theS states.

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is due to the fact that the instanton interaction can be weain our model since we include in our Hamiltonian an attrative Coulomb-like interaction.

As one can see from Table I, the values of the first separameters are rather satisfactory (e is expected to be neazero, see Ref.@11#!, while one can see that the Coulomb-likparameterk is rather strong and that the size of thes quark isalmost zero. So small a value for the strange quark size cappear troublesome but this does not cause any numedifficulties, as mentioned above. Good meson and barspectra can be computed withgs around 0.5gn and reason-able values for the parametersms

0 and ds . But the betterbaryon spectra are obtained with small values ofgs . As it isexpected we foundg8,g @9#. Note that when the parameteare fitted only to baryons, the factor 2 in front of the operaP[ns] can be simulated by a redefinition of the parameterg8.But in this case, we haveg8.g. It is also clear from thistable that some of the best parameters for baryons are dent from the corresponding best parameters for mesonparticular quantities related to the quark masses. Moreothe value of the parameterk is higher for baryon, while thevalue of the confinement slopea is lower. With so manydifferences in these two sets of parameters, it is not surping that mesons and baryons cannot be well reproducedgether. Some physics is clearly missing. We discusspoint in the next section.

IV. CONCLUDING REMARKS

Several works have been devoted to the study of thestanton interaction in the framework of semirelativisticrelativistic models for mesons@3–10#. A few of these modelshave been applied to baryons. Our purpose here was to cpute baryon spectra with the semirelativistic instanton bamodel for mesons we have developed in Ref.@11# and withthe same underlying fundamental ingredients. Whenmodel is directly applied to baryons, the spectra obtainednot good. It is necessary to change all the parametercompute a more relevant spectra. The natural link betwmeson and baryon is then broken, and the baryon speobtained are not very different from those yielded by mod~i! with similar complexity but based on one-gluon exchanprocess~see, for instance, Ref.@21#!, ~ii ! relying on covariant

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equation with only an instanton induced interaction suppmenting the confinement@8–10#.

Our semirelativistic model Hamiltonian contains a potetial part written in the lowest order, that is to say, thatrelativistic correction is included. In particular, the spin-spterm—responsible of the low pion mass in most potenmodels ~see, for instance, Ref.@22#!—is not present hereThe instanton induced forces are assumed to take intocount all spin effects. This is probably an approximation tcrude. Both interactions, instanton induced one and spin-term, have very similar contributions for nonmixed flavmesons. Nevertheless, to include the two interactions iHamiltonian means a complete new fitting of parametersparticular the parameterk that measures the strength of thCoulomb-like potential and of the spin-spin interaction. Thcould modify appreciably the spectra of baryons. It is nsure that the inclusion of relativistic corrections in the modcould cure all its defects. In particular, the relative positiof positive and negative parity excitations of the nucleon iproblem for all models based on the one-gluon exchadominance. This puzzle is solved with the meson exchapotential proposed more recently@20#. Within this model, thequarks interact by exchanging pseudoscalar mesons, cpletely ruling out the one-gluon exchange process. Dessome serious critics@23#, one is forced to ascertain that spetra of light baryons are remarkably improved. It is thus posible that the meson exchange process could be one okeys to explaining baryon spectra and could supplementstanton induced interaction. Such a study is in progress.us note that the Hamiltonian described in Ref.@20# cannotreproduce meson and baryon spectra with the same separameters. It thus suffers the same drawback as our mo

This work clearly shows that the instanton induced forccannot explain alone both meson and baryon spectra. Cobutions coming from one-gluon and meson exchange pcesses are probably necessary. Nevertheless, as the ’t Hinteraction solves naturally thep-K-h-h8 problem withoutany additional assumptions, it must certainly be an unavoable ingredient of potential models.

ACKNOWLEDGMENTS

C.S. would thank the F.N.R.S. for financial support. F.would thank the I.I.S.N. for financial support.

. J.

l.

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ith

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to

r-


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Baryon spectra with instanton induced forces

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