A convincing, uncontroversial observation of string breaking, when the static
potential is extracted from Wilson loops only, is still missing. This failure
can be understood if the overlap of the Wilson loop with the broken string is
exponentially small. In that case, the broken string ground state will only be
seen if the Wilson loop is long enough. Our preliminary results show string
breaking in the context of the 3d SU(2) adjoint static potential, using the
L\"uscher-Weisz exponential variance reduction approach. As a by-product, we
measure the fundamental SU(2) static potential with improved accuracy and see
clear deviations from Casimir scaling.Comment: Lattice2002(topology), AMS-LaTeX v1.2, 3 pages with 2 figures; added
  reference

Aoki

Bali

Bernard

Bolder

de Forcrand

DeTar

Drummond

Gliozzi

Kallio

Knechtli

Laermann

Luscher

Majumdar

Parisi

Pennanen

Philippe de Forcrand

Philipsen

Poulis

Slavo Kratochvila

Stephenson

English

arXiv

Crossref

String breaking with Wilson loops?

A convincing, uncontroversial observation of string breaking, when the static potential is extracted from Wilson loops only, is still missing. This failure can be understood if the overlap of the Wilson loop with the broken string is exponentially small. In that case, the broken string ground state will only be seen if the Wilson loop is long enough. Our preliminary results show string breaking in the context of the 3d SU(2) adjoint static potential, using the L\"uscher-Weisz exponential variance reduction approach. As a by-product, we measure the fundamental SU(2) static potential with improved accuracy and see clear deviations from Casimir scaling

Kratochvila, S

De Forcrand, Philippe

CERN Document Server

1String breaking with Wilson loops?Slavo KratochvilaaETHZ]Institute for Theoretical Physics, ETH Zu¨rich, CH-8093 Zu¨rich, Switzerlandand Philippe de Forcrandb[ETHZ]cCERN]Theory Division, CERN, CH-1211 Geneva 23, Switzerlanda[c[A convincing, uncontroversial observation of string breaking, when the static potential is extracted from Wilsonloops only, is still missing. This failure can be understood if the overlap of the Wilson loop with the broken stringis exponentially small. In that case, the broken string ground state will only be seen if the Wilson loop is longenough. Our preliminary results show string breaking in the context of the 3d SU(2) adjoint static potential,using the Lu¨scher-Weisz exponential variance reduction approach. As a by-product, we measure the fundamentalSU(2) static potential with improved accuracy and see clear deviations from Casimir scaling.1. INTRODUCTIONThe breaking of a long flux tube between twoquarks into a matter-antimatter pair is one of themost fundamental phenomena in QCD. Three ap-proaches have been used to measure the static po-tential: (i) Correlation of Polyakov Loops, at -nite T [1]. (ii) Variational Ansatz using two typesof operators: One for the string-like state and onefor the broken string state [2,3]. (iii) Wilson loops[4,5].String breaking has been seen using the rsttwo methods, but no clear signal using the thirdone has been observed. The failure of the Wil-son loop method seems to be mainly due to thepoor overlap of the operators with the brokenstate. The strong coupling model [6] uses theheavy quark expansion to show that there is anexponentially small overlap. Based on a topo-logical argument, [7] suggests that there may beno overlap at all. We show that there is a smalloverlap but not nearly as small as predicted.2. STATIC POTENTIALThe static potential helps to characterize con-ning forces. Assume that a static charge anda static anticharge are separated by a distance∗Talk presented by S. KratochvilaR < Rb, where Rb is the string breaking dis-tance. If R is increased to a value larger than Rb,a matter-antimatter pair will be created whichscreens the static charges. A further increase ofR has no eect on the static potential - it remains constant, this indicates string breaking. In ourcase, we deal with adjoint static charges which arescreened by the gluons of the gauge eld. The ob-ject adjoint charge-gluon is called a gluelump.In a Hamiltonian formulation, a static chargeand a static anticharge are created at Euclideantime 0, forming a state jφi, and annihilated attime T . This can be expressed as the Wilson loopW (R, T ):hφje−TH jφi = W (R, T ) (1)On the other hand, one can expand the left sidein (1) in the eigenbasis Ψ(n) of the Hamiltonian.At large T , only the ground state survives:hφje−TH jφi =1∑n=0jhΨ(n)jφij2e−TEnT!1jhΨ(0)jφij2e−TV (R) (2)By using (1) and (2), the static potential V (R),which is dened as the energy of the ground stateE0, can be connected with the Wilson loop:V (R) = − limT!11Tlog W (R, T ) (3)2This holds as long as the overlap hφjΨ(0)i doesnot vanish.In section 4, we t the energy of the unbrokencase using the Ansatz:V (R)  V0− cR+σR (4)where σ is the string tension.3. METHODWe are using a 3d-lattice with extent 482  64at β = 4ag2 = 6.0. The gauge group is SU(2).We measure adjoint static charges in order toavoid costly dynamical quarks. The price to payis that the signal decays much faster becauseVadj  83Vfund (Casimir scaling).We make use of Lu¨scher-Weisz exponentialvariance reduction [8]. This method general-izes Multihit [9] from single links to link-link-correlators T(na). The possibility to use a hier-archical scheme allows us to measure long Wilsonloops up to the desired accuracy. Using the no-tation introduced in [8], we actually implementthe following scheme:W (R, T ) = L(0)γf  [[[T(na)][T((n+1)a)]][[T((n+2)a)][T((n+3)a)]]]   gγL(T )(5)The parameters are the following: The inner-most average [T(na)] was calculated from 10sets of time-like links, each obtained after 10updates (1HB/4OR). The second-level average[[T(na)][T((n + 1)a)]]was calculated from 160averages of [T(na)] separated by 200 updates ofthe spatial links on the time-slice [(n + 1)a]. Fi-nally, the third-level average was calculated from39 second-level averages separated by 200 updatesof the spatial links on the time-slice [(n + 2)a].To reduce contributions from excited statesΨ(n6=0), we replace the simple spatial transportersL(0)γ and L(T ) with "staples", which areconstructed in the following way: (i) After eachcalculation of second level averages, , we formthe smeared adjoint spatial links, , at thetime-slice (n + 2)a. (ii) We multiply them withthe second level averages to obtain the staples, . (iii) This procedure is repeated 39 timesto provide an error reduction in the new spatialtransporters.After processing 40 congurations as above, weobtain the following preliminary results.4. RESULTSWe measure both the adjoint and the funda-mental static potential (Fig. 1). The spatial sep-aration R starts at 2 and is increased up to 12. Inthe adjoint case, the Ansatz (4) works well for theenergy V 0(R) of the unbroken string in the wholerange of R. However, at R  10 the minimalenergy remains constant, V (R  10) = 1.95(10).Therefore, we observe string breaking at a dis-tance Rb  10.The dotted line at 2.06(1) represents twice theenergy of a gluelump using a direct measurement[4] on our lattice. This value is close to thegroundstate potential for R > Rb (See Fig.1).Both Rb  10 [3], as the string breaking distance,and 2E(Qg) [3,4], as twice the gluelump-energy,are in agreement with the literature.In the fundamental case, the Ansatz (4) againworks very well. An important issue is the ful-lment of the Casimir scaling law. At R . 4,VadjVfund 83 holds as required by perturbation the-ory. However at large distances, the slopes of thepotentials are in the ratio 2.31(1) < 83 .0.60.811.21.41.61.822.22.42.62 4 6 8 10 12V(R)RADJPOTENSCFUPOTEEQgLABELVPRLABELFig. 1. The adjoint and 83fundamental static potentialsV (R) vs R. The horizontal line at 2.06(1) represents twice3the energy of a gluelump.The value of the static potential V (R) at axed R, e.g. R = 12, is determined by theslope of the logarithm of the Wilson loop (3). InFig. 2 it can be seen that a single exponentialt is not sucient in the adjoint case. At smalltemporal extent T we get a larger slope than atlarge T . This can be explained as follows: Atsmall T the signal is dominated by the unbrokenstring state. The broken string state can onlybe observed after T is large. The way to t alldata points is to use the double-exponential t c0e−V ′(R)T + c1e−2E(Qg¯)T . V 0(R) correspondsto the energy of the unbroken string, E(Qg) to theenergy of the gluelump. The ratio of the ampli-tudes is c0c1  104, the turning point is at Tt  20.1e-351e-301e-251e-201e-151e-101e-0510 5 10 15 20 25 30 35W(R,T)Tc0c1turpointbroken stringunbroken string double exp fit8 to 20 fit20 to 28 fitFig. 2. Adjoint Wilson loop W (R, T ) vs T , for R = 12.The double-exponential fit matches all data points. Single-exponential fits (dotted lines) do not. The unbroken stringstate can be fitted by using data from T = 8 up to 20, thebroken string state by using data from T = 20 up to 28.Note how small a signal can be measured.5. CONCLUSIONIn the context of the 3d SU(2) adjoint staticpotential, string breaking has been observed usingWilson loops only.It is a necessary condition that both R and Thave to be larger than the string breaking dis-tance Rb. Note, however, that the variational ap-proach allows to bypass this requirement.The strong coupling model [6] is too pes-simistic. A necessary condition to be able to mea-sure the broken state is that the signal shouldbe larger than the one from the unbroken state:c1e−2MQgT  c0e−V (R)T . The heavy quark ex-pansion results in the ratio c0c1  eMQgR, whichwould imply T  43 at R = 12, whereas in oursimulation the turning point was Tt  20.A similar observation using Wilson loops in fullQCD should be dicult but feasible [10].REFERENCES1. E. Laermann, C. DeTar, O. Kaczmarek andF. Karsch, Nucl. Phys. Proc. Suppl. 73 (1999)447; C. DeTar, U. M. Heller and P. Lacock,Nucl. Phys. Proc. Suppl. 83 (2000) 310;2. O. Philipsen and H. Wittig, Phys. Rev.Lett. 81 (1998) 4056 [Erratum-ibid. 83(1999) 2684]; F. Knechtli and R. Sommer[ALPHA collaboration], Phys. Lett. B 440(1998) 345; O. Philipsen and H. Wittig,Phys. Lett. B 451 (1999) 146; P. de For-crand and O. Philipsen, Phys. Lett. B 475(2000) 280; K. Kallio and H. D. Trot-tier, arXiv:hep-lat/0001020; P. Pennanenand C. Michael [UKQCD Collaboration],arXiv:hep-lat/0001015; C. W. Bernard etal. [MILC Collaboration], Nucl. Phys. Proc.Suppl. 94 (2001) 546.3. P. W. Stephenson, Nucl. Phys. B 550 (1999)427.4. G. I. Poulis and H. D. Trottier, Phys. Lett. B400 (1997) 358.5. S. Aoki et al. [CP-PACS Collaboration], Nucl.Phys. Proc. Suppl. 73 (1999) 216.6. I. T. Drummond, Phys. Lett. B 434 (1998)92.7. F. Gliozzi and P. Provero, Nucl. Phys. Proc.Suppl. 83 (2000) 461.8. M. Luscher and P. Weisz, JHEP 0109 (2001)010; P. Majumdar, arXiv:hep-lat/0208068.9. G. Parisi, R. Petronzio and F. Rapuano,Phys. Lett. B 128 (1983) 418.10. H. D. Trottier and K. Y. Wong, arXiv:hep-lat/0209048, these proceedings.

String breaking with Wilson loops?

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