We perform a detailed study of the adjoint static potential in the
pseudoparticle approach, which is a model for SU(2) Yang-Mills theory. We find
agreement with the Casimir scaling hypothesis and there is clear evidence for
string breaking. At the same time the potential in the fundamental
representation is linear for large separations. Our results are in qualitative
agreement with results from lattice computations.Comment: 5 pages, 2 figures, talk given at "Quark Confinement and the Hadron
  Spectrum VIII", September 1 - 6 2008, Mainz, German

Szasz, Christian

Wagner, Marc

English

arXiv

We perform a detailed study of the adjoint static potential in the pseudoparticle approach, which is a model for SU(2) Yang-Mills theory. We find agreement with the Casimir scaling hypothesis and there is clear evidence for string breaking. At the same time the potential in the fundamental representation is linear for large separations. Our results are in qualitative agreement with results from lattice computations

Hochschulschriftenserver - Universität Frankfurt am Main

PoS(Confinement8)059The adjoint potential in the pseudoparticleapproach: string breaking and Casimir scalingChristian SzaszUniversity of Erlangen-Nürnberg, Institute for Theoretical Physics III, Staudtstraße 7,91058 Erlangen, GermanyE-mail: christian.szasz@theorie3.physik.uni-erlangen.deMarc Wagner∗Humboldt University Berlin, Department of Physics, Newtonstraße 15, 12489 Berlin, GermanyE-mail: mcwagner@physik.hu-berlin.deWe perform a detailed study of the adjoint static potential in the pseudoparticle approach, whichis a model for SU(2) Yang-Mills theory. We ﬁnd agreement with the Casimir scaling hypothesisand there is clear evidence for string breaking. At the same time the potential in the fundamentalrepresentation is linear for large separations. Our results are in qualitative agreement with resultsfrom lattice computations.8th Conference Quark Conﬁnement and the Hadron SpectrumSeptember 1-6, 2008Mainz, Germany∗Speaker.c ￿ Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/PoS(Confinement8)059The adjoint potential in the pseudoparticle approach: string breaking and Casimir scaling Marc Wagner1. IntroductionA common approach to gain a better understanding of Yang-Mills theory, in particular themechanism of conﬁnement, is to restrict the full path integral to a small subset of gauge ﬁeldconﬁgurations, which are supposed to be of physical importance. Examples are instanton gas andliquid models (cf. [1] and references therein), ensembles of regular gauge instantons and merons[2, 3, 4], the pseudoparticle approach [5, 6, 7, 8], calorons with non-trivial holonomy [9, 10], andmodels based on center vortices (cf. e.g. [11, 12, 13, 14, 15]).In this paper we apply the pseudoparticle approach to SU(2) Yang-Mills theory and perform adetailed study of the static potential for various representations.2. The pseudoparticle approach in SU(2) Yang-Mills theoryThe basic idea of the pseudoparticle approach is to approximate the Yang-Mills path integralDOE=1ZZDAO[A]e−S[A] , S[A] =14g2Zd4xFamnFamn, (2.1)where Famn = ¶mAan −¶nAam +eabcAbmAcn, with a small number of physically relevant degrees offreedom. To this end, the integration over all gauge ﬁeld conﬁgurations in (2.1) is restricted to asmall subset, which can be written as a linear superposition of a ﬁxed number of pseudoparticles1:Aam(x) = åjA (j)C ab(j)abm(x−z(j)), (2.2)where j is the pseudoparticle index and A (j) ∈ R, C ab(j) ∈ SO(3) and z(j) ∈ R4 are the ampli-tude, the color orientation and the position of the j-th pseudoparticle respectively. The functionalintegration over all gauge ﬁeld conﬁgurations is deﬁned as the integration over pseudoparticle am-plitudes and color orientations:ZDA ... =Z  ÕjdA (j)dC(j)!... (2.3)For the results presented in this work we have used 625 “long range pseudoparticles”, whichfall off as 1/distance, inside a hypercubic spacetime region (for details regarding this setup cf. [8]):aam,inst.(x) =hamnxnx2+l2 , aam,antiinst.(x) =¯ hamnxnx2+l2 , aam,akyron(x) =da1xmx2+l2. (2.4)The ﬁrst two types generate transverse gauge ﬁeld components and are similar to regular gaugeinstantons and antiinstantons, while the third type, the so-called akyron [6], is responsible forlongitudinal gauge ﬁeld components. We would like to stress that gauge ﬁeld conﬁgurations (2.2)are in general not even close to solutions of the classical Yang-Mills equations of motion, i.e. thepseudoparticle approach is not a semiclassical model. The idea is rather to approximate physicallyrelevant gauge ﬁeld conﬁgurations by a small number of degrees of freedom.1In this paper the term pseudoparticle refers to any gauge ﬁeld conﬁguration aam, which is localized in space and intime, not only to solutions of the classical Yang-Mills equations of motion.2PoS(Confinement8)059The adjoint potential in the pseudoparticle approach: string breaking and Casimir scaling Marc Wagner3. Casimir scaling and adjoint string breakingIn the following the potential associated with a pair of static color charges f(J) and (f(J))† inspin-J-representation at separation R is denoted by V(J)(R). In pure Yang-Mills theory there is nostring breaking, when the charges are in the fundamental representation (J = 1/2). For charges inthe adjoint representation (J = 1) the situation is different: gluons are able to screen such chargesand the connecting gauge string is expected to break, when the charges are separated adiabaticallybeyond a certain distance; a pair of essentially non-interacting gluelumps is formed.The starting point to extract the static potential in spin-J-representation are “string trial states”S(J)(x,y)|W  = (f(J)(x))†U(J)(x;y)f(J)(y)|W  , |x−y| = R, (3.1)where U(J) denotes a spatial parallel transporter. We compute temporal correlation functionsC(J)string(T) =  W|￿S(J)(x,y,T)￿†S(J)(x,y,0)|W  µDW(J)(R,T)E(3.2)and determine the corresponding potential values from their exponential fall-off (for details cf. [8]).The numerical result for the fundamental potential is shown in Figure 1a (here and in the fol-lowing we have used the value g =12.5 for the coupling constant). It is linear for large separations,i.e. there is conﬁnement. We set the physical scale by ﬁtting V(1/2)(R) =V0+sR and by identify-ing the string tension s with sphysical = 4.2/fm2. This amounts to a spacetime region of extensionL4 = (1.85fm)4.Numerical results for higher representation potentials (J = 1,...,5/2) are shown in Figure 1b.According to the Casimir scaling hypothesis these potentials are supposed to fulﬁllV(1/2)(R) ≈V(1)(R)8/3≈V(3/2)(R)5≈V(2)(R)8≈V(5/2)(R)35/3(3.3)for intermediate separations. Figure 1c shows that this is the case for the adjoint potential, whilethere are slight deviations for J ≥ 3/2. This is in agreement with what has been observed in 4dSU(2) lattice gauge theory [16]. 0 0.5 1 1.5 2 2.5 0  0.5  1  1.5  2  2.5  3  3.5  4  4.5V(1/2)Ra)   V(1/2) as a function of Rfitting range 0 500 1000 1500 2000 2500 3000 0  0.2  0.4  0.6  0.8  1  1.2  1.4  1.6V(J) in MeVR in fmb)   V(J) as a function of R for different JJ = 5/2...J = 1J = 1/2 0 2 4 6 8 10 12 0  0.2  0.4  0.6  0.8  1  1.2  1.4  1.6V(J) / V(1/2)R in fmc)   V(J) / V(1/2) as a function of R for different JJ=5/2J=2J=3/2J=1Figure 1: a) The fundamentalstatic potentialV(1/2) as a function of the separation R. b) “Pure Wilson loopstatic potentials”V(J) for different representations as functions of the separation R. c) Ratios V(J)/V(1/2) asfunctions of the separation R compared to the Casimir scaling expectation.3PoS(Confinement8)059The adjoint potential in the pseudoparticle approach: string breaking and Casimir scaling Marc WagnerNote that there is no sign of string breaking for the adjoint potential even for separationsR>∼1.6fm. This is, because string trial states have poor overlap to the ground state, which is ex-pected to resemble a two gluelump state. The solution to overcome this problem is to use a wholeset of trial states containing not only string trial states (3.1), but also “two-gluelump trial states”åj=x,y,zGj(x)Gj(y)|W  , Gj(x) = Tr￿f(1)(x)Bj(x)￿, |x−y| = R. (3.4)We extract the adjoint potential from the corresponding correlation matrices by solving a general-ized eigenvalue problem and by computing effective masses (for details cf. [8]). Results are shownin Figure 2a. The potential saturates at around two times the magnetic gluelump mass (which is≈ 1000MeV at g = 12.5 in this regularization [8]) at separation Rsb ≈ 1.0fm. This string breakingdistance as well as the observed level ordering (the ﬁrst excited state is an excited string state forsmall separations, then becomes a two gluelump state and ﬁnally a string state again, etc.) is inagreement with results from lattice computations [17, 18].To investigate, whether the gluonic string really breaks, when two static charges are separatedadiabatically, we perform a mixing analysis. During the computation of effective masses we obtainapproximations of the ground state and the ﬁrst excited state,|0  ≈ a0string|string +a02g-lump|2g-lump  , |1  ≈ a1string|string +a12g-lump|2g-lump ,(3.5)where |string  and |2g-lump  are normalized trial states. The overlaps |aj...|2 are shown as functionsof the separation in Figure 2b and 2c. The transition between string and two-gluelump states israpid but smooth indicating that string breaking is present in the pseudoparticle approach.4. Conclusions and outlookWe have computed static potentials for various representations within the pseudoparticle ap-proach. While the fundamental static potential is linear for large separations, we clearly observestring breaking for the adjoint representation. Both the string breaking distance Rsb ≈ 1.0fm andthe level ordering are in agreement with lattice results, and a mixing analysis indicates a rapid, butsmooth transition between a string and a two gluelump state, when two static charges are separated 0 500 1000 1500 2000 2500 3000 0  0.2  0.4  0.6  0.8  1  1.2  1.4  1.6V(1) in MeVR in fma)   string and two-gluelump trial statessecond excited statefirst excited stateground state 0 0.2 0.4 0.6 0.8 1 0  0.2  0.4  0.6  0.8  1  1.2  1.4  1.6|a...0|2R in fmb)   overlaps of the ground state approximationstringtwo-gluelump 0 0.2 0.4 0.6 0.8 1 0  0.2  0.4  0.6  0.8  1  1.2  1.4  1.6|a...1|2R in fmc)   overlaps of the first excited state approximationstringtwo-gluelumpFigure 2: a) The adjoint static potential V(1) and its ﬁrst two excitations as functions of the separation R.b) Overlaps of the ground state approximationto the trial states as functions of the separation R. c) Overlapsof the ﬁrst excited state approximation to the trial states as functions of the separation R.4PoS(Confinement8)059The adjoint potential in the pseudoparticle approach: string breaking and Casimir scaling Marc Wagneradiabatically. Moreover, higher representation potentials exhibit Casimir scaling. We concludethat the pseudoparticle approach is a model, which is able to reproduce many essential features ofSU(2) Yang-Mills theory.Currently our efforts are focused on applying the pseudoparticle approach to fermionic theo-ries. First steps in this direction have been successful [19, 20]. Now we intend to consider QCD,where a cheap computation of exact all-to-all propagators should be possible due to the small num-ber of degrees of freedom involved. Another appealing possibility is an application to supersym-metric theories, where an exact realization of supersymmetry might be possible due to translationalinvariance present in pseudoparticle ensembles.AcknowledgmentsMW would like to thank M. Faber, J. Greensite and M. Polikarpov for the invitation to thisconference. Moreover, we acknowledge useful conversations with M. Ammon, G. Bali, P. de For-crand, H. Hofmann, E.-M. Ilgenfritz, F. Lenz and M. Müller-Preussker.References[1] T. Schäfer and E. V. Shuryak, Rev. Mod. Phys. 70 (1998) 323 [arXiv:hep-ph/9610451].[2] F. Lenz, J. W. Negele and M. Thies, Phys. Rev. D 69, 074009 (2004) [arXiv:hep-th/0306105].[3] J. W. Negele, F. Lenz and M. Thies, Nucl. Phys. Proc. Suppl. 140, 629 (2005)[arXiv:hep-lat/0409083].[4] F. Lenz, J. W. Negele and M. Thies, Annals Phys. 323, 1536 (2008) [arXiv:0708.1687[hep-ph]].[5] M. Wagner and F. Lenz, PoS LAT2005, 315 (2006) [arXiv:hep-lat/0510083].[6] M. Wagner, Phys. Rev. D 75, 016004 (2007) [arXiv:hep-ph/0608090].[7] M. Wagner, AIP Conf. Proc. 892, 231 (2007) [arXiv:hep-ph/0610291].[8] C. Szasz and M. Wagner, Phys. Rev. D 78, 036006 (2008) [arXiv:0806.1977[hep-ph]].[9] P. Gerhold, E. M. Ilgenfritz and M. Müller-Preussker, Nucl. Phys. B 760, 1 (2007)[arXiv:hep-ph/0607315].[10] P. Gerhold, E. M. Ilgenfritz, M. Müller-Preussker, B. V. Martemyanov and A. I. Veselov, AIP Conf.Proc. 892, 213 (2007) [arXiv:hep-ph/0611161].[11] M. Faber, J. Greensite and S. Olejnik, Phys. Rev. D 57, 2603 (1998) [arXiv:hep-lat/9710039].[12] M. Engelhardt and H. Reinhardt, Nucl. Phys. B 585, 591 (2000) [arXiv:hep-lat/9912003].[13] M. Engelhardt, M. Quandt and H. Reinhardt, Nucl. Phys. B 685, 227 (2004) [arXiv:hep-lat/0311029].[14] S. Raﬁbakhsh and S. Deldar, AIP Conf. Proc. 892, 527 (2007).[15] M. Faber, D. Neudecker, poster at “Quark Conﬁnement and the Hadron Spectrum VIII” (2008).[16] C. Piccioni, Phys. Rev. D 73, 114509 (2006) [arXiv:hep-lat/0503021].[17] I. H. Jorysz and C. Michael, Nucl. Phys. B 302, 448 (1988).[18] P. de Forcrand and O. Philipsen, Phys. Lett. B 475, 280 (2000) [arXiv:hep-lat/9912050].[19] M. Wagner, Phys. Rev. D 76, 076002 (2007) [arXiv:0704.3023[hep-lat]].[20] M. Wagner, PoS LATTICE2007, 339 (2006) [arXiv:0708.2359[hep-lat]].5

The adjoint potential in the pseudoparticle approach: string breaking and Casimir scaling

Abstract

Similar works

Full text

Available Versions

Hochschulschriftenserver - Universität Frankfurt am Main