The adjoint potential in the pseudoparticle approach: string breaking and Casimir scaling
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The adjoint potential in the pseudoparticleapproach: string breaking and Casimir scaling
Christian Szasz
University of Erlangen-Nürnberg, Institute for Theoretical Physics III, Staudtstraße 7,91058 Erlangen, GermanyE-mail: [email protected]
Marc Wagner ∗ Humboldt University Berlin, Department of Physics, Newtonstraße 15, 12489 Berlin, GermanyE-mail: [email protected]
We 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. ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ he adjoint potential in the pseudoparticle approach: string breaking and Casimir scaling
Marc Wagner
1. Introduction
A 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 theory
The basic idea of the pseudoparticle approach is to approximate the Yang-Mills path integral D O E = Z Z DA O [ A ] e − S [ A ] , S [ A ] = g Z d x F a mn F a mn , (2.1)where F a mn = ¶ m A a n − ¶ n A a m + e abc A b m A c n , 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 pseudoparticles : A a m ( x ) = (cid:229) j A ( j ) C ab ( j ) a b m ( x − z ( j )) , (2.2)where j is the pseudoparticle index and A ( j ) ∈ R , C ab ( j ) ∈ SO(3) and z ( j ) ∈ R 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: Z DA . . . = Z (cid:213) j d A ( j ) d C ( 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]): a a m , inst. ( x ) = h a mn x n x + l , a a m , antiinst. ( x ) = ¯ h a mn x n x + l , a a m , akyron ( x ) = d a x m x + l . (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. In this paper the term pseudoparticle refers to any gauge field configuration a a m , which is localized in space and intime, not only to solutions of the classical Yang-Mills equations of motion. he adjoint potential in the pseudoparticle approach: string breaking and Casimir scaling Marc Wagner
3. Casimir scaling and adjoint string breaking
In 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 = / 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 i = ( f ( J ) ( x )) † U ( J ) ( x ; y ) f ( J ) ( y ) | W i , | x − y | = R , (3.1)where U ( J ) denotes a spatial parallel transporter. We compute temporal correlation functions C ( J ) string ( T ) = h W | (cid:16) S ( J ) ( x , y , T ) (cid:17) † S ( J ) ( x , y , ) | W i (cid:181) D W ( 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 = . V ( / ) ( R ) = V + s R and by identify-ing the string tension s with s physical = . / fm . This amounts to a spacetime region of extension L = ( .
85 fm ) .Numerical results for higher representation potentials ( J = , . . . , /
2) are shown in Figure 1b.According to the Casimir scaling hypothesis these potentials are supposed to fulfill V ( / ) ( R ) ≈ V ( ) ( R ) / ≈ V ( / ) ( R ) ≈ V ( ) ( R ) ≈ V ( / ) ( R ) / J ≥ /
2. This is in agreement with what has been observed in 4dSU(2) lattice gauge theory [16]. V ( / ) R a) 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 1.6 V ( J ) i n M e V R in fm b) 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 1.6 V ( J ) / V ( / ) R in fm c) V (J) / V (1/2) as a function of R for different JJ=5/2J=2J=3/2J=1 Figure 1: a)
The fundamental static potential V ( / ) 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 ( / ) asfunctions of the separation R compared to the Casimir scaling expectation. he adjoint potential in the pseudoparticle approach: string breaking and Casimir scaling Marc Wagner
Note that there is no sign of string breaking for the adjoint potential even for separations R > ∼ . (cid:229) j = x , y , z G j ( x ) G j ( y ) | W i , G j ( x ) = Tr (cid:16) f ( ) ( x ) B j ( x ) (cid:17) , | 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 ≈ g = . R sb ≈ . | i ≈ a | string i + a | i , | i ≈ a | string i + a | i , (3.5)where | string i and | i are normalized trial states. The overlaps | a j ... | 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 outlook
We 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 R sb ≈ . V ( ) i n M e V R in fm a) 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 ... | R in fm b) 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 ... | R in fm c) overlaps of the first excited state approximationstringtwo-gluelump Figure 2: a)
The adjoint static potential V ( ) and its first two excitations as functions of the separation R . b) Overlaps of the ground state approximation to 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 . he adjoint potential in the pseudoparticle approach: string breaking and Casimir scaling Marc Wagner adiabatically. 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.
Acknowledgments
MW 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.
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