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Computing the static potential using non-string-like trial states Lattice 2016 - Southhampton Tobias Neitzel Goethe Universit at Frankfurt am Main, Institut f ur Theoretische Physik neitzel@th.physik.uni-frankfurt.de In collabaration

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Computing the static potential using non-string-like trial states

Lattice 2016 - Southhampton Tobias Neitzel Goethe Universit¨ at Frankfurt am Main, Institut f¨ ur Theoretische Physik neitzel@th.physik.uni-frankfurt.de In collabaration with Janik K¨ amper, Owe Philipsen, Marc Wagner July 25, 2016

Tobias Neitzel Computing the static potential using non-string-like trial states July 25, 2016 1 / 20

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Motivation

Tobias Neitzel Computing the static potential using non-string-like trial states July 25, 2016 2 / 20

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  • For calculating the static potential with a high resolution we have to work

with off axis separated quarks.

  • e.g. matching the lattice QCD potential with the perturbative potential to

determine ΛMS in Fourier space.

[F. Karbstein, A. Peters and M. Wagner, JHEP 1409, 114 (2014) [arXiv:1407.7503 [hep-ph]] Tobias Neitzel Computing the static potential using non-string-like trial states July 25, 2016 3 / 20

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  • For calculating the static potential with a high resolution we have to work

with off axis separated quarks.

  • e.g. matching the lattice QCD potential with the perturbative potential to

determine ΛMS in Fourier space.

[F. Karbstein, A. Peters and M. Wagner, JHEP 1409, 114 (2014) [arXiv:1407.7503 [hep-ph]]

  • The quantity of interest is the Wilson loop, which connects the two quarks

like a string.

Tobias Neitzel Computing the static potential using non-string-like trial states July 25, 2016 3 / 20

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  • To compute the spatial part of the Wilson loop one has to go over stair-like

paths through the lattice.

Tobias Neitzel Computing the static potential using non-string-like trial states July 25, 2016 4 / 20

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  • To compute the spatial part of the Wilson loop one has to go over stair-like

paths through the lattice.

  • These stair-like paths are causing a big computational effort for a large

number of lattice points.

Tobias Neitzel Computing the static potential using non-string-like trial states July 25, 2016 4 / 20

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  • To compute the spatial part of the Wilson loop one has to go over stair-like

paths through the lattice.

  • These stair-like paths are causing a big computational effort for a large

number of lattice points.

  • Idea: Substitute the spatial part of the Wilson loop by an other object to

avoid the calculation of stair-like paths.

Tobias Neitzel Computing the static potential using non-string-like trial states July 25, 2016 4 / 20

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The Technical Part

Tobias Neitzel Computing the static potential using non-string-like trial states July 25, 2016 5 / 20

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  • The spatial Wilson line is needed to ensure gauge invariance of the q¯

q trial state.

Tobias Neitzel Computing the static potential using non-string-like trial states July 25, 2016 6 / 20

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  • The spatial Wilson line is needed to ensure gauge invariance of the q¯

q trial state.

  • Transformation behavior required: U′(x, y) = G(x)U(x, y)G †(y)

Tobias Neitzel Computing the static potential using non-string-like trial states July 25, 2016 6 / 20

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  • The spatial Wilson line is needed to ensure gauge invariance of the q¯

q trial state.

  • Transformation behavior required: U′(x, y) = G(x)U(x, y)G †(y)
  • We explore an idea, which has been used in the context of Polyakov loops

and the static potential at finite temperature.

[O. Jahn and O. Philipsen, Phys. Rev. D 70, 074504 (2004) [hep-lat/0407042]] [O. Philipsen, Phys. Lett. B 535, 138 (2002) [hep-lat/0203018]] Tobias Neitzel Computing the static potential using non-string-like trial states July 25, 2016 6 / 20

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  • The spatial Wilson line is needed to ensure gauge invariance of the q¯

q trial state.

  • Transformation behavior required: U′(x, y) = G(x)U(x, y)G †(y)
  • We explore an idea, which has been used in the context of Polyakov loops

and the static potential at finite temperature.

[O. Jahn and O. Philipsen, Phys. Rev. D 70, 074504 (2004) [hep-lat/0407042]] [O. Philipsen, Phys. Lett. B 535, 138 (2002) [hep-lat/0203018]]

  • Consider the covariant lattice Laplace operator:

∆f = 1 a2

  • U†

1(x − a, y, z)f (x − a, y, z) − 2f (x) + U1(x, y, z)f (x + a, y, z)

  • + 1

a2

  • U†

2(x, y − a, z)f (x, y − a, z) − 2f (x) + U2(x, y, z)f (x, y + a, z)

  • + 1

a2

  • U†

3(x, y, z − a)f (x, y, z − a) − 2f (x) + U3(x, y, z)f (x, y, z + a)

  • Tobias Neitzel

Computing the static potential using non-string-like trial states July 25, 2016 6 / 20

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  • The spatial Wilson line is needed to ensure gauge invariance of the q¯

q trial state.

  • Transformation behavior required: U′(x, y) = G(x)U(x, y)G †(y)
  • We explore an idea, which has been used in the context of Polyakov loops

and the static potential at finite temperature.

[O. Jahn and O. Philipsen, Phys. Rev. D 70, 074504 (2004) [hep-lat/0407042]] [O. Philipsen, Phys. Lett. B 535, 138 (2002) [hep-lat/0203018]]

  • Consider the covariant lattice Laplace operator:

∆f = 1 a2

  • U†

1(x − a, y, z)f (x − a, y, z) − 2f (x) + U1(x, y, z)f (x + a, y, z)

  • + 1

a2

  • U†

2(x, y − a, z)f (x, y − a, z) − 2f (x) + U2(x, y, z)f (x, y + a, z)

  • + 1

a2

  • U†

3(x, y, z − a)f (x, y, z − a) − 2f (x) + U3(x, y, z)f (x, y, z + a)

  • Transformation behavior: ∆′ = G(x)∆G †(x)

Tobias Neitzel Computing the static potential using non-string-like trial states July 25, 2016 6 / 20

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  • Writing f (x) as a vector in position space ∆ can be written as a matrix.

Tobias Neitzel Computing the static potential using non-string-like trial states July 25, 2016 7 / 20

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  • Writing f (x) as a vector in position space ∆ can be written as a matrix.
  • Consider f (x) is now an eigenvector of the covariant Laplace operator.

∆f (x) = λf (x)

Tobias Neitzel Computing the static potential using non-string-like trial states July 25, 2016 7 / 20

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  • Writing f (x) as a vector in position space ∆ can be written as a matrix.
  • Consider f (x) is now an eigenvector of the covariant Laplace operator.

∆f (x) = λf (x) ∆′f ′(x) = λf ′(x)

  • Apply an gauge transformation on the eigenvector-equation.

Tobias Neitzel Computing the static potential using non-string-like trial states July 25, 2016 7 / 20

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  • Writing f (x) as a vector in position space ∆ can be written as a matrix.
  • Consider f (x) is now an eigenvector of the covariant Laplace operator.

∆f (x) = λf (x) ∆′f ′(x) = λf ′(x) G(x)∆G †(x)f ′(x) = λf ′(x)

  • Apply an gauge transformation on the eigenvector-equation.

Tobias Neitzel Computing the static potential using non-string-like trial states July 25, 2016 7 / 20

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  • Writing f (x) as a vector in position space ∆ can be written as a matrix.
  • Consider f (x) is now an eigenvector of the covariant Laplace operator.

∆f (x) = λf (x) ∆′f ′(x) = λf ′(x) G(x)∆G †(x)f ′(x) = λf ′(x) ∆G †(x)f ′(x) = λG †(x)f ′(x)

  • Apply an gauge transformation on the eigenvector-equation.

Tobias Neitzel Computing the static potential using non-string-like trial states July 25, 2016 7 / 20

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  • Writing f (x) as a vector in position space ∆ can be written as a matrix.
  • Consider f (x) is now an eigenvector of the covariant Laplace operator.

∆f (x) = λf (x) ∆′f ′(x) = λf ′(x) G(x)∆G †(x)f ′(x) = λf ′(x) ∆G †(x)f ′(x) = λG †(x)f ′(x)

  • Apply an gauge transformation on the eigenvector-equation.
  • We see: G †(x)f ′(x) is again eigenvector to the covariant Laplace operator.

Tobias Neitzel Computing the static potential using non-string-like trial states July 25, 2016 7 / 20

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  • Now we know: f (x) and G †(x)f ′(x) are members of the same eigenspace.

Tobias Neitzel Computing the static potential using non-string-like trial states July 25, 2016 8 / 20

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  • Now we know: f (x) and G †(x)f ′(x) are members of the same eigenspace.
  • In SU(3) the eigenvalues are in general nondegenerate. This means:

f (x)eiφ = G †(x)f ′(x)

Tobias Neitzel Computing the static potential using non-string-like trial states July 25, 2016 8 / 20

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  • Now we know: f (x) and G †(x)f ′(x) are members of the same eigenspace.
  • In SU(3) the eigenvalues are in general nondegenerate. This means:

f (x)eiφ = G †(x)f ′(x)

  • In SU(2) however, the eigenvalues are always two fold degenrate. This means:

αf1(x) + βf2(x) = G †(x)f ′(x)

  • Where f1 and f2 are an orthonormal basis of the corresponding eigenspace.

Tobias Neitzel Computing the static potential using non-string-like trial states July 25, 2016 8 / 20

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Transformation law for SU(3): Transformation law for SU(2): f (x)eiφ = G †(x)f ′(x) αf1(x) + βf2(x) = G †(x)f ′(x) Wilson Line: U′(x, y) = G(x)U(x, y)G †(y)

  • Now it is easy to create an object with the needed transformation behavior.

Tobias Neitzel Computing the static potential using non-string-like trial states July 25, 2016 9 / 20

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Transformation law for SU(3): Transformation law for SU(2): f (x)eiφ = G †(x)f ′(x) αf1(x) + βf2(x) = G †(x)f ′(x) Wilson Line: U′(x, y) = G(x)U(x, y)G †(y) SU(3) - Case: f ′(x)f ′†(y) = G(x)f (x)f †(y)G †(y)

  • Now it is easy to create an object with the needed transformation behavior.

Tobias Neitzel Computing the static potential using non-string-like trial states July 25, 2016 9 / 20

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Transformation law for SU(3): Transformation law for SU(2): f (x)eiφ = G †(x)f ′(x) αf1(x) + βf2(x) = G †(x)f ′(x) Wilson Line: U′(x, y) = G(x)U(x, y)G †(y) SU(3) - Case: f ′(x)f ′†(y) = G(x)f (x)f †(y)G †(y) SU(2) - Case:

2

  • i=1

f ′

i (x)f ′† i (y) = G(x)

  • 2
  • i=1

fi(x)f †

i (y)

  • G †(y)
  • Now it is easy to create an object with the needed transformation behavior.
  • Where f1 and f2 are an orthonormal basis of the corresponding eigenspace.

Tobias Neitzel Computing the static potential using non-string-like trial states July 25, 2016 9 / 20

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  • We found objects with the required transformation behavior given by

f (x)f †(y) for SU(3) and

2

  • i=1

fi(x)f †

i (y)

for SU(2).

Tobias Neitzel Computing the static potential using non-string-like trial states July 25, 2016 10 / 20

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  • We found objects with the required transformation behavior given by

f (x)f †(y) for SU(3) and

2

  • i=1

fi(x)f †

i (y)

for SU(2).

  • With these new objects it is not necessary to distinguish a certain path

between x and y.

Tobias Neitzel Computing the static potential using non-string-like trial states July 25, 2016 10 / 20

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  • We found objects with the required transformation behavior given by

f (x)f †(y) for SU(3) and

2

  • i=1

fi(x)f †

i (y)

for SU(2).

  • With these new objects it is not necessary to distinguish a certain path

between x and y. Advantages: The computation of stair-like paths is not longer needed.

Tobias Neitzel Computing the static potential using non-string-like trial states July 25, 2016 10 / 20

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  • We found objects with the required transformation behavior given by

f (x)f †(y) for SU(3) and

2

  • i=1

fi(x)f †

i (y)

for SU(2).

  • With these new objects it is not necessary to distinguish a certain path

between x and y. Advantages: The computation of stair-like paths is not longer needed. Price to pay: One has to compute the eigenvectors of ∆ first.

Tobias Neitzel Computing the static potential using non-string-like trial states July 25, 2016 10 / 20

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Figure : Runtime of the eigenvector calculation and the remaining computations using the new method

Tobias Neitzel Computing the static potential using non-string-like trial states July 25, 2016 11 / 20

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Results

Tobias Neitzel Computing the static potential using non-string-like trial states July 25, 2016 12 / 20

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Figure : Effective mass in units of the lattice spacing for the ordinary Wilson loop - using 100 basically independent SU(2) gaugelink configurations with β = 2.5 (≈ 0.089fm) on a 24x24 Lattice

Tobias Neitzel Computing the static potential using non-string-like trial states July 25, 2016 13 / 20

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Figure : Effective mass in units of the lattice spacing for the new method - using 100 basically independent SU(2) gaugelink configurations with β = 2.5 (≈ 0.089fm) on a 24x24 Lattice

Tobias Neitzel Computing the static potential using non-string-like trial states July 25, 2016 14 / 20

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Figure : Comparison of the effective masses from the ordinary Wilson loop and the new method - using 100 basically independent SU(2) gaugelink configurations with β = 2.5 (≈ 0.089fm) on a 24x24 Lattice

Tobias Neitzel Computing the static potential using non-string-like trial states July 25, 2016 15 / 20

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Figure : Potential for the static q¯ q pair in units of the lattice spacing - using the ordinary Wilson loop on 100 basically independent SU(2) gaugelink configurations with β = 2.5 (≈ 0.089fm) on a 24x24 Lattice

Tobias Neitzel Computing the static potential using non-string-like trial states July 25, 2016 16 / 20

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Figure : Potential for the static q¯ q pair in units of the lattice spacing - using the new method on 100 basically independent SU(2) gaugelink configurations with β = 2.5 (≈ 0.089fm) on a 24x24 Lattice

Tobias Neitzel Computing the static potential using non-string-like trial states July 25, 2016 17 / 20

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Figure : Off-axis potential for the static q¯ q pair in units of the lattice spacing - using the new method on 100 basically independent SU(2) gaugelink configurations with β = 2.5 (≈ 0.089fm) on a 24x24 Lattice

Tobias Neitzel Computing the static potential using non-string-like trial states July 25, 2016 18 / 20

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Figure : On-axis potential for the static q¯ q pair in units of the lattice spacing - using the new method on 60 basically independent SU(3) gaugelink configurations with β = 3.9 on a 48x24 Lattice

Tobias Neitzel Computing the static potential using non-string-like trial states July 25, 2016 19 / 20

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Summary

  • In the ordinary approach the calculation of the static q¯

q-potential, for off-axis separations, requires the computation of time consuming stair-like paths.

  • These stair-like paths come from the Wilson loop, an object that ensures

gauge invariance of the used q¯ q trial state.

  • By using the eigenvectors of the covariant Laplace operator we were able to

substitute the spatial part of the Wilson loop by a new object.

  • Advantages: Fast computation times for off axis calculations and nearly

similar quality of the results (error bars).

  • Possible application: The potential with fine resolution can be used for better

modeling and comparison with perturbative theories (ΛMS-determination, b¯ b-spectrum).

Tobias Neitzel Computing the static potential using non-string-like trial states July 25, 2016 20 / 20