measurement of the gluon magnetic mass in external abelian magnetic field

Measurement of the Gluon Measurement of the Gluon Magnetic Mass in external Magnetic Mass in external

Abelian magnetic fieldAbelian magnetic field

S.S. AntropovS.S. Antropov, V.I. Demchik, V.V. Skalozub, V.I. Demchik, V.V. Skalozub

Measurement of the Gluon Magnetic Mass Measurement of the Gluon Magnetic Mass S.S. AntropovS.S. Antropov, V.I. Demchik, V.V. , V.I. Demchik, V.V. SkalozubSkalozub

in external Abelian magnetic fieldin external Abelian magnetic field

• Introduction.Introduction.

• Action on a lattice.Action on a lattice.

• Data and simulation analysis.Data and simulation analysis.

• Discussion.Discussion.

• Conclusions.Conclusions.

IntroductioIntroductionn• The goal of our study is to measure a The goal of our study is to measure a

gluon magnetic mass which appears as gluon magnetic mass which appears as an inverse screening length in a system an inverse screening length in a system in colour external magnetic field.in colour external magnetic field.

• The object of our study is SU(2) gluon The object of our study is SU(2) gluon fields on a lattice. fields on a lattice.

• For our purposes we used method of For our purposes we used method of Monte-Carlo calculation.Monte-Carlo calculation.



ˆx a

x, 0

x

U Pexp ig A ( )d ,

where x ) coordinates of the side, ) shift in -th direction, a ) where x ) coordinates of the side, ) shift in -th direction, a ) lattice spacing, glattice spacing, g00 ) coupling constant, P ) mere. ) coupling constant, P ) mere.

• We start with associating link variables (which is We start with associating link variables (which is SU(2) matrixes) with a vector potential of gluon field SU(2) matrixes) with a vector potential of gluon field as follows:as follows:

ˆˆW x, x a , x,x a ,x

1S 1 Tr U U U U ,

2

• The Wilson lattice action readsThe Wilson lattice action reads

where is inverse temperature. where is inverse temperature.

Action on a latticeAction on a lattice



• To represent To represent Abelian external Abelian external magnetic field the magnetic field the following boundary following boundary conditions conditions were were usedused**

* * it is published in paperit is published in paper::• G. 't Hooft (1979).G. 't Hooft (1979).



Three possibilities exist: • mmagnetic flux tubes are formed (Meissner effect),

• magnetic fields have Coulomb behavior,

• magnetic fields are screened.

In case of flux tubes plaquette average* for cubic spatial part of the lattice (with linear size NS)

has to show an effect proportional to . If magnetic fields are

Coulombic ) ) proportional to , for screening ) ) to ,

where m - magnetic mass.

x xNoTwist Twist

TrU (x) TrU (x)

2S

1

N

4S

1

N

2 2Sm Ne

*Plaquette average have sense of internal energy.



Monte-Carlo data and fit curves for Abelian magnetic field.Monte-Carlo data and fit curves for Abelian magnetic field.



Data and simulation analysis.Data and simulation analysis.

Fit results for Abelian magnetic field.Fit results for Abelian magnetic field.

• Many possible fit has been tested.Many possible fit has been tested. • BestBest fits for dependencesfits for dependences , and . , and .

• We see that for case of Abelian magnetic field the magnetic We see that for case of Abelian magnetic field the magnetic mass is 0 and the screening is absent.mass is 0 and the screening is absent.

mr1e

r1

r

2 2m r1e

r



Discussion.Discussion.For Abelian magnetic field: For Abelian magnetic field:

• Best fits for dependencesBest fits for dependences , and corresponds , and corresponds

to appearance of a magnetic flux tubes with increasing field to appearance of a magnetic flux tubes with increasing field

strength. strength.

• Because values of for best Because values of for best dependences is almost equal and dependences is almost equal and because the screening parameters for that dependences are small we can because the screening parameters for that dependences are small we can conclude that magnetic mass is 0 for that case.conclude that magnetic mass is 0 for that case.

mr1e

r1

r

2 2m r1e

r

2



Conclusions.Conclusions.For Abelian magnetic field: For Abelian magnetic field:

• Magnetic flux tubes with increasing field strength appear. Magnetic flux tubes with increasing field strength appear.

• MMagnetic mass is equal to 0.agnetic mass is equal to 0.

Thank you for your Thank you for your attentionattention

Fit results for non-Abelian magnetic field. 3 pointsFit results for non-Abelian magnetic field. 3 points

Fit results for non-Abelian magnetic field. 17 pointsFit results for non-Abelian magnetic field. 17 points

exp( )C mr 0.06 20.062.44 10m

2 2exp( )C m r 0.02 20.021.57 10m

/C r/ exp( )C r mr52 6541.25 10m

2 2/ exp( )C r m r5951.2 1057842.4 10m

2/C r 2 2 2/ exp( )C r m r 53.3 10m 4/C r 4 4 4/ exp( )C r m r0m

Fit results for Abelian magnetic fieldFit results for Abelian magnetic field

measurement of the gluon magnetic mass in external abelian magnetic field

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