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The thesis is about the improvement of the static-light axial current on the lattice and was performed at the University of Wuppertal in 2008, in the framework of the master's course Computer simulation in Science (CSIS)

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- 1. Introduction The static-light current on the lattice The static-light current in the Lattice SF Determination of cA (1) stat Minimisation of the self energy Summary Master thesis in CSiSImprovement of the static-light axial current on the lattice Alois Grimbach Institut fuer Theoretische Physik Bergische Universitt Wuppertal Author Short Paper Title
- 2. Introduction The static-light current on the lattice The static-light current in the Lattice SF Determination of cA (1) stat Minimisation of the self energy SummaryOutline 1 Introduction 2 The static-light current on the lattice Actions and Currents O(a) improvement HYP smearing 3 The static-light current in the Lattice SF The Schrdinger Functional Pertubation Theory in the SF HYP smearing in the SF stat (1) 4 Determination of cA 5 Minimisation of the self energy 6 Summary Author Short Paper Title
- 3. Introduction The static-light current on the lattice The static-light current in the Lattice SF Determination of cA (1) stat Minimisation of the self energy SummaryIntroduction Lattice QCD Strong interaction is described by SU(3) colour group Pertubation theory successful at small distances (High energy regime) 1974, Wilson: Lattice QCD (Low energy regime): hadronic spectra and matrix elements between hadronic states can be investigated Principle: Euclidean (Wick-rotated) hypercubic lattice with lattice spacing a allows application of statistical methods - Investigation of static-light axial current is helpful in b-physics - Improvement accelerates approach to continuum limit Author Short Paper Title
- 4. Introduction The static-light current on the lattice The static-light current in the Lattice SF Determination of cA (1) stat Minimisation of the self energy SummaryIntroduction Lattice QCD Strong interaction is described by SU(3) colour group Pertubation theory successful at small distances (High energy regime) 1974, Wilson: Lattice QCD (Low energy regime): hadronic spectra and matrix elements between hadronic states can be investigated Principle: Euclidean (Wick-rotated) hypercubic lattice with lattice spacing a allows application of statistical methods - Investigation of static-light axial current is helpful in b-physics - Improvement accelerates approach to continuum limit Author Short Paper Title
- 5. Introduction The static-light current on the lattice The static-light current in the Lattice SF Determination of cA (1) stat Minimisation of the self energy SummaryIntroduction Lattice QCD Strong interaction is described by SU(3) colour group Pertubation theory successful at small distances (High energy regime) 1974, Wilson: Lattice QCD (Low energy regime): hadronic spectra and matrix elements between hadronic states can be investigated Principle: Euclidean (Wick-rotated) hypercubic lattice with lattice spacing a allows application of statistical methods - Investigation of static-light axial current is helpful in b-physics - Improvement accelerates approach to continuum limit Author Short Paper Title
- 6. Introduction The static-light current on the lattice The static-light current in the Lattice SF Determination of cA (1) stat Minimisation of the self energy SummaryIntroduction Lattice QCD Strong interaction is described by SU(3) colour group Pertubation theory successful at small distances (High energy regime) 1974, Wilson: Lattice QCD (Low energy regime): hadronic spectra and matrix elements between hadronic states can be investigated Principle: Euclidean (Wick-rotated) hypercubic lattice with lattice spacing a allows application of statistical methods - Investigation of static-light axial current is helpful in b-physics - Improvement accelerates approach to continuum limit Author Short Paper Title
- 7. Introduction The static-light current on the lattice The static-light current in the Lattice SF Determination of cA (1) stat Minimisation of the self energy SummaryIntroduction Lattice QCD Strong interaction is described by SU(3) colour group Pertubation theory successful at small distances (High energy regime) 1974, Wilson: Lattice QCD (Low energy regime): hadronic spectra and matrix elements between hadronic states can be investigated Principle: Euclidean (Wick-rotated) hypercubic lattice with lattice spacing a allows application of statistical methods - Investigation of static-light axial current is helpful in b-physics - Improvement accelerates approach to continuum limit Author Short Paper Title
- 8. Introduction The static-light current on the lattice Actions and Currents The static-light current in the Lattice SF O(a) improvement Determination of cA (1) stat HYP smearing Minimisation of the self energy SummaryActionsGauge Action Gauge Action Action consists of gauge action and fermionic action S = SG [U] + SF [U, , ] Gauge links U (x) - connect x with x + a - are members of SU(3) group Gauge Action is described by sum over plaquettes 1 SG [U] = g 2 tr {1 U(p)} 0 p Formulation is gauge invariant and yields Yang-Mills theory in the continuum limit Author Short Paper Title
- 9. Introduction The static-light current on the lattice Actions and Currents The static-light current in the Lattice SF O(a) improvement Determination of cA (1) stat HYP smearing Minimisation of the self energy SummaryActionsGauge Action Gauge Action Action consists of gauge action and fermionic action S = SG [U] + SF [U, , ] Gauge links U (x) - connect x with x + a - are members of SU(3) group Gauge Action is described by sum over plaquettes 1 SG [U] = g 2 tr {1 U(p)} 0 p Formulation is gauge invariant and yields Yang-Mills theory in the continuum limit Author Short Paper Title
- 10. Introduction The static-light current on the lattice Actions and Currents The static-light current in the Lattice SF O(a) improvement Determination of cA (1) stat HYP smearing Minimisation of the self energy SummaryActionsGauge Action Gauge Action Action consists of gauge action and fermionic action S = SG [U] + SF [U, , ] Gauge links U (x) - connect x with x + a - are members of SU(3) group Gauge Action is described by sum over plaquettes 1 SG [U] = g 2 tr {1 U(p)} 0 p Formulation is gauge invariant and yields Yang-Mills theory in the continuum limit Author Short Paper Title
- 11. Introduction The static-light current on the lattice Actions and Currents The static-light current in the Lattice SF O(a) improvement Determination of cA (1) stat HYP smearing Minimisation of the self energy SummaryActionsFermion Action Two theories for light and static quarks Light quarks Fermionic Action for light quarks Sl [l , l ] = a4 l (x)(D + m0 )l (x) x D is Wilson-Dirac operator 1 D = 2 ( + ) a Wilson term - removes fermion doublers - vanishes in the continuum limit a 0 - breaks chiral symmetry for massless fermions Author Short Paper Title
- 12. Introduction The static-light current on the lattice Actions and Currents The static-light current in the Lattice SF O(a) improvement Determination of cA (1) stat HYP smearing Minimisation of the self energy SummaryActionsFermion Action Two theories for light and static quarks Light quarks Fermionic Action for light quarks Sl [l , l ] = a4 l (x)(D + m0 )l (x) x D is Wilson-Dirac operator 1 D = 2 ( + ) a Wilson term - removes fermion doublers - vanishes in the continuum limit a 0 - breaks chiral symmetry for massless fermions Author Short Paper Title
- 13. Introduction The static-light current on the lattice Actions and Currents The static-light current in the Lattice SF O(a) improvement Determination of cA (1) stat HYP smearing Minimisation of the self energy SummaryActionsFermionic Action Heavy quarks Heavy quarks are described by HQET - static approximation at m0 - higher contributions organised as powers of inverse quark mass Static quarks Static quarks - have only temporal dynamics - are described by decoupled pair of fermion elds Sh [h , h ] = a4 x h (x) h (x) 0 Sh [, ] = a4 x (x)0 (x) h h h h Author Short Paper Title
- 14. Introduction The static-light current on the lattice Actions and Currents The static-light current in the Lattice SF O(a) improvement Determination of cA (1) stat HYP smearing Minimisation of the self energy SummaryActionsFermionic Action Heavy quarks Heavy quarks are described by HQET - static approximation at m0 - higher contributions organised as powers of inverse quark mass Static quarks Static quarks - have only temporal dynamics - are described by decoupled pair of fermion elds Sh [h , h ] = a4 x h (x) h (x) 0 Sh [, ] = a4 x (x)0 (x) h h h h Author Short Paper Title
- 15. Introduction The static-light current on the lattice Actions and Currents The static-light current in the Lattice SF O(a) improvement Determination of cA (1) stat HYP smearing Minimisation of the self energy SummaryStatic-light Axial Current Axial Current Isovector Axial Current for SU(2) isospin A (x) = (x) 5 1 (x) 2 Static-light Axial Current - is dened by Astat = l (x)0 5 h (x) 0 - is induced by a static quark and a light anti-quark Author Short Paper Title
- 16. Introduction The static-light current on the lattice Actions and Currents The static-light current in the Lattice SF O(a) improvement Determination of cA (1) stat HYP smearing Minimisation of the self energy SummaryStatic-light Axial Current Axial Current Isovector Axial Current for SU(2) isospin A (x) = (x) 5 1 (x) 2 Static-light Axial Current - is dened by Astat = l (x)0 5 h (x) 0 - is induced by a static quark and a light anti-quark Author Short Paper Title
- 17. Introduction The static-light current on the lattice Actions and Currents The static-light current in the Lattice SF O(a) improvement Determination of cA (1) stat HYP smearing Minimisation of the self energy SummaryO(a) improvementSymanzik improvement scheme Discretisation error proportional to lattice spacing a - can be improved to O(a2 ) Symanzik improvement scheme - consider momentum cutoff as scale of new physics - describe lattice action by continuum effective theory Seff = d 4 x L0 (x) + ak Lk (x) k =1 - lowest order describes continuum eld theory - cancel term proportional to a by counterterms Author Short Paper Title
- 18. Introduction The static-light current on the lattice Actions and Currents The static-light current in the Lattice SF O(a) improvement Determination of cA (1) stat HYP smearing Minimisation of the self energy SummaryO(a) improvementSymanzik improvement scheme Discretisation error proportional to lattice spacing a - can be improved to O(a2 ) Symanzik improvement scheme - consider momentum cutoff as scale of new physics - describe lattice action by continuum effective theory Seff = d 4 x L0 (x) + ak Lk (x) k =1 - lowest order describes continuum eld theory - cancel term proportional to a by counterterms Author Short Paper Title
- 19. Introduction The static-light current on the lattice Actions and Currents The static-light current in the Lattice SF O(a) improvement Determination of cA (1) stat HYP smearing Minimisation of the self energy SummaryO(a) improvementSymanzik improvement scheme Discretisation error proportional to lattice spacing a - can be improved to O(a2 ) Symanzik improvement scheme - consider momentum cutoff as scale of new physics - describe lattice action by continuum effective theory Seff = d 4 x L0 (x) + ak Lk (x) k =1 - lowest order describes continuum eld theory - cancel term proportional to a by counterterms Author Short Paper Title
- 20. Introduction The static-light current on the lattice Actions and Currents The static-light current in the Lattice SF O(a) improvement Determination of cA (1) stat HYP smearing Minimisation of the self energy SummaryO(a) improvementAction and Current Counterterms can be found by - considering dimensions and symmetries - taking into account EOM Action - counterterm is Sheikoleslami Wohlert clover term - proportional constant cSW Static-light axial current - counterterm Astat = l j 5 1 ( j + )h 0 2 j stat - proportional constant cA may be expanded in PT by stat stat (k ) 2k cA = cA g0 k =0 Author Short Paper Title
- 21. Introduction The static-light current on the lattice Actions and Currents The static-light current in the Lattice SF O(a) improvement Determination of cA (1) stat HYP smearing Minimisation of the self energy SummaryO(a) improvementAction and Current Counterterms can be found by - considering dimensions and symmetries - taking into account EOM Action - counterterm is Sheikoleslami Wohlert clover term - proportional constant cSW Static-light axial current - counterterm Astat = l j 5 1 ( j + )h 0 2 j stat - proportional constant cA may be expanded in PT by stat stat (k ) 2k cA = cA g0 k =0 Author Short Paper Title
- 22. Introduction The static-light current on the lattice Actions and Currents The static-light current in the Lattice SF O(a) improvement Determination of cA (1) stat HYP smearing Minimisation of the self energy SummaryO(a) improvementAction and Current Counterterms can be found by - considering dimensions and symmetries - taking into account EOM Action - counterterm is Sheikoleslami Wohlert clover term - proportional constant cSW Static-light axial current - counterterm Astat = l j 5 1 ( j + )h 0 2 j stat - proportional constant cA may be expanded in PT by stat stat (k ) 2k cA = cA g0 k =0 Author Short Paper Title
- 23. Introduction The static-light current on the lattice Actions and Currents The static-light current in the Lattice SF O(a) improvement Determination of cA (1) stat HYP smearing Minimisation of the self energy SummaryO(a) improvementAction and Current Counterterms can be found by - considering dimensions and symmetries - taking into account EOM Action - counterterm is Sheikoleslami Wohlert clover term - proportional constant cSW Static-light axial current - counterterm Astat = l j 5 1 ( j + )h 0 2 j stat - proportional constant cA may be expanded in PT by stat stat (k ) 2k cA = cA g0 k =0 Author Short Paper Title
- 24. Introduction The static-light current on the lattice Actions and Currents The static-light current in the Lattice SF O(a) improvement Determination of cA (1) stat HYP smearing Minimisation of the self energy SummarySmearing techniques - APE consider gauge links APE smearing APE smearing - decorate the gauge link with staples - parameter weigthing the staples Author Short Paper Title
- 25. Introduction The static-light current on the lattice Actions and Currents The static-light current in the Lattice SF O(a) improvement Determination of cA (1) stat HYP smearing Minimisation of the self energy SummarySmearing techniques - HYP HYP smearing - 3 levels of recursive APE smearing - use only links that stay within the hypercubes attached to the original link - project onto SU(3) after each step - parameters 1 , 2 , 3 weigthing the smearing steps Author Short Paper Title
- 26. Introduction The static-light current on the lattice Actions and Currents The static-light current in the Lattice SF O(a) improvement Determination of cA (1) stat HYP smearing Minimisation of the self energy SummaryHYP smearing - Properties - preserves locality - improves signal-to-noise ratio - origin: reduction of static self-energy statcA for HYP smeared action stat - estimated values for cA known from hybrid methods 4 - error (O)(g0 ), but unknownsought quantities sought(1): stat - one-loop expansion of cA with HYP smearing sought(2): - mimimum of self-energy w.r.t. smearing parameters Author Short Paper Title
- 27. Introduction The static-light current on the lattice Actions and Currents The static-light current in the Lattice SF O(a) improvement Determination of cA (1) stat HYP smearing Minimisation of the self energy SummaryHYP smearing - Properties - preserves locality - improves signal-to-noise ratio - origin: reduction of static self-energy statcA for HYP smeared action stat - estimated values for cA known from hybrid methods 4 - error (O)(g0 ), but unknownsought quantities sought(1): stat - one-loop expansion of cA with HYP smearing sought(2): - mimimum of self-energy w.r.t. smearing parameters Author Short Paper Title
- 28. Introduction The static-light current on the lattice Actions and Currents The static-light current in the Lattice SF O(a) improvement Determination of cA (1) stat HYP smearing Minimisation of the self energy SummaryHYP smearing - Properties - preserves locality - improves signal-to-noise ratio - origin: reduction of static self-energy statcA for HYP smeared action stat - estimated values for cA known from hybrid methods 4 - error (O)(g0 ), but unknownsought quantities sought(1): stat - one-loop expansion of cA with HYP smearing sought(2): - mimimum of self-energy w.r.t. smearing parameters Author Short Paper Title
- 29. Introduction The static-light current on the lattice Actions and Currents The static-light current in the Lattice SF O(a) improvement Determination of cA (1) stat HYP smearing Minimisation of the self energy SummaryHYP smearing - Properties - preserves locality - improves signal-to-noise ratio - origin: reduction of static self-energy statcA for HYP smeared action stat - estimated values for cA known from hybrid methods 4 - error (O)(g0 ), but unknownsought quantities sought(1): stat - one-loop expansion of cA with HYP smearing sought(2): - mimimum of self-energy w.r.t. smearing parameters Author Short Paper Title
- 30. Introduction The static-light current on the lattice The Schrdinger Functional The static-light current in the Lattice SF Pertubation Theory in the SF Determination of cA (1) stat HYP smearing in the SF Minimisation of the self energy SummaryTool: The Schrdinger Functional (SF) The SF - sketch P (x)|x0 =T = (x) Uk (x)|x0 =T = Wk (x) U (x)|x0 >T = 1 (x)|x0 >T = 0 x0 = T x0 = 0 U (x)|x0 u,v,y,z - expand them in pertubation theory Author Short Paper Title
- 35. Introduction The static-light current on the lattice The Schrdinger Functional The static-light current in the Lattice SF Pertubation Theory in the SF Determination of cA (1) stat HYP smearing in the SF Minimisation of the self energy SummaryPertubation Theory in the SF-1 Pertubation theory - approach - describe link variable by gauge vector eld q (x) U (x) = exp(g0 aq (x)) - expand in terms of coupling constant g0 correlation functions dene correlation functions fA (x0 ) = a6 stat 1 stat 2 A0 (x)h (y)5 l (z) and y,z 1 12 stat f1 = 2 a6 L < l (u)5 h (v)h (y)5 ( z) > u,v,y,z - expand them in pertubation theory Author Short Paper Title
- 36. Introduction The static-light current on the lattice The Schrdinger Functional The static-light current in the Lattice SF Pertubation Theory in the SF Determination of cA (1) stat HYP smearing in the SF Minimisation of the self energy Summary statFeynman Diagrams for fA at one-loop ordersetting-sun tadpoles gluon exchange Author Short Paper Title
- 37. Introduction The static-light current on the lattice The Schrdinger Functional The static-light current in the Lattice SF Pertubation Theory in the SF Determination of cA (1) stat HYP smearing in the SF Minimisation of the self energy SummaryFeynman Diagrams for f1stat at one-loop ordersetting-sun tadpoles gluon exchange Author Short Paper Title
- 38. Introduction The static-light current on the lattice The Schrdinger Functional The static-light current in the Lattice SF Pertubation Theory in the SF Determination of cA (1) stat HYP smearing in the SF Minimisation of the self energy SummaryHYP links in the SF - 1 sought: relation between HYP link and original thin link in time-momentum space known - result on the full torus in momentum space: (3) B (p) = f (p)q (p) + O(g0 ) solution: - anti FT in time - is feasible du to Dirichlet BC Author Short Paper Title
- 39. Introduction The static-light current on the lattice The Schrdinger Functional The static-light current in the Lattice SF Pertubation Theory in the SF Determination of cA (1) stat HYP smearing in the SF Minimisation of the self energy SummaryHYP links in the SF - 2 result (3) B0 (x0 ; p) = 6 h0;i (p)q(i) (x0 + as(i); p) i=0 with i H (i) sH (i) h0;i (p) 1 3 0 0 0 1 6 k =1 a2 pk 0k (p) 2 1,2,3 i 0 + i1 api 0i (p) 6 4,5,6 i 3 1 i1 ap(i) 0(i) (p) 6 - result was checked by direct spatial FT - computation of spatial HYP links is more involved in publication Author Short Paper Title
- 40. Introduction The static-light current on the lattice The static-light current in the Lattice SF Determination of cA (1) stat Minimisation of the self energy SummaryDetermination of cA (1) stat stat fA L take the ratio X (g0 , a , T , ) = stat L f1 stat (1) cA can be extracted eliminates divergent part m of the self-energy - wave function renormalistion constants at the boundaries cancel continuum extrapolation stat (1) cA may be extracted from the computed correlation functions as L2 (1) L (1) L lim a 0 2a (+ )Xlat ( a )|ct =1 lim a 0 ct (1) LXb ( a ) stat (1) cA = L lim a 0 LXA (0) ( a )L L L Author Short Paper Title
- 41. Introduction The static-light current on the lattice The static-light current in the Lattice SF Determination of cA (1) stat Minimisation of the self energy SummaryDetermination of cA (1) stat stat fA L take the ratio X (g0 , a , T , ) = stat L f1 stat (1) cA can be extracted eliminates divergent part m of the self-energy - wave function renormalistion constants at the boundaries cancel continuum extrapolation stat (1) cA may be extracted from the computed correlation functions as L2 (1) L (1) L lim a 0 2a (+ )Xlat ( a )|ct =1 lim a 0 ct (1) LXb ( a ) stat (1) cA = L lim a 0 LXA (0) ( a )L L L Author Short Paper Title
- 42. Introduction The static-light current on the lattice The static-light current in the Lattice SF Determination of cA (1) stat Minimisation of the self energy SummaryDetermination of cA (1) stat HYP1 HYP2 stat(1) cstat(1) for the HYP1 action cA for the HYP2 action A 0.07 0.1 0.06 0.09 0.05 Theta=0.5 0.08 0.04 Theta=0.5 stat(1) cstat(1) 0.03 0.07 cA A 0.02 0.06 0.01 0.05 Theta=1.0 0 Theta=1.0 0.01 0.04 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 a/L a/L stat (1) stat (1) cA HYP1 = 0.0025(3) cA HYP2 = 0.0516(3)) Author Short Paper Title
- 43. Introduction The static-light current on the lattice The static-light current in the Lattice SF Determination of cA (1) stat Minimisation of the self energy SummaryResults for the self-energy self-energy - The self energy can be determined by summing up the 1-loop Feynman diagrams - comparison with known results provides a check of the diagrams results Action e(1) EH 0.168502(1) HYP1 0.048631(1) HYP2 0.035559(1) - results differ less than 0.3% from the linear divergent contribution to the static propagator at 1-loop order in M. Della Morte, A. Shindler and R. Sommer, [arXiv:hep-lat/0506008] Author Short Paper Title
- 44. Introduction The static-light current on the lattice The static-light current in the Lattice SF Determination of cA (1) stat Minimisation of the self energy Summarysmearing parameters- The self-energy has a functional dependence upon the smearingparameters, i.e. 2 (1) k k ke(1) = ek1 k2 k3 11 22 33 k 1,k 2,k 3=0- coefcients can be determined out of the of the Feynman diagrams- coefcients have a triangular structure, only for0 k3 k2 k1 2 non-zero- Results are align with the one-loop expansion of the staticself-energy won from the static potential by R.Hoffmann Author Short Paper Title
- 45. Introduction The static-light current on the lattice The static-light current in the Lattice SF Determination of cA (1) stat Minimisation of the self energy SummaryFunction of e(1) w.r.t the smearing parameters3D plot of e(1) at 1 = 1 Minimum at = (1 , , 2 , 3 ) 0.08 0.075 = (1.0000, 0.9011, 0.5196) 0.07 0.065 0.06 with (1) e(1) ( ) = 0.03520(1) e 0.055 0.05 0.045 0.04 1.2 1 loop result for HYP2 1 1.2 0.8 1 0.6 0.8 0.4 0.6 0.4 e(1) (HYP2 ) = 0.03544(1) 0.2 0.2 0 0 0.2 0.2 3 2 Author Short Paper Title
- 46. Introduction The static-light current on the lattice The static-light current in the Lattice SF Determination of cA (1) stat Minimisation of the self energy SummarySummary Theoretical topic of PT in the SF was reviewed Feynman rules were extended to HYP smearing (1) stat cA HYPx at 1-loop order was determined The self-energy was minimised w.r.t. the HYP parameters Author Short Paper Title
- 47. Introduction The static-light current on the lattice The static-light current in the Lattice SF Determination of cA (1) stat Minimisation of the self energy SummarySummary Theoretical topic of PT in the SF was reviewed Feynman rules were extended to HYP smearing (1) stat cA HYPx at 1-loop order was determined The self-energy was minimised w.r.t. the HYP parameters Author Short Paper Title
- 48. Introduction The static-light current on the lattice The static-light current in the Lattice SF Determination of cA (1) stat Minimisation of the self energy SummarySummary Theoretical topic of PT in the SF was reviewed Feynman rules were extended to HYP smearing (1) stat cA HYPx at 1-loop order was determined The self-energy was minimised w.r.t. the HYP parameters Author Short Paper Title
- 49. Introduction The static-light current on the lattice The static-light current in the Lattice SF Determination of cA (1) stat Minimisation of the self energy SummarySummary Theoretical topic of PT in the SF was reviewed Feynman rules were extended to HYP smearing (1) stat cA HYPx at 1-loop order was determined The self-energy was minimised w.r.t. the HYP parameters Author Short Paper Title

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