CSIS thesis Alois Grimbach

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 CSiS Improvement of the static-light axial current on the lattice Alois Grimbach Institut fuer Theoretische Physik Bergische Universität 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 Summary Outline 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 Schrödinger 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 Summary Introduction 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 Summary Actions Gauge Action Gauge Action Action consists of gauge action and fermionic action S = SG [U] + SF [U, Ψ, Ψ] Gauge links Uµ (x) - connect x with x + aˆ µ - 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 Summary Actions Fermion Action Two theories for light and static quarks Light quarks Fermionic Action for light quarks ¯ ¯ Sl [ψl , ψl ] = a4 Ψl (x)(D + m0 )Ψl (x) x D is Wilson-Dirac operator 1 D = 2 γµ (∇∗ + ∇µ ) − a∇∗ ∇µ µ µ 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 Summary Actions Fermion Action Two theories for light and static quarks Light quarks Fermionic Action for light quarks ¯ ¯ Sl [ψl , ψl ] = a4 Ψl (x)(D + m0 )Ψl (x) x D is Wilson-Dirac operator 1 D = 2 γµ (∇∗ + ∇µ ) − a∇∗ ∇µ µ µ 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 Summary Actions Fermionic 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 ﬁelds Sh [ψh , ψ h ] = a4 x ψ h (x)∇∗ ψh (x) 0 Sh [ψ—, ψ¯ ] = −a4 x ψ¯ (x)∇0 ψ—(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 Summary Actions Fermionic 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 ﬁelds Sh [ψh , ψ h ] = a4 x ψ h (x)∇∗ ψh (x) 0 Sh [ψ—, ψ¯ ] = −a4 x ψ¯ (x)∇0 ψ—(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 Summary Static-light Axial Current Axial Current Isovector Axial Current for SU(2) isospin Aα (x) = Ψ(x)γµ γ5 1 τ α Ψ(x) µ 2 Static-light Axial Current - is deﬁned by Astat = Ψl (x)γ0 γ5 Ψh (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 Summary Static-light Axial Current Axial Current Isovector Axial Current for SU(2) isospin Aα (x) = Ψ(x)γµ γ5 1 τ α Ψ(x) µ 2 Static-light Axial Current - is deﬁned by Astat = Ψl (x)γ0 γ5 Ψh (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 Summary O(a) improvement Symanzik 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 ﬁeld 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 Summary O(a) improvement Action 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 ← ← − − δAstat = Ψl γj γ5 1 ( ∇ j + ∇ ∗ )Ψh 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 Summary Smearing techniques - APE consider gauge links APE smearing APE smearing - decorate the gauge link with staples - parameter α 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 Summary Smearing 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 α1 , α2 , α3 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 Summary HYP smearing - Properties - preserves locality - improves signal-to-noise ratio - origin: reduction of static self-energy stat cA for HYP smeared action stat - estimated values for cA known from hybrid methods 4 - error ∝ (O)(g0 ), but unknown sought 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 Schrödinger 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 Tool: The Schrödinger Functional (SF) The SF - sketch P− ψ(x)|x0 =T = ρ′ (x) ′ Uk (x)|x0 =T = Wk (x) Uµ (x)|x0 >T = 1 ψ(x)|x0 >T = 0 x0 = T x0 = 0 Uµ (x)|x0 <0 = 1 ψ(x)|x0 <0 = 0 P+ ψ(x)|x0 =0 = ρ(x) Author Short Paper Title Uk (x)|x0 =0 = Wk (x)

31. Introduction The static-light current on the lattice The Schrödinger 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 O(a) improvement in the SF The Schrödinger Functional - Dirichlet boundary conditions for fermionic ﬁelds at x0 = 0 and x0 = T - PBC in spatial directions described by a phase shift Θk O(a) improvement in the SF - contains an additionally boundary term for the light action: Wilson Dirac operator in the SF δD = δDV + δDb - static quark action does not contain boundary term due to EOM - static axial current does not contain a boundary term stat (0) - free theory is already O(a) improved → cA =0 Author Short Paper Title

32. Introduction The static-light current on the lattice The Schrödinger 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 O(a) improvement in the SF The Schrödinger Functional - Dirichlet boundary conditions for fermionic ﬁelds at x0 = 0 and x0 = T - PBC in spatial directions described by a phase shift Θk O(a) improvement in the SF - contains an additionally boundary term for the light action: Wilson Dirac operator in the SF δD = δDV + δDb - static quark action does not contain boundary term due to EOM - static axial current does not contain a boundary term stat (0) - free theory is already O(a) improved → cA =0 Author Short Paper Title

33. Introduction The static-light current on the lattice The Schrödinger 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 Correlation functions in the SF expectation value of operator O 1 < O >= Z fields Oe−S - integrate over fermionic and gluonic fields fermionic fields - compute fermionic fields analytically - correlation functions can be reduced to basic correlation functions for light and static quarks by Wick contraction gluonic fields - gluonic fields can be evaluated in pertubation theory Author Short Paper Title

34. Introduction The static-light current on the lattice The Schrödinger 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 Pertubation Theory in the SF-1 Pertubation theory - approach - describe link variable by gauge vector ﬁeld qµ (x) Uµ (x) = exp(g0 aqµ (x)) - expand in terms of coupling constant g0 correlation functions deﬁne correlation functions fA (x0 ) = −a6 stat 1 stat ¯ 2 A0 (x)ζh (y)γ5 ζl (z) and y,z 1 12 ¯ ¯ stat f1 = −2 a6 L < ζl′ (u)γ5 ζh (v)ζh (y)γ5 ζ( z) > ′ u,v,y,z - expand them in pertubation theory Author Short Paper Title

35. Introduction The static-light current on the lattice The Schrödinger 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 Pertubation Theory in the SF-1 Pertubation theory - approach - describe link variable by gauge vector ﬁeld qµ (x) Uµ (x) = exp(g0 aqµ (x)) - expand in terms of coupling constant g0 correlation functions deﬁne correlation functions fA (x0 ) = −a6 stat 1 stat ¯ 2 A0 (x)ζh (y)γ5 ζl (z) and y,z 1 12 ¯ ¯ stat f1 = −2 a6 L < ζl′ (u)γ5 ζh (v)ζh (y)γ5 ζ( z) > ′ u,v,y,z - expand them in pertubation theory Author Short Paper Title

36. Introduction The static-light current on the lattice The Schrödinger 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 stat Feynman Diagrams for fA at one-loop order setting-sun tadpoles gluon exchange Author Short Paper Title

37. Introduction The static-light current on the lattice The Schrödinger 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 Feynman Diagrams for f1stat at one-loop order setting-sun tadpoles gluon exchange Author Short Paper Title

38. Introduction The static-light current on the lattice The Schrödinger 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 HYP 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) ˜ Bµ (p) = ν fµν (p)qν (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 Schrödinger 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 HYP links in the SF - 2 result ˜ (3) B0 (x0 ; p) = 6 ˜ h0;i (p)qµ(i) (x0 + as(i); p) i=0 with i µH (i) sH (i) h0;i (p) α1 3 0 0 0 1− 6 k =1 a2 pk Ω0k (p) ˆ2 1,2,3 i 0 + iα1 api Ω0i (p) 6 ˆ 4,5,6 i −3 1 − iα1 apµ(i) Ω0µ(i) (p) 6 ˆ - 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 Summary Determination of cA (1) stat stat fA L take the ratio X (g0 , a , T , Θ) = √ stat L f1 stat (1) cA can be extracted eliminates divergent part δm 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 LXδA (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 Summary Determination of cA (1) stat stat fA L take the ratio X (g0 , a , T , Θ) = √ stat L f1 stat (1) cA can be extracted eliminates divergent part δm 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 LXδA (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 Summary Determination 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 Summary Results 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 Summary smearing parameters - The self-energy has a functional dependence upon the smearing parameters, i.e. 2 (1) k k k e(1) = ek1 k2 k3 α11 α22 α33 k 1,k 2,k 3=0 - coefﬁcients can be determined out of the of the Feynman diagrams - coefﬁcients have a triangular structure, only for 0 ≤ k3 ≤ k2 ≤ k1 ≤ 2 non-zero - Results are align with the one-loop expansion of the static self-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 Summary Function of e(1) w.r.t the smearing parameters 3D plot of e(1) at α1 = 1 Minimum at α∗ = (α1 , , α2 , α3 ) ∗ ∗ ∗ 0.08 0.075 = (1.0000, 0.9011, 0.5196) 0.07 0.065 0.06 with (1) e(1) (α∗ ) = 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) (αHYP2 ) = 0.03544(1) 0.2 0.2 0 0 −0.2 −0.2 α3 α 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 Summary Summary 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

CSIS thesis Alois Grimbach

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