Topological Charge in Lattice QCD and the Chiral Condensate

Transcription

1 Topological Charge in Lattice QCD and the Chiral Condensate Andreas Athenodorou The Cyprus Institute May 31, 218 Gauge Topology 3, ECT* Trento, Italy data

2 Topological Charge in Lattice QCD and the Chiral Condensate Based on the articles: Topological susceptibility from twisted mass fermions using spectral projectors and the gradient flow, [arxiv: ]. Comparison of topological charge definitions in Lattice QCD, [arxiv: ]. Topological charge using cooling and the gradient flow, [arxiv: ]. Gluon Green functions free of Quantum fluctuation, [arxiv: ]. Neutron electric dipole moment using N f = twisted mass fermions, [arxiv: ]. Work in progress with P. Boucaud, F. De Soto, J. Rodríguez-Quinterio and S. Zafeiropoulos In collaboration with : Constantia Alexandrou (University of Cyprus, The Cyprus Institute) Philippe Boucaud (CNRS, Orsay) Krzysztof Cichy (Goethe-Universität Frankfurt am Main, Adam Mickiewicz University) Martha Constantinou (Temple University) Feliciano De Soto (University Pablo de Olavide) Arthur Dromard (Regensburg University) Derek P. Horkel (Temple University) Karl Jansen (NIC, DESY Zeuthen) Giannis Koutsou (The Cyprus Institute) José Rodríguez-Quinterio (Universidad de Huelva) Urs Wenger (University of Bern) Falk Zimmermann (Universität Bonn) Savvas Zafeiropoulos (University of Heidelberg)

3 Overview of Topics Why Studying Topological Charge The Neutron Electric Dipole Moment depends on the Topological Charge Instanton dependence of α MOM (k) (Feliciano s presentation) Interested in the QCD Vaccuum There is not only one unique way of extracting the topological charge (corrections of O(a)) Each different method on the Lattice is accompanied with pros and cons To better understand the different methods: Comparison of Topological charge definitions which belong to the two categories g Gluonic f Fermionic Comparison of Topological Susceptibility: gluonic Vs. fermionic Pion mass dependence of the Top. Susceptibility provides The Chiral Condensate What we learn? Is there Universality?

4 Investigation Several definitions of the topological charge: f fermionic (Index, Spectral flow, Spectral Projectors). g gluonic with UV fluctuations removed via smoothing (gradient flow, cooling, smearing,...).? How are these definitions numerically related? Gradient Flow is a well defined smoothing scheme with good renormalizability properties. M. Lüscher [arxiv: ]! Gradient Flow is numerically equivalent to cooling! C. Bonati and M. D Elia [arxiv: ] and C. Alexandrou, AA and K. Jansen, [arxiv: ]? Can this be applied to other smoothing schemes? C. Alexandrou, el al, [arxiv: ] Comparison of different definitions presented by Krzysztof Cichy et. al... K. Cichy et. al, [arxiv: ]! Most definitions are highly correlated.! The topological susceptibilities are in the same region. Not meaningfull comparison.

5 Gluonic Definition of the Topological Charge g Topological charge can be defined as: Q = d 4 xq(x), with q(x) = 1 32π 2ǫ µνρσtr{f µν F ρσ }. g Discretizations of q(x) on the lattice: Plaquette (leading correction term of O(a 2 )) q plaq L (x) = 1 ( ) 32π 2ǫ µνρσtr Cµν plaq Cplaq ρσ Clover (leading correction term of O(a 2 )) ql clov (x) = 1 ( ) 32π 2ǫ µνρσtr Cµν clov Cclov ρσ, with C plaq µν (x) = Im ( ), with C clov µν (x) = 1 4 Im ( ).. Improved (leading correction term of O(a 4 )) q imp L (x) = b q clov L (x)+b 12qL rect (x), with Crect µν (x) = 1 8 Im +. g Smoothing Remove the ultraviolet fluctuations

6 Gluonic Definition of the Topological Charge g Topological charge can be defined as: Q = d 4 xq(x), with q(x) = 1 32π 2ǫ µνρσtr{f µν F ρσ }. g Discretizations of q(x) on the lattice: Plaquette (leading correction term of O(a 2 )) q plaq L (x) = 1 ( ) 32π 2ǫ µνρσtr Cµν plaq Cplaq ρσ Clover (leading correction term of O(a 2 )) ql clov (x) = 1 ( ) 32π 2ǫ µνρσtr Cµν clov Cclov ρσ, with C plaq µν (x) = Im ( ), with C clov µν (x) = 1 4 Im ( ).. Improved (leading correction term of O(a 4 )) q imp L (x) = b q clov L (x)+b 12qL rect (x), with Crect µν (x) = 1 8 Im +. g Smoothing Remove the ultraviolet fluctuations

7 Smoothing Schemes for Gluonic Topological Charge g The Gradient Flow M. Lüscher, JHEP 18 (21) 71 g Cooling. M. Teper, Phys. Lett. B 162 (1985) 357. g APE smearing. M. Albanese et al. [APE Coll.], Phys. Lett. B 192 (1987) 163. g Stout smearing. C. Morningstar and M. J. Peardon, Phys. Rev. D69 (24) 5451 g HYP smearing. A. Hasenfratz and F. Knechtli, Phys. Rev. D64 (21) 3454 g Several other modified versions of smearing. Credits to Ed Bennett (213 Royal Society Award) Finding needles in four-dimensional haystacks

8 Smoothing Schemes for Gluonic Topological Charge g The Gradient Flow M. Lüscher, JHEP 18 (21) 71 g Cooling. M. Teper, Phys. Lett. B 162 (1985) 357. g APE smearing. M. Albanese et al. [APE Coll.], Phys. Lett. B 192 (1987) 163. g Stout smearing. C. Morningstar and M. J. Peardon, Phys. Rev. D69 (24) 5451 g HYP smearing. A. Hasenfratz and F. Knechtli, Phys. Rev. D64 (21) 3454 g Several other modified versions of smearing. C.f De Soto s talk

9 The Flagship smoother: The Gradient Flow Defined by a local diffusion equation that evolves the Gauge field as a function of the flow time Gauge field generated by the gradient flow does not require renormalization Solution of the evolution equations: V µ (x,τ) = g 2 [ x,µs G (V(τ))]V µ (x,τ) With link derivative defined as: x,µ S G (U) = i a The flown fields are given by: B µ (τ,x) = V µ (x,) = U µ (x), T a d ( ds S G e isy a ) U (x y) 2 d 4 y e 4τ (4πτ) 2 A µ(x) Iterative process for gradient flow time τ One can define a reference flow time t such that t 2 E(t) t=t =.3 s= i a T a (a) x,µ S G(U), smoothing radius : 8τ with t = a 2 τ and E(t) = 1 2V Tr{F µν (x,t)f µν (x,t)} x

10 The Flagship smoother: The Gradient Flow Gradient flow depends on the detais of the Smoothing Action (c +8c 1 = 1) : S g = β N Wilson: c 1 =, ( x c 4 µ,ν=1 1 µ<ν { 1 ReTr(U 1 1 x,µ,ν ) } +c 1 4 Symanzik tree-level improved: c 1 = 1/12, Iwasaki: c 1 =.331, µ,ν=1 µ ν ) { 1 ReTr(U 1 2 } x,µ,ν ), Different Actions are expected to lead to different Instanton (with scale λ) behaviour { S Lat (a,λ) = S cont 1+(a/λ) 2 a 2 +(a/λ) 4 a 4 +O(a/λ) 6} Wilson: a 2 = 1/5 Symanzik tree-level improved: a 2 =,a 4 = 17/21 Iwasaki: a 2 = /5 Stable instanton solutions require a lattice action which increases by decreasing the scale parameter λ Only for Iwasaki we expect stable solutions However, the most-frequently used action for smoothing is the Wilson

11 The Flagship smoother: The Gradient Flow Gradient Flow provides a concrete theoretical framework for smoothing The evolution of the fields is processing in gradient flow time steps ǫ (typically ǫ.1) This means that it requires many iterations to reach a given gradient flow time (τ = n int ǫ) Other theoretically weaker smoothing techniques provide similar results? how fast are they?

12 Gradient Flow Solution of the evolution equations: The Wilson flow Vs. Cooling V µ (x,τ) = g 2 [ x,µs G (V(τ))]V µ (x,τ) V µ (x,) = U µ (x), With link derivative defined as: x,µ S G (U) = i a i a T a d ( ds S G e isy a ) U T a (a) x,µ S G(U), s= Total gradient flow time: τ Cooling Cooling U µ (x) SU(N): U old µ (x) U new µ (x) with Choose a U new µ (x) that maximizes: One full cooling iteration n c = 1 P(U) e (lim β β 1 N ReTrX µ Uµ). ReTr{U new µ (x)x µ (x)}.

13 Perturbative expansion of links A link variable which has been smoothed can been written as: U µ (x,j sm ) 1+i a u a µ(x,j sm )T a. Simple staples are written as: per space-time slice, thus: X µ (x,j sm ) 6 1+i a w a µ(x,j sm )T a. For the Wilson flow with Ω µ (x) = U µ (x)x µ(x) g 2 x,µs G (U)(x) = 1 (Ω µ (x) Ω µ ) 2 (x) 1 ) (Ω 6 Tr µ (x) Ω µ (x). where g 2 x,µs G (U) i a [ 6u a µ (x,τ) w a µ (x,τ)] T a.

14 Wilson flow Vs. Cooling Evolution of the Wilson flow by an infinitesimally small flow time ǫ: u a µ (x,τ +ǫ) ua µ (x,τ) ǫ[ 6u a µ (x,τ) wa µ (x,τ)]. where U µ (x,τ +ǫ) 1+i a ua µ(x,τ +ǫ)t a After a cooling step: u a µ(x,n c +1) wa µ (x,n c) 6 Wilson flow would evolve the same as cooling if ǫ = 1/6. + Cooling has an additional speed up of two.! As we saw before, cooling has the same effect as the Wilson flow if: τ n c 3. C. Bonati and M. D Elia, Phys. Rev. D89 (214), 155 [arxiv: ] We generalized of this result for smoothing actions with rectangular terms (b 1 ): n c τ. 3 15b 1 C. Alexandrou, AA and K. Jansen, Phys. Rev. D92 (215), [arxiv: ] What happens for other smoothing schemes??.

15 Gradient flow Vs. Cooling Test the matching conditions for N f = 2 twisted mass fermions Produced with the Iwasaki Gluonic Action Lattice spacing a.85fm (r /a = 5.35(4)) Pion Mass m π 34 MeV Lattice size L/a = 16 N f = twisted mass fermions, Produced with the Iwasaki Gluonic Action Lattice spacing a =.823(1) fm (r /a = 5.71(41)) Pion Mass m π 37 MeV Lattice size L/a = 32 N f = Pure Gauge, Produced with the Symanzik tree-level improved Lattice spacing a =.285(1) fm Lattice size L/a = 32

16 Gradient flow Vs. Cooling Matching condition: τ n c 3. Define function τ(n c ) such as τ and n c change the action by the same amount. 2 τ(n c ) τ = n c /3.1 Wilson Flow Cooling 15 τ 1 Average Action Density n c τ or n c /3

17 Topological Charge: Wilson flow Vs. Cooling For smoothing actions with rectangular terms: n c τ. 3 15b 1 Symanzik tree-level: n c = 4.25τ Iwasaki: n c = 7.965τ 1 gradient flow cooling 1 gradient flow cooling.1.1 SG.1 SG e-5.1 Symanzik tree-level 1 n c or 4.25 τ 1 1e-5.1 Iwasaki 1 n c or τ 1

18 Wilson flow Vs. APE According to the APE operation: [ U (n APE+1) µ (x) = Proj SU(3) (1 α APE )U (n APE) µ (x)+ α ] APE X (n APE) µ (x) 6. Evolution of the Wilson flow by an infinitesimally small flow time ǫ is expressed as: u a µ(x,τ +ǫ) u a µ(x,τ) ǫ [ 6u a µ(x,τ) w a µ(x,τ) ]. Evolution of the APE smearing with parameter α APE is expressed as: u a µ (x,n APE +1) u a µ (x,n APE) α APE 6 [ 6u a µ (x,n APE ) w a µ (x,n APE) ]. Hence, APE has the same effect as the Wilson flow if: τ α APE 6 n APE.

19 Wilson flow Vs. APE Matching condition: τ α APE 6 n APE. Define function τ(α APE,n APE ) such as τ and n APE changes action by the same amount τ τ(.6,n APE ) τ(.5,n APE ) τ(.4,n APE ).6 6 n APE.5 6 n APE.4 6 n APE Average Action Density.1.1 Wilson Flow APE with α APE =.4 APE with α APE =.5 APE with α APE = n APE τ or α APE 6 n APE

20 Wilson flow Vs. stout According to the stout smearing operation: with Q µ (x) = i 2 U (n st+1) µ (x) = exp ( iq n st µ (x) ) U (n st) µ (x). ( Ξ µ(x) Ξ µ (x) ) i ( ) 6 Tr Ξ µ(x) Ξ µ (x), with Ξ µ (x) = ρ st X µ (x)u µ(x) Evolution of the Wilson flow by an infinitesimally small flow time ǫ is expressed as: u a µ (x,τ +ǫ) ua µ (x,τ) ǫ[ 6u a µ (x,τ) wa µ (x,τ)]. Evolution of the stout smearing with parameter ρ st is expressed as: u a µ(x,n st +1) u a µ(x,n st ) ρ st [ 6u a µ (x,n st ) w a µ(x,n st ) ]. Hence, stout smearing has the same effect as the Wilson flow if τ ρ st n st.

21 Wilson flow Vs. stout Matching condition: τ ρ st n st. Define function τ(ρ st,n st ) such as τ and n st changes action by the same amount τ τ(.1,n st ) τ(.5,n st ) τ(.1,n st ).1 n st.5 n st.1 n st Average Action Density.1.1 Wilson Flow Stout with ρ =.1 Stout with ρ =.5 Stout with ρ = n st τ or ρ st n st

22 Wilson flow Vs. HYP We considered HYP smearing with parameters: α HYP1 =.75, α HYP2 =.6 α HYP3 =.3 HYP staples not the same as X µ (x) (A. Hasenfratz and F. Knechtli, Phys. Rev. D64 (21) 3454). a) b) Define function τ HYP (n HYP ) and fit using the ansatz: τ HYP (n HYP ) = A n HYP +B n 2 HYP +C n3 HYP. with A =.25447(32), B =.1312(9), C = 1.217(91) 1 5 Hence, HYP smearing has the same effect as the Wilson flow if τ τ HYP (n HYP ).

23 Wilson flow Vs. HYP We considered HYP smearing with parameters: α HYP1 =.75, α HYP2 =.6 α HYP3 =.3 HYP staples not the same as X µ (x) (A. Hasenfratz and F. Knechtli, Phys. Rev. D64 (21) 3454). a) b) Define function τ HYP (n HYP ) and fit using the ansatz: τ HYP (n HYP ) = A n HYP +B n 2 HYP +C n3 HYP. with A =.25447(32), B =.1312(9), C = 1.217(91) 1 5 Hence, HYP smearing has the same effect as the Wilson flow if τ τ HYP (n HYP ).

24 Wilson flow Vs. HYP Numerical matching condition: τ HYP (n HYP ) = A n HYP +B n 2 HYP +C n3 HYP. Define function τ(n HYP ) such as τ and n HYP changes action by the same amount τ(n HYP ).245 n HYP τ HYP (n HYP ).1 Wilson Flow HYP smearing τ Average Action Density n HYP τ or τ HYP (n HYP )

25 Topological Susceptibility - The Wilson flow The Wilson flow time t 2.5a 2.5 χ = Q2 V.45 rχ 1/ Gradient flow 8 1 τ

26 Topological Susceptibility - Cooling The Wilson flow time t 2.5a 2 n c = 7.5 cooling steps.5 χ = Q2 V.45 rχ 1/ Gradient flow Cooling 8 1 τ or n c /3

27 Topological Susceptibility - APE The Wilson flow time t 2.5a 2 n APE = 37.5 APE smearing steps for α APE =.4.5 χ = Q2 V.45 rχ 1/ Gradient flow APE smearing α APE = τ or α APE n APE /6

28 Topological Susceptibility - APE The Wilson flow time t 2.5a 2 n APE = 3 APE smearing steps for α APE =.5.5 χ = Q2 V.45 rχ 1/ Gradient flow APE smearing α APE =.4 APE smearing α APE = τ or α APE n APE /6

29 Topological Susceptibility - APE The Wilson flow time t 2.5a 2 n APE = 25 APE smearing steps for α APE =.6.5 χ = Q2 V.45 rχ 1/ Gradient flow APE smearing α APE =.4 APE smearing α APE =.5 APE smearing α APE = τ or α APE n APE /6

30 Topological Susceptibility - stout The Wilson flow time t 2.5a 2 n st = 25 stout smearing steps for ρ st =.1.5 χ = Q2 V.45 rχ 1/ Gradient flow stout smearing ρ st = τ or ρ st n st

31 Topological Susceptibility - stout The Wilson flow time t 2.5a 2 n st = 5 stout smearing steps for ρ st =.5.5 χ = Q2 V.45 rχ 1/ Gradient flow stout smearing ρ st =.1 stout smearing ρ st = τ or ρ st n st

32 Topological Susceptibility - stout The Wilson flow time t 2.5a 2 n st = 25 stout smearing steps for ρ st =.1.5 χ = Q2 V.45 rχ 1/ Gradient flow stout smearing ρ st =.1 stout smearing ρ st =.5 stout smearing ρ st = τ or ρ st n st

33 Topological Susceptibility - HYP The Wilson flow time t 2.5a 2 n st = 1 HYP smearing steps.5 χ = Q2 V.45 rχ 1/ Gradient flow HYP smearing 8 1 τ or τ HYP

34 Correlation between different smoothers Let us have a look at the correlation coefficient c Q1,Q 2 = ( Q1 Q 1 )( Q2 Q 2 ) (Q1 Q 1 ) 2 (Q2 Q 2 ) 2. WF, t cool, t APE, t stout, t HYP, t WF, t 1.().97() 1.() 1.().97() cool, t.97() 1.().97().97().94() APE, t 1.().97() 1.() 1.().97() stout, t 1.().97() 1.() 1.().97() HYP, t.97().94().97().97() 1.() Topological charges are highly correlated! In the continuum all numbers become 1.

35 Fixing the smoothing scale One can fix a physical flow time: λ S 8t. Similarly for cooling 12 faster than Gradient Flow: 8nc λ S a 3. For the APE smearing 2 faster than Gradient Flow: 4αAPE n APE λ S a. 3 For the stout smearing 3 faster than Gradient Flow λ S a 8ρ st n st. Similar procedure can be applied to t.

36 Level of agreement on the topological charge Why topological susceptibility has such a high level of agreement? Q Wilson flow cooling HYP APE with α APE =.5 stout with ρ st = τ

37 Fermionic Definitions of the Topological Charge f Index definition with different steps of HYP smearing. M. F. Atiyah and I. M. Singer, Annals Math. 93 (1971) f Spectral-flow with different steps of HYP smearing. S. Itoh, Y. Iwasaki and T. Yoshie, Phys. Rev. D 36 (1987) 527 f Spectral projectors with different cutoffs M 2. L. Giusti and M. Lüscher, JHEP 93 (29) 13 and M. Lüscher and F. Palombi, JHEP 19 (21) 11

38 Index definition Chiral Symmetry can be realized on the Lattice! Modified definition of chiral symmetry on the lattice Dirac operator satisfies the Ginsparg-Wilson relation is chirally symmetric γ 5 D +Dγ 5 = aγ 5 Dγ 5 Overlap Dirac operator, introduced by Neuberger H. Neuberger, Phys. Lett. B427 (1998) , [hep-lat/98131]. Atiyah-Singer index theorem: relates the number of Zero modes to topological charge Q = n n + It gives integer values of Q. Massless Overlap Operator D = 1 ( 1 A ) a A A, A = 1+s ad W, s can be tuned to optimize the locality of the Overlap operator In the continuum limit the dependence on s vanishes Several orders of magnitude slower than Gradient Flow

39 Spectral Projectors L. Giusti and M. Lüscher, [arxiv: ] & M. Lüscher and F. Palombi, [arxiv:18.732]. Introduce the projector P M to the subspace of eigenmodes of the Hermitian Dirac operator D D with eigenvalues below M 2. P M can be calculated stochastically vs. explicit computation of eigenmodes pros. For explicit computation of eigenmodes, comp. cost drops from O(V 2 ) to O(V) cons. Introducing stochastic noise To avoid the stochastic noise we opted for an explicit computation of eigenmode. The bare topological charge is given by The topological susceptibility Q = λ i <M 2 With renormalized quantities ( ) ZS Q = Q, χ = Z P i R i, R i = u i γ 5u i χ = Q2 V. ( ZS Z P ) 2 χ, M = Z 1 P M.

40 Spectral Projectors For chirally symmetric fermions Q = Tr{γ 5 P M }. This definition is then equivalent to the index definition Wilson twisted mass fermions lead to a shift of O(a 2 ) Results depend on the renormalized spectral threashold M 2 The 4 lowest eigenmodes allow M up to 16 MeV for all the ensembles Investigate the dependence in M M should not be too small because of large cutoff effects M should not be too large because of enhanced noise

41 Comparison of topological Susceptibility Comparison of results for the topological susceptibility..1.8 nosmear s=.4 nosmear s= HYP1 s= HYP1 s= HYP1 s=.75 HYP5 s= HYP5 s=.5 M 2 =.1 M 2 =.4 M 2 =.15 nosmear tlsym Wplaq t 3t t 3t Iwa t 3t stout t 3t cool Wplaq t 3t APE t 3t HYP t 3t a χ 1/4.6 M 2 = index of overlap spectral flow spectral projectors Using N f = 2 twisted mass configuration with: cft nosmear cft GF cft stout definition cft cooling cft APE cft HYP β = 3.9, a.85fm, r /a = 5.35(4), m π 34 MeV, m π L = 2.5, L/a = 16

42 Correlation Coefficient Comparison of the correlation coefficient between fermionic and gluonic definitions. cft HYP (GF Wplaq t) 16 cft APE.5 (GF Wplaq t) 15 cft stout.1 (GF Wplaq t) 14 cft cool (GF Iwa t) definition 2 cft cool (GF tlsym t) 12 cft cool (GF Wplaq t) 11 cft GF Iwa t 1 cft GF tlsym t 9 cft GF Wplaq t 8 cft nonsmear 7 spec. proj. M 2 =.1 6 spec. proj. M 2 =.4 5 SF HYP5 s=. 4 SF HYP1 s=. 3 index HYP1 s= index nonsmear s= definition 1

43 Fermionic Vs. Gluonic - Continuum limit Correlation for a fermionic and gluonic definitions as we approach the continuum limit index with HYP1, s= vs. the Wilson flow at t correlation a/r

44 Comparison in Neutron Electric Dipole Moment The Electric Dipole Moment is given by d N = lim q θ F 3(q 2 ) 2m N. From Nucleon-Nucleon matrix element of the electromagnetic current J em µ we obtain CP-odd electromagnetic Form-Factor F 3 (Q 2 ): N( p f,s f ) J em µ N( p i,s i ) = ū( p f,s f ) [ +θ F 3(Q 2 ] ) Q ν σ µν γ 5 + u( p i,s i ). 2m N To extract the CP-odd Form Factor we need to calculate J N ( p f,t f )J em µ ( q,t)j N( p i,t i )Q. = f(f 3,α 1 ) The phase α 1 appears due to the CP-breaking in the θ-vacuum.

45 Comparison in Neutron Electric Dipole Moment The phase α 1 (blue: cooling, red: Gradient Flow): -.5 Wilson Symanzik tree-level Iwasaki α n c or 3 τ n c or 4.25 τ n c or τ The CP-odd form factor: F3()/2mN [e fm] Wilson Symanzik Iwasaki

46 From Feliciano De Soto s presentation: Comparison in α mom (k) after smoothing

47 What we learned? Topological susceptibilities are in the same ballpark: aχ 1/4 [.8,.9]. Correlation coefficient appears to increase towards to 1 as a. Different definitions influenced by different lattice artifacts. Most correlation coefficients are above 8 %. Cooling, APE smearing, stout smearing are numerically equivalent if matched: n APE τ n c 3, τ α APE τ ρ st n st faster, 2 faster 3 faster.

48 What we learned? Topological susceptibilities are in the same ballpark: aχ 1/4 [.8,.9]. Correlation coefficient appears to increase towards to 1 as a. Different definitions influenced by different lattice artifacts. Most correlation coefficients are above 8 %. Cooling, APE smearing, stout smearing are numerically equivalent if matched: n APE τ n c 3, τ α APE τ ρ st n st faster, 2 faster 3 faster.

49 Topological Susceptibility - What we want to learn Topological susceptibilities are in the same ballpark: aχ 1/4 [.8,.9]. Different definitions influenced by different lattice artifacts. Comparison between Fermionic Definition Gradient flow Do both approaches lead to compatible results? Pion mass dependence Universality of Topological Susceptibility Universality in Pure SU(3) M. Lüscher and F. Palombi, JHEP 19 (21) 11 This has never been demonstrated before for Full QCD. Work with N f = twisted mass fermions with β a [fm] Z P Z P /Z S r /a (36).529(9).699(13) 5.231(38) (3).54(5).697(7) 5.71(41) (18).514(3).74(5) 7.538(58) This leads to the extraction of the Chiral Condensate

50 Topological Susceptibility - What we want to learn Topological susceptibilities are in the same ballpark: aχ 1/4 [.8,.9]. Different definitions influenced by different lattice artifacts. Comparison between Fermionic Definition Spectral Projectors Gradient flow Do both approaches lead to compatible results? Pion mass dependence Universality of Topological Susceptibility Universality in Pure SU(3) M. Lüscher and F. Palombi, JHEP 19 (21) 11 This has never been demonstrated before for Full QCD. Work with N f = twisted mass fermions with β a [fm] Z P Z P /Z S r /a (36).529(9).699(13) 5.231(38) (3).54(5).697(7) 5.71(41) (18).514(3).74(5) 7.538(58) This leads to the extraction of the Chiral Condensate

51 Topological Susceptibility in Chiral Perturbation Theory We consider the partition function of QCD with non-zero θ (Mao and Chiu, ): Z(θ) = DA µ DψDψ e is {QCD,θ=} iθq[u]. Q Topological Susceptibility is the second derivative of the energy density at θ = χ = 2 ε(θ) θ 2 = Q2 V, ε(θ) = log(z). V θ= The lowest order, two flavor χpt Lagrangian can be written as: L (2) χpt = f2 [ π 4 Tr µ U(x) µ U(x) ] ( [ ]) + ΣRe Tr MU(x), The partition function can be written in terms of L (2) χpt : Z(θ) = DUe i d 4 xl (2) (U(x),θ). Topological susceptibility yields to: χ = 2 ε(θ) θ 2 r r = Σ µ l 2, θ= 3-parameter fit (plus O(a 2 )) r 4 χ = [r3 Σ] r ( ) µ l,r a 2 [ ] ( ) α a 2 + [c] + r µ l,r, 2 r

52 r 4 χ = f(r µ l,r ) Cross sections for the three lattice spacings: a =.89 fm a =.82 fm a =.62 fm

53 Global fits r 4 χ = f(r µ l,r )

54 Chiral Condensate: Σ 1/3 = f(m) Comparison of Σ 1/3 for Gradient Flow and Spectral Projectors Σ 1/3 as a function of M

55 Total discretization error From the 3-parameters fit: r 4 χ = [r3 Σ] r µ l,r 2 ( ) a 2 [ ] α + [c] + r r r µ l,r ( a r ) 2, Total discretisation error: [ ] α [c] + r µ l,r, r Global fits for gradient Flow µ l,r = 12.6 MeV µ l,r = 18.5 MeV µ l,r = 26. MeV

56 Fit parameters M [MeV] Σ 1/3 [MeV] r Σ 1/3 c α/r N ens χ 2 /d.o.f (18).76(44).121(32) -4.7(1.4) (21).764(5).191(46) -4.6(1.8) (23).764(55).249(54) -4.3(2.) (3).764(73).291(6) -3.9(2.5) 6.66 GF 289(49).69(12) 1.15(12).7(3.8) Spectral projector method has larger mass dependent discretization effects Spectral projector method has smaller mass independent discretization effects Value of Chiral Condensate agrees between Gradient Flow and Spectral Projectors Universality is confirmed

57 χ(a ) LO Fit Alternative fit For each ensemble we fix the quark mass and extract χ We perform a linear fit of r 4 χ as a function of (a/r ) 2 using r 4 χ(a) = [K](a/r ) 2 +[r 4 χ(a = )], To calculate the chiral condensate, we fit the continuum values of r 4 χ(a = ) as a function of r µ l,r using r 4 χ(a = ) = [r3 Σ]r µ l 2.

58 r 4 χ = f(a 2 /r 2 ) Continuum extrapolations: µ l,r = 12.6 MeV µ l,r = 26. MeV µ l,r = 18.5 MeV

59 χ(a ) LO Fit Alternative fit For each ensemble we fix the quark mass and extract χ We perform a linear fit of r 4 χ as a function of (a/r ) 2 using r 4 χ(a) = [K](a/r ) 2 +[r 4 χ(a = )], To calculate the chiral condensate, we fit the continuum values of r 4 χ(a = ) as a function of r µ l,r using r 4 χ(a = ) = [r3 Σ]r µ l 2.

60 r 4 χ = f(µ l,r ) Continuum fit to the lowest order χpt expression for topological susceptibility.

61 χ(a ) LO Fit Values of the chiral condensate M [MeV] Σ 1/3 [MeV] r Σ 1/3 χ 2 /d.o.f (1).87(23) (11).815(26) (12).89(29).387 GF 38(26).74(62).171 Agreement between the Spectral Projectors and Gradient-Flow Total discretization error µ l,r = 12.6 MeV µ l,r = 18.5 MeV µ l,r = 26. MeV

62 What we learned by investigating the topological susceptibility? The topological susceptibility computed using spectral projectors has much smaller discretization effects than Gradient-Flow. Universality has been confirmed Spectral cutoff as small as M 3 MeV is sufficient for extracting fit quantities. We can minimize the cutoff effects by choosing the right M (depending on the quark mass) We use M = 12 MeV to quote r Σ 1/3 =.754(5) stat (26) sys, Σ 1/3 = 318(21) stat (11) sys MeV. This result is in agreement with other investigations (Banks-Casher relation) This is highly relevant for calculating topological charge dependent quantities when there is only one lattice spacing available and a continuum extrapolation cannot be performed.

63 What we learned by investigating the topological susceptibility? The topological susceptibility computed using spectral projectors has much smaller discretization effects than Gradient-Flow. Universality has been confirmed Spectral cutoff as small as M 3 MeV is sufficient for extracting fit quantities. We can minimize the cutoff effects by choosing the right M (depending on the quark mass) We use M = 12 MeV to quote r Σ 1/3 =.754(5) stat (26) sys, Σ 1/3 = 318(21) stat (11) sys MeV. This result is in agreement with other investigations (Banks-Casher relation) This is highly relevant for calculating topological charge dependent quantities when there is only one lattice spacing available and a continuum extrapolation cannot be performed.

64 THANK YOU!

65 The nedm: General Considerations The structure of the Standard Model Based on the role of the discrete symmetries C, P, T Possible experimental observation of nedm Flags violation of P and T Due to CPT symmetry violation of CP Best experimental bound: d N < e fm [C. A. Baker et al, Phys. Rev. Lett. 97, (26) [arxiv:hep-ex/622]]

66 The nedm: General Considerations The structure of the Standard Model Based on the role of the discrete symmetries C, P, T Possible experimental observation of nedm Flags violation of P and T Due to CPT symmetry violation of CP Best experimental bound: d N < e fm [C. A. Baker et al, Phys. Rev. Lett. 97, (26) [arxiv:hep-ex/622]]

67 The nedm: General Considerations Supersymmetric predictions: d N ( )e fm [M. Pospelov and A. Ritz, [hep-ph/54231]] Weak Interactions prediction: d N 1 19 e fm [S. Dar, [arxiv:hep-ph/8248]] Missing the neutron electric dipole moment resulting from QCD. Only from ab initio Lattice Calculations

68 The nedm: General Considerations Supersymmetric predictions: d N ( )e fm [M. Pospelov and A. Ritz, [hep-ph/54231]] Weak Interactions prediction: d N 1 19 e fm [S. Dar, [arxiv:hep-ph/8248]] Missing the neutron electric dipole moment resulting from QCD. Only from ab initio Lattice Calculations QCD d N /θ

69 The nedm: General Considerations Consider violation of CP in strong interactions... QCD Lagrangian density with no additional insertions: L QCD (x) = 1 2g 2Tr[G µν (x)g µν (x)]+ f is invariant under C, P and T transformations. Hence, it cannot induce a non-vanishing nedm. We need to insert the CP-violating Cherns-Simons (CS) term: L CS (x) iθ We, now, consider QFT with Lagrangian density: ψ f (x)(iγ µ D µ +m f )ψ f (x), 1 32π 2ǫ µνρσtr[g µν (x)g ρσ (x)]. L(x) = L QCD (x)+l CS (x). Model dependend studies as well as ChPT predictions: θ O ( ) d N θ O ( ) e fm.

70 The nedm: General Considerations Interaction between nucleon fields u N (x) and the electromagnetic field tensor F µν : 1 2 θf 3(q 2 ) 2m N u N (p f )σ µν γ 5 u N (p i )F µν. In the static limit (q 2 = (p f p i ) 2 ) the above gives the Electric Dipole Moment of the nucleon d N = lim θ F 3(q 2 ). q 2m N Hence, what needs to be done is to Extract CP-odd Form-Factor F 3 () for θ from expectation values such as: O(x 1,...,x n ) θ = 1 d[u]d[ψ f ]d[ ψ f ] O(x 1,...,x n ) e S QCD+iθ q(x)d 4x. Z θ How?? External Electric Field Method? Analytical Continuation to Imaginary θ? Extract F 3 () from perturbative expansion in θ: O(x 1,...,x n ) θ = O(x 1,...,x n ) θ= + O(x 1,...,x n ) ( iθ ) d 4 xq(x) θ= +O(θ 2 ).

71 The nedm: General Considerations Interaction between nucleon fields u N (x) and the electromagnetic field tensor F µν : 1 2 θf 3(q 2 ) 2m N u N (p f )σ µν γ 5 u N (p i )F µν. In the static limit (q 2 = (p f p i ) 2 ) the above gives the Electric Dipole Moment of the nucleon d N = lim θ F 3(q 2 ). q 2m N Hence, what needs to be done is to Extract CP-odd Form-Factor F 3 () for θ from expectation values such as: O(x 1,...,x n ) θ = 1 d[u]d[ψ f ]d[ ψ f ] O(x 1,...,x n ) e S QCD+iθ q(x)d 4x. Z θ How? X External Electric Field Method? Analytical Continuation to Imaginary θ? Extract F 3 () from perturbative expansion in θ: O(x 1,...,x n ) θ = O(x 1,...,x n ) θ= + O(x 1,...,x n ) ( iθ ) d 4 xq(x) θ= +O(θ 2 ).

72 The nedm: General Considerations Interaction between nucleon fields u N (x) and the electromagnetic field tensor F µν : 1 2 θf 3(q 2 ) 2m N u N (p f )σ µν γ 5 u N (p i )F µν. In the static limit (q 2 = (p f p i ) 2 ) the above gives the Electric Dipole Moment of the nucleon d N = lim θ F 3(q 2 ). q 2m N Hence, what needs to be done is to Extract CP-odd Form-Factor F 3 () for θ from expectation values such as: O(x 1,...,x n ) θ = 1 d[u]d[ψ f ]d[ ψ f ] O(x 1,...,x n ) e S QCD+iθ q(x)d 4x. Z θ How? X External Electric Field Method X Analytical Continuation to Imaginary θ? Extract F 3 () from perturbative expansion in θ: O(x 1,...,x n ) θ = O(x 1,...,x n ) θ= + O(x 1,...,x n ) ( iθ ) d 4 xq(x) θ= +O(θ 2 ).

73 The nedm: General Considerations Interaction between nucleon fields u N (x) and the electromagnetic field tensor F µν : 1 2 θf 3(q 2 ) 2m N u N (p f )σ µν γ 5 u N (p i )F µν. In the static limit (q 2 = (p f p i ) 2 ) the above gives the Electric Dipole Moment of the nucleon d N = lim θ F 3(q 2 ). q 2m N Hence, what needs to be done is to Extract CP-odd Form-Factor F 3 () for θ from expectation values such as: O(x 1,...,x n ) θ = 1 d[u]d[ψ f ]d[ ψ f ] O(x 1,...,x n ) e S QCD+iθ q(x)d 4x. Z θ How? X External Electric Field Method X Analytical Continuation to Imaginary θ Extract F 3 () from perturbative expansion in θ: O(x 1,...,x n ) θ = O(x 1,...,x n ) θ= + O(x 1,...,x n ) ( iθ ) d 4 xq(x) θ= +O(θ 2 ).

74 The nedm: General Considerations This is a first test on the applicability of the implemented methods! Our final goal is to Extract the nedm for physical pion mass continuum limit Extract the nedm on configurations produced with N f = twisted mass fermions, Produced with Iwasaki Gauge Action. Lattice spacing a.82 fm. L/a = 32. B β = 1.95, a =.823(1) fm aµ =.55, M π =.372 GeV r /a = 5.71(41) , L = 2.6 fm No. of confs = 4695

75 The nedm: Lattice Calculation From Nucleon-Nucleon matrix element of the electromagnetic current J em µ we obtain CP-odd electromagnetic Form-Factor F 3 (Q 2 ): N( p f,s f ) J em µ N( p i,s i ) = ū( p f,s f ) Assuming a theory where CP symmetry is violated... Problem: CP-odd Form-Factor Q k F 3 (Q 2 ) Parametrize F 3 (Q 2 ) in Q 2 and peform a Dipole fit. Use the Application of the derivative to the ratio technique. [ +θ F 3(Q 2 ] ) Q ν σ µν γ 5 + u( p i,s i ). 2m N Use the The elimination of the momentum in the plateau region technique. Now let s extract F 3 (Q 2 )!!! What are the desired quantities without treating the CS term perturbatively? How 3-pt functions modify if we treat the CS term perturbatively. Equate the above two: perturbatively non-perturbatively (in θ). [ E. Shintani et al, arxiv:hep-lat/5522]

76 The nedm: non-perturbative treatment (in θ) Consider the 3-pt function: G µ,(θ) 3pt ( q,t f,t i,t) J N ( p f,t f )J em µ ( q,t)j N( p i,t i ) θ. By inserting two complete sets of states etc... G µ,(θ) 3pt ( q,t f,t i,t) e Ef N θ(t f t) e Ei N θ(t t i ) J N N( p f,s f ) θ N( p f,s f ) J em µ N( p i,s i ) θ N( p i,s i ) J N. s f,s i with E f N θ = p 2 f +m2 N θ and Ei N θ = p 2 i +m 2 N θ. with J θ N Nθ ( p,s) θ = Z θ N uθ N ( p,s), and Nθ ( p,s) J θ N θ = (Z θ N ) ū θ N ( p,s). and ( iγ µ p µ +m N θe i2α(θ)γ 5 ) u θ N ( p,s) = ūθ N ( p,s)( iγ µ p µ +m N θe i2α(θ)γ 5 ) =. The phase e i2α(θ)γ 5 appears due to the CP-breaking in the θ-vacuum.

77 The nedm: non-perturbative treatment (in θ) The phase e i2α(θ)γ 5 appears due to the CP-breaking in the θ-vacuum: CP-even: m N θ = m N +O(θ 2 ), Z θ N = Z N +O(θ 2 ). CP-odd: α(θ) = α 1 θ +O(θ 3 ). Form-Factors: with N θ ( p f,s f ) J em µ Nθ ( p i,s i ) θ = ū θ N ( p f,s f )W θ µ (Q)uθ N ( p i,s i ), W θ µ (Q) = g(θ2 ) W even µ (Q) + iθ h(θ 2 ) W odd µ (Q). and g(θ 2 ) = 1+O(θ 2 ) and h(θ 2 ) = 1+O(θ 2 ) W even µ (Q) = γ µ F 1 (Q 2 ) i F 2(Q 2 ) 2m N Q ν σ νµ, W odd µ (Q) = i F 3(Q 2 ) 2m N Q ν σ νµ γ 5 +F A (Q 2 ) ( Q µ /Q γ µ Q 2) γ 5. G µ,(θ) 3pt ( q,t f,t i,t) = Z N 2 e Ef N (t f t) e Ei N (t t i ) i /p f + m N (1 + 2iα 1 θγ 5 ) 2E f N [ ] W even µ (Q) + iθw odd µ (Q) i/p i + m N (1 + 2iα 1 θγ 5 ) 2EN i + O(θ 2 ).

78 The nedm: non-perturbative treatment (in θ) The phase e i2α(θ)γ 5 appears due to the CP-breaking in the θ-vacuum: CP-even: m N θ = m N +O(θ 2 ), Z θ N = Z N +O(θ 2 ). CP-odd: α(θ) = α 1 θ +O(θ 3 ). Form-Factors: with N θ ( p f,s f ) J em µ Nθ ( p i,s i ) θ = ū θ N ( p f,s f )W θ µ (Q)uθ N ( p i,s i ), W θ µ (Q) = g(θ2 ) W even µ (Q) + iθ h(θ 2 ) W odd µ (Q). and g(θ 2 ) = 1+O(θ 2 ) and h(θ 2 ) = 1+O(θ 2 ) W even µ (Q) = γ µ F 1 (Q 2 ) i F 2(Q 2 ) 2m N Q ν σ νµ, W odd µ (Q) = i F 3(Q 2 ) 2m N Q ν σ νµ γ 5 +F A (Q 2 ) ( Q µ /Q γ µ Q 2) γ 5. G µ,(θ) 3pt ( q,t f,t i,t) = Z N 2 e Ef N (t f t) e Ei N (t t i ) i /p f + m N (1 + 2iα 1 θγ 5 ) 2E f N [ ] W even µ (Q) + iθw odd µ (Q) i/p i + m N (1 + 2iα 1 θγ 5 ) 2EN i + O(θ 2 ).

79 The nedm: non-perturbative treatment (in θ) Correction came from Abramczyk, et al., arxiv: CP-even Dirac Equation: (iγ µ p µ +m N )u N ( p) Parity Operator: γ 4 Under Parity u N ( p) γ 4 u N ( p) CP-odd Dirac Equation: ( iγ µ p µ +m N θe i2α(θ)γ 5 ) u θ N ( p) Parity Operator: e i2α(θ)γ 5γ 4 Under Parity u θ N ( p) ei2α(θ)γ 5γ 4 u θ N ( p) γ 4 is no longer parity operator of neutron state. This accounts to rotating: F 2 = cos(2α)f 2 sin(2α)f 3 F 3 = sin(2α)f 2 +cos(2α)f 3

80 The nedm: perturbative treatment (in θ) Perturbative treatment of the CS term according to: e iθ d 4 xq(x) e iθq = 1 + iθq + O(θ 2 ) Let us apply the perturbative expansion in 3-pt functions: where: G µ,(θ) 3pt ( q,t f,t i,t) = J N ( p f,t f )J em µ ( q,t)j N( p i,t i ) θ = G µ,() 3pt ( q,t f,t i,t) + iθg µ,() 3pt,Q ( q,t f,t i,t) + O G µ,() 3pt ( q,t f,t i,t) = J N ( p f,t f )J em µ ( q,t)j N( p i,t i ), G µ,() 3pt,Q ( q,t f,t i,t) = J N ( p f,t f )J em µ ( q,t)j N( p i,t i )Q. ( θ 2), What is new here is the topological charge Q

81 The nedm: equate non-perturbative with perturbative in θ Equating the results from the two treatments we obtain: G () 3pt ( q,t f,t i,t) = Z N 2 e Ef N (t f t) e Ei N (t t i ) i /p f + m N 2E f N G () 3pt,Q ( q,t f,t i,t) = Z N 2 e Ef N (t f t) e Ei N (t t i )[ i /p f + m N 2E f N W even µ (Q) i /p i + m N 2E i N W odd µ (Q) i /p i + m N 2E i N, + 2α 1 m N 2E f N Similarly for 2-pt functions we obtain: γ 5 W even µ (Q) i /p i + m N 2E i N + i/p f + m N 2E f N W even µ (Q) 2α1 m N 2E i N γ 5 ]. where G (θ) 2pt ( q,t f,t i ) = G () 2pt ( q,t f,t i ) + iθg () 2pt,Q ( q,t f,t i ) + O ( θ 2), G () 2pt ( q,t f,t i ) = J N ( q,t f )J N ( q,t i ) = Z N 2 e E N t i/q + m N 2E N, G () 2pt,Q ( q,t f,t i ) = J N ( q,t f )J N ( q,t i )Q = Z N 2 e E N t 2α1 m N 2E N γ 5. We need to measure the topological charge Q via: Cooling Gradient-Flow

82 The nedm: Topological charge We calculate the topological charge: Q = d 4 xq(x), with topological charge density: q(x) = 1 32π 2ǫ µνρσtr{g µν G ρσ }. We use the improved definition with leading lattice artifacts (clasically) of O(a 4 ): with c = 5/3 and c 1 = 1/12 as well as where q clov L (x) = 1 ( ) 32π 2ǫ µνρσtr C clov µν Cclov ρσ C clov µν (x) = 1 4 Im ( ) q(x) = c q clov L (x) + c 1q rect L (x), and q rect L (x) = 2 ( ) 32π 2ǫ µνρσtr C rect µν Crect ρσ and C rect µν (x) = 1 8 Im +., We need to smooth out the ultraviolet fluctuations...

83 The nedm: Topological charge Cooling Cooling U µ (x) SU(N): U old µ (x) U new µ (x) with Choose a U new µ (x) that maximizes: P(U) e (lim β β 1 N ReTrX µ Uµ). ReTr{U new µ (x)x µ (x)}. One full cooling iteration is noted as n c = 1. Gradient Flow Solution of the evolution equations: V µ (x,τ) = g 2 [ x,µs G (V(τ))]V µ (x,τ) V µ (x,) = U µ (x), With link derivative defined as: x,µ S G (U) = i a i a T a d ( ds S G e isy a ) U T a (a) x,µ S G(U), s= Total gradient flow time is expressed as τ = n int ǫ.

84 The nedm: Topological charge - Equivalence between smoothers Recently shown that cooling exhibits equivalence with gradient flow: Smoothing with Wilson Action Perturbatively τ n c /3 [C. Bonati and M. D Elia, Phys. Rev. D 89, 155 (214) [arxiv: ]] Configurations produced with Iwasaki Gauge Action Generalize the equivalence for Symanzik improved actions with rectangles Smoothing with: S g = β N ( x with c +8c 1 = 1 c 4 µ,ν=1 1 µ<ν { 1 ReTr(U 1 1 x,µ,ν ) } +c 1 4 Shown that perturbatively τ n c /(3 15c 1 ) Consider µ,ν=1 µ ν Wilson: c 1 =, τ n c /3 Symanzik tree-level improved: c 1 = 1/12, τ n c /4.25 Iwasaki: c 1 =.331, τ n c /7.965 { 1 ReTr(U 1 2 x,µ,ν ) } ), [C. Alexandrou, A. A and K. Jansen, Phys. Rev. D 92, (215) [arxiv: ]].

85 The nedm: Topological charge - Equivalence between smoothers 1 gradient flow cooling 1 gradient flow cooling 1 gradient flow cooling SG.1 SG.1 SG e-5.1 Wilson 1 n c or 3 τ 1 Symanzik tree-level 1e n c or 4.25 τ 1 1e-5.1 Iwasaki 1 n c or τ r χ 1/4 r χ 1/4 r χ 1/ Wilson 1 2 gradient flow cooling 3 n c or 3 τ gradient flow Symanzik tree-level cooling n c or 4.25 τ Iwasaki 1 2 gradient flow cooling 3 4 n c or τ 5 6

86 The nedm: Topological charge - Equivalence between smoothers We read an observable at a fixed value of a 8τ = O(.3fm). [M. Lüscher, JHEP 18:71 21 [arxiv: v3]]. For practical reasons we choose a value of a 8τ.6fm. This corresponds to: Smoothing action c c 1 n c /τ n c (a 8τ.6fm) τ (n c ) Wilson Symanzik tree-level improved Iwasaki

87 The nedm: Topological charge - History 4 gradient flow cooling 2 Q -2-4 Iwasaki (n c = 5,τ = 6.3) # conf

88 The nedm: Topological charge - Distribution 3 25 gradient flow cooling Iwasaki (n c = 5,τ = 6.3) 2 confs Q

89 The nedm: Topological charge - Distribution 3 gradient flow 25 Iwasaki (n c = 5,τ = 6.3) 2 confs Q

90 The nedm: Topological charge - Distribution 3 25 gradient flow cooling Iwasaki (n c = 5,τ = 6.3) 2 confs Q

91 The nedm: 2-pt functions We calculate the 2-pt function: ( x f,t f ) ( x i =,t i = ) In practice we evaluate the 2-pt functions: With projectors We determine the value of α 1 by: G 2pt ( q,t f,t i,γ ) Z N 2 e E N (t f t i ) αβ [ ] Γ Λ 1/2 ( q), αβ G 2pt,Q ( q,t f,t i,γ 5 ) Z N 2 e E N (t f t i ) Γ 5 αβ [ α 1 m N E N γ 5 ] αβ. Γ = 1 4 (1 + γ ), Γ 5 = γ 5 4, R 2pt (α 1,t f,t i ) = G ( ) 2pt,Q,tf,t i,γ 5 ( ). G 2pt,tf,t i,γ

92 The nedm: 2-pt Plateaus We calculate the plateau: Π 2pt (α 1 ) = lim R 2pt(α 1,t f,t i ) = α 1. t f t i Example of a plateau for Iwasaki action, gradient flow at τ = α t f /a

93 The nedm: α 1 - Gradient Flow VS Cooling α 1 via cooling (n c ) and gradient flow (τ) for Wilson smoothing action -.5 Wilson Symanzik tree-level Iwasaki α n c or 3 τ n c or 4.25 τ n c or τ

94 The nedm: 3-pt functions Calculate: J em µ q = p f p i = p i ( x, t) ( x f,t f ) ( x i =,t i = ) We extract the CP-odd Form-Factors F 3 (q 2 ) from: R µ 3pt,Q ( q,tf,t i,t,γ k ) = G µ 3pt,Q ( q,t f,t i,t,γ k ) G 2pt ( q,t f t i,γ ) G 2pt( q,t f t,γ )G 2pt (,t t i,γ )G 2pt (,t f t i,γ ) G 2pt (,t f t,γ )G 2pt ( q,t t i,γ )G 2pt ( q,t f t i,γ ). We calculate the plateau: Π µ 3pt (Γ k) = lim t f t lim ( ) t t i Rµ 3pt q,tf,t i,t,γ k. Projectors: Γ k = i 4 (1 + γ )γ 5 γ k (k = 1,2,3).

95 The nedm: 3-pt functions The above plateau gives: Π 3pt,Q (Γ k) = i 2m 2 [ N α 1 E N (E N + m N ) Q F 1 (Q 2 ) k + (E N + 3m N )α 1 F 2 (Q 2 ) 2m N 4m 2 N + (E N + m N )F 3 (Q 2 ] ) 4m 2 N. We elliminate F 1 and F 2 through the CP-even plateaus: Π k F 3 = iπ 3pt,Q (Γ k) + iα 1 Π k 3pt (Γ ) + α i,j=1 ǫ jki Π j 3pt (Γ i), = EN + m N 8E N m 2 N Q k F 3 (Q 2 ). Momentum Q k hinders a direct evaluation of F 3 (Q 2 = ) Usual parametrization in Q 2 and then fit with Dipole ansatz Application of the derivative to the ratio technique. [D. Guadagnoli et al, JHEP 34 (23) 19, [arxiv:hep-lat/2144]] The elimination of the momentum in the plateau region technique [C. Alexandrou et al, PoS LATTICE214 (214) 75, [arxiv: ]]

96 The nedm: Extraction of F 3 () via Dipole Fit We extract the plateau Π k F 3 ( Q 2,Γ k ) for momentum transfer Q 2 : (Averaging over all momentum directions and index k...) for t f t i = 1a, t f t i = 12a and t f t i = 14a (we chose results for t f t i = 12a) We use the value of α 1 to build the ratio which leads to F 3 (Q 2 ) via Π k F 3. We parametrize F 3 (Q 2 ) in momentum transfer Q 2. We finally perform a dipole fit: F 3 ( Q 2 ) = F 3 () ( ) Q2 m 2 F 3

97 The nedm: Extraction of F 3 () via Dipole fit Example of a plateau for Iwasaki action, gradient flow at τ = 6.3 and Q 2.17GeV 2 : t f = 1a, t i = ( Q 2.17GeV 2 ) R t f = 12a, t i = t f = 14a, t i = (t t f /2)/a

98 The nedm: Extraction of F 3 () via Dipole fit Example of a dipole fit for Iwasaki action, gradient flow at τ = 6.3 and Q 2 1GeV 2 : F3(Q 2 ) Q 2 [Gev 2 ]

99 The nedm: Extraction of F 3 () via Dipole fit F 3 ()/2am N via cooling (n c ) and gradient flow (τ): -.2 Wilson Symanzik tree-level Iwasaki F3()/2amN n c or 3 τ n c or 4.25 τ n c or τ The three smoothing actions give consistent results!

100 The nedm: Extraction of F 3 () via Derivative on the ratio Assuming continuous momenta one can remove the Q k dependence in front of F 3 (Q 2 ): lim Q 2 Q j Π k F 3 ( Q) = EN + m N 8E N m 2 N δ kj F 3 (). We build ratios such as: lim Q 2 Q j R µ 3pt,Q ( q,t f,t i,t,γ k ) = 1 G 2pt (,t f,t i,γ ) L/2 a x j = L/2+a Requires position space correlators G () µj N Jµ emqj ( x,t f,t N i,t,γ k ) In finite volume this approximates the derivative of δ-function: L/2 a a 3 x j = L/2+a L a ix j G µ 3pt,Q ( x,t f,t i,t,γ k ) = 1 V x i = i j L/2 a a3 k x j = L/2+a L 1 (2π) 3 L a x i = i j L a x i = i j,i=1,2,3 ix j G µ 3pt,Q ( x,t f,t i,t,γ k ). ( ix j exp i ) k x Gµ 3pt,Q ( k,t f,t i,t,γ k ) d 3 k k j δ (3) ( k)g µ 3pt,Q ( k,t f,t i,t,γ k ). Residual t-dependence G µ,() 3pt,Q ( q,t f,t i,t,γ k ) exp( E N t), with E N for L [W. Wilcox, Phys.Rev. D66 (22) 1752, [arxiv:hep-lat/2424]]

101 The nedm: Extraction of F 3 () via Derivative on the ratio Example of a plateau for Iwasaki action and gradient flow at τ = 6.3: R3pt(Q 2 = ) exp( E N t) (t t f /2)/a t f = 1a, t i = t f = 12a, t i = t f = 14a, t i =

102 The nedm: Extraction of F 3 () via Momentum Elimination Define a suitable ratio such that: Π k F 3 = iπ 3pt,Q (Γ k) + iα 1 Π k 3pt (Γ ) + α 1 1 On-axis momenta e.g. q = (± Q,,) T 2 3 i,j=1 ǫ jki Π j 3pt (Γ i) = EN + m N 8E N m 2 N Q k F 3 (Q 2 ). Fourier transform on Π( Q) Π(y) in position space; with (n = y/a) Π(y) = { +Π(n), n =,...,N/2 Π(N n), n = N/2 + 1,...,N 1, N = L/a, Average over pos. and neg. y we get Π(n) Finally transform back and introduce continuous momenta k: Π(k) = [ ] exp(ikn)π(n) n=, N/2 + 2i N/2 1 n=1 Π(n) sin ( ) k 2 (2n) We define ˆk 2sin ( k 2) and Pn (ˆk2 ) = P n (( 2sin ( k 2)) 2 ) = sin(nk)/sin ( k 2) and obtain: Π(ˆk) Π() = i N/2 1 n=1 ˆkP n (ˆk2 ) Π(n). P n (ˆk2 ) is related to Chebyshev polynomials of the 2 nd kind

103 The nedm: Extraction of F 3 () via Momentum Elimination By applying the derivative in respect to ˆk: F 3 (ˆk 2 ) 2m N = i N/2 1 n=1 P n (ˆk 2 )Π(n), Generalize to off-axis momentum classes } M( Q,Q 2 off { q ) = q = {± Q,Q 1,Q 2 }, Q 2 1 +Q2 2 = q2 off where {± Q,Q 1,Q 2 } denotes all possible permutations of ± Q, Q 1 and Q 2 To combine results for F 3 (Q 2 )/(2m N ) for different momentum classes Q off and arrive at (Euclidean) Q 2 (ˆk,Q 2 off ) = : Analytic Continuation: Final formular takes similar form: k iκ ˆk iˆκ = 2sinh (κ 2 P n (ˆk 2 2 ) P n (ˆκ ) = sinh(nκ)/sinh ( κ) 2 F 3 (ˆκ 2 ) 2m N = i N/2 1 n=1 P n (ˆκ 2 )Π(n), ) Combine results from different sets of M( Q,Q 2 off ) by taking the error weighted average.

104 The nedm: Extraction of F 3 () via Elimination of momentum F 3 (Q 2 )/2m N via gradient flow (τ = 6.3) with the Iwasaki smoothing action extracted for Q 2 offmax = 5(2π/L)2 :.25 direct computation F 3 (Q 2 = )/(2m N ) standard method F3(Q 2 )/(2mN)[e fm] Q 2 [GeV 2 ]

105 The nedm: Comparison Weighted results for the three different smoothing actions with Q extracted via cooling (open symbols) and the gradient flow (closed symbols): F3()/2mN [e fm] Wilson Symanzik Iwasaki

106 The nedm: Comparison Our result ( ) compared to other works: F3()/2mN [e fm] ChPT Ottnad et al. arxiv: Our result.45(6).1 N f = 2+1 DWF Shintani et al. Lattice N f = 2 Clover Shintani et al. arxiv: m 2 π[gev 2 ] ( ) Ottnad et. al. ChPT, [arxiv: [hep-ph]], ( ) Shintani et al, ( ) Shintani et al [arxiv:83.797].

107 The nedm: Corrected Values Without rotation Phase e iαγ 5 there is mixing of F 2 and F 3. [Abramczyk, et al, arxiv: ] F 2 = cos(2α)f 2 sin(2α)f 3 F 3 = sin(2α)f 2 +cos(2α)f 3

108 The nedm: What we learn? Calculate the nedm for B55.32 at M π 37 MeV. Implemented three momentum dependence treating techniques: Dipole fit. Application of the derivative to the ratio. Elimination of the momentum in the plateau region. Implemented two smoothers and demonstrated equivalence: Cooling. Gradient Flow. Used three smoothing actions: Wilson. Symanzik tree-level improved. Iwasaki. All combinations give similar results! Our result agrees with older estimations! We are looking forward to the physical point continuum limit.