Pion couplings to the scalar B meson. Antoine Gérardin

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1 Antoine Gérardin 1 Pion couplings to the scalar B meson Pion couplings to the scalar B meson Antoine Gérardin In collaboration with B. Blossier and N. Garron Based on [arxiv: ] LPT Orsay January 15, Rencontre de Physique des Particules

2 Motivations Antoine Gérardin 2 Pion couplings to the scalar B meson Lattice simulations are often done at unphysical light quark masses ( m π > m phys π ) Computing the quark propagator at small pion mass is numerically difficult solution : use different quark masses and extrapolate to the physical pion mass

3 Motivations Antoine Gérardin 3 Pion couplings to the scalar B meson Example: chiral extrapolation of the B mesons decay constant [ALPHA Collaboration] : fb δ (y, a)/gev f δ B s (y β = 5.2 β = 5.3 β = y = m2 π 8πf 2 π y exp.4 y y exp.4

4 Motivations Antoine Gérardin 4 Pion couplings to the scalar B meson Lattice simulations are often done at unphysical light quark masses ( m π > m phys π ) Computing the quark propagator at small pion mass is numerically difficult solution : use different quark masses and extrapolate to the physical pion mass Heavy-light mesons: Light quark dynamics is well described by chiral perturbation theory (χpt) Heavy quarks are described by the Heavy Quark Effective Theory (HQET) } HMχPT = HQET + χpt Use HMχPT (Heavy Meson Chiral Perturbation Theory) to extrapolate lattice data HMχPT is parametrized by three couplings g, h and g at leading order

5 Motivations Antoine Gérardin 5 Pion couplings to the scalar B meson L HMχPT = f π 2 8 µ Σ ab µ Σ ba + ig H b γ µ γ 5 A µ ba H a + i H b v µ D µba H a }{{}}{{} negative parity states Goldstone bosons + i g S b γ µ γ 5 A µ ba S a + i S b v µ D µba S a }{{} positive parity states + ih S b γ µ γ 5 A µ ba H a

6 Motivations Antoine Gérardin 5 Pion couplings to the scalar B meson L HMχPT = f π 2 8 µ Σ ab µ Σ ba + ig H b γ µ γ 5 A µ ba H a + i H b v µ D µba H a }{{}}{{} negative parity states Goldstone bosons + i g S b γ µ γ 5 A µ ba S a + i S b v µ D µba S a }{{} positive parity states + ih S b γ µ γ 5 A µ ba H a

7 Motivations Antoine Gérardin 5 Pion couplings to the scalar B meson L HMχPT = f π 2 8 µ Σ ab µ Σ ba + ig H b γ µ γ 5 A µ ba H a + i H b v µ D µba H a }{{}}{{} negative parity states Goldstone bosons + i g S b γ µ γ 5 A µ ba S a + i S b v µ D µba S a }{{} positive parity states + ih S b γ µ γ 5 A µ ba H a

8 Motivations Antoine Gérardin 6 Pion couplings to the scalar B meson Lattice simulations are often done at unphysical light quark masses ( m π > m phys π ) Computing the quark propagator at small pion mass is numerically difficult solution : use different quark masses and extrapolate to the physical pion mass Heavy-light mesons: Light quark dynamics is well described by chiral perturbation theory (χpt) Heavy quarks are described by the Heavy Quark Effective Theory (HQET) } HMχPT = HQET + χpt Use HMχPT (Heavy Meson Chiral Perturbation Theory) to extrapolate lattice data HMχPT is parametrized by three couplings g, h and g at leading order only g is actually considered in chiral extrapolations = m B E Bπ is usually not m π on the lattice the coupling h is large h and g may have important contributions

9 Antoine Gérardin 7 Pion couplings to the scalar B meson Determination of g

10 The soft coupling g g ɛ i = B ( ) ψ l γ i γ 5 ψ l B 1(ɛ i, ) Three-point correlation function: A µ = ψ l (x)γ µ γ 5 ψ l (x) (axial current) C (3) (t, t 1 ) = Z A 1 V 3 x, y, z 1 3 A k( z, t) A k ( y, t 1 ) S ( x, ) A k (x) = ψ h (x)γ A k ψ l(x) S(x) = ψ h (x)γ S ψ l (x) Z A was determined non-perturbatively by the ALPHA Collaboration [Nucl.Phys. B865 (212) ] Two-point correlation functions C (2) S (t) = S( x, t)s ( y, ), C (2) A (t) = x, y x, y A k ( x, t)a k ( y, ) Finally, the following ratio converge to the coupling g : C (3) (t, t 1 ) R(t) = C (2) S (t)c(2) A (t) t 1 t =t a g + excited states Antoine Gérardin 8 Pion couplings to the scalar B meson

11 Excited states : summed GEVP Antoine Gérardin 9 Pion couplings to the scalar B meson Use N = 3 interpolating operators with different overlaps with excited states matrix of correlators Solve the Generalized Eigenvalue Problem (GEVP): [Michael, 85; Lüscher and Wolff, 9] C (2) S (t)v n(t, t ) = λ n (t, t )C (2) S (t )v n (t, t ) Eigenvectors (v n ) and eigenvalues (λ n ) can be used to construct the summed ratio M sgevp nn (t, t ): M sgevp nn (t, t ) = t (v n (t, t ), [K(t, t )/λ n (t, t ) K(t, t )] v n (t, t )) ) 1/2 ( v n (t, t ), C (2) A (t )v n (t, t ) ( v n (t, t ), C (2) S (t )v n (t, t ) with : K ij (t, t ) = t 1 C (3) ij (t, t 1) summed GEVP [JHEP 121 (212) 14] ) 1/2 N6 M sgevp t 1 11 (t) t =t 1 g + O ( te N+1,nt ) N+1,n = E N+1 E n (excited states contribution is reduced) M sgevp (t, t) t/a

12 Extrapolation to the physical point Antoine Gérardin 1 Pion couplings to the scalar B meson -.5 g small dependence on the lattice spacing small dependence on the pion mass β=5.2 β=5.3 β=5.5 g =.122(8)(6) m π g = α [ g2 (4πf π ) 2 m2 π log(m 2 π) + h2 m 2 ( π (4πf π ) g g ) ] m 2 π log(m 2 π) + Cm 2 π (HMχP T )

13 Antoine Gérardin 11 Pion couplings to the scalar B meson Determination of h

14 Lattice computation of h Antoine Gérardin 12 Pion couplings to the scalar B meson Continuum g B Bπ = π(k)b(p ) B (p) = m B m B The decay rate Γ is given by: (static limit) m 2 B m2 B m B f π h π(k) B (p) B(p ) Γ(B Bπ) = k ( ) ( ) 8πm 2 gb 2 B Bπ m 2 B (m B + m π) 2 m 2 B (m B m π) 2 k = 2m B Lattice: Fermi Golden rule [Phys.Rev. D63 (21)] (McNeile et al.) Γ (B Bπ) = (2π) x 2 ρ, x = B Bπ ρ is the density of final states on the lattice: ρ = L3 ke π 2π 2 Γ (B Bπ) k = 1 π ( ) L 3 (ae π ) (ax) 2 a Conclusion: one can access to h through x, computed on the lattice

15 Lattice computation: x = B Bπ Antoine Gérardin 13 Pion couplings to the scalar B meson Two-point correlation function: x = B Bπ C B Bπ(t) = O Bπ (t)o B () = t 1 ÔB B e m B t 1 }{{} x Bπ ÔBπ e E Bπ(t t 1 ) }{{} Near threshold (m B E Bπ ) C B Bπ(t) ÔB B x Bπ ÔBπ t e Et Consider the following ratio R(t) = C (2) B ( Bπ(t) ) C (2) B (t)c (2) 1/2 xt + excited states B Bπ Bπ (t)

16 Lattice computation: x = B Bπ Antoine Gérardin 13 Pion couplings to the scalar B meson Two-point correlation function: x = B Bπ C B Bπ(t) = O Bπ (t)o B () = t 1 ÔB B e m B t 1 }{{} x Bπ ÔBπ e E Bπ(t t 1 ) }{{} Near threshold (m B E Bπ ) C B Bπ(t) ÔB B x Bπ ÔBπ t e Et Consider the following ratio R(t) = C (2) B ( Bπ(t) ) C (2) B (t)c (2) 1/2 xt + excited states B Bπ Bπ (t) Or the GEVP ratio to reduce the excited states contribution R GEVP (t, t ) = ( vb (t, t ), C B Bπ(t) v Bπ (t, t ) ) (vb (t, t ), C B B (t) v B (t, t ) ) (v Bπ (t, t ), C Bπ Bπ (t) v Bπ (t, t )) xt + excited states x eff (t) = t R GEVP (t) t 1 t =t a x

17 Lattice computation: x = B Bπ Antoine Gérardin 14 Pion couplings to the scalar B meson We have considered the corrections coming from = m B E Bπ We have considered the contribution from excited states x eff (t) x eff (t) t/a t/a a =.65 fm, m π = 44 MeV a =.48 fm, m π = 34 MeV m π 28 MeV 31 MeV 34 MeV 435 MeV ax.156(4).159(3).174(6).241(1) h.86(4).86(3).85(4).84(5)

18 h: chiral and continuum extrapolations Antoine Gérardin 15 Pion couplings to the scalar B meson 1.1 β = β = 5.3 β = 5.5 small dependence on the lattice spacing.9 small dependence on the quark mass h.8.7 h =.86(4) stat (2) χ m 2 π [ NLO HMχPT : h = h 1 3 3ĝ g 2 + 2ĝ g 4 (4πf π ) 2 ( m 2 π log(m 2 π) (m exp π ) 2 log((m exp π ) 2 ) )] + C ( m 2 π (m exp π ) 2)

19 Conclusion Antoine Gérardin 16 Pion couplings to the scalar B meson We have computed the soft pion couplings h and g with N f = 2 dynamical quarks h =.84(3)(2), g =.122(8)(6) Good control under/over systematics: in both case, the continuum and chiral extrapolations are performed we used the GEVP to reduce the contamination from excited states These couplings can be used in the chiral extrapolations of relevant heavy-light meson properties h ĝ : B meson orbital excitations cannot be neglected in chiral loops [Becirevic et al. (212)]: h =.66(1)(6) They used three-point correlation functions to obtain the form factor A + ( 2 ) They compute the radial density to obtain the form factor in the correct limit A + ( 2 ) A + () g B Bπ PDG: Γ D = 267(4) MeV, m D = 2318(29) MeV h =.74(16) 1/m c corrections are expected to be sizable for D meson

20 Antoine Gérardin 17 Pion couplings to the scalar B meson

21 Lattice setup Lattice discretization N f = 2 O(a) improved Wilson-Clover Fermions HYP1-2 discretization for the static quark action Discretization effects 3 lattice spacings a : (.48,.65,.75) <.1 fm Light quark mass chiral extrapolations different pion masses in the range [28 MeV, 44 MeV] total of 4 ensembles Antoine Gérardin 18 Pion couplings to the scalar B meson

22 Correlation functions Antoine Gérardin 19 Pion couplings to the scalar B meson quark-antiquark interpolating operators O B Γ,n(t) = 1 V x [ ] d (n) (x)γb(x) meson-meson interpolating operators 2 3 π+ ()B () O Bπ Γ,n = 1 V 2 x i 1 3 π ()B () 2 [ d(x1 )Γu(x 1 ) ] [ u (n) (x 2 )Γb(x 2 )] 3 local (Γ = γ, γ 5 ) and derivative (Γ = γ i γ γ 5 i, γ i i ) interpolating operators 4 levels of gaussian smearing 1 [ u(x1 )Γu(x 1 ) d(x 1 )Γd(x 1 ) ] 6 [ ] d (n) (x 2 )Γb(x 2 ) y, Γ 2 x, Γ 1 y 2, γ 5 x, Γ 2 y 1 x 1 y 1 x 1 y 1, γ 5 y 2 x 2 y 2 x 2

23 Systematic errors on h: corrections from = m B E Bπ Neglecting excited states, at threshold we have: C B Bπ(t) = t 1 ÔB B x Bπ ÔBπ e m B t1 e E Bπ(t t 1 ) ÔB B x Bπ ÔBπ t e Et when the threshold condition is only approximately fulfilled, the linear time dependance becomes: t 2 ( ) sinh 2 t = t + 2 t O( 4 ) m π 28 MeV 31 MeV 34 MeV 435 MeV ( 3t 2 2 ) 1 for t [ 2] a.36(4).26(8).1(3).12(6) 24 Table: = m B E Bπ x eff (t) x eff (t) t/a a =.65 fm, m π = 44 MeV a =.48 fm, m π = 34 MeV Antoine Gérardin 2 Pion couplings to the scalar B meson t/a

24 Systematic errors on h: Excited states contribution Antoine Gérardin 21 Pion couplings to the scalar B meson C B Bπ(t) = O B (t)o Bπ () = t 1 ÔB B x Bπ ÔBπ e m B t1 e E Bπ(t t 1 ) When excited states are taken into account (X 1 = B, X 2 = Bπ) : Assumptions: C B Bπ(t) = nm x nm = X n X m ÔB Xn x nm X m ÔBπ e Ent 1 e Em(t t 1) t 1 I consider only the contribution from the first excited state X 3 : m B E Bπ < E X3 < E X4 < X 3 has a non-negligible overlap with O B Ô B X3 and ÔBπ X 3 the symmetric case (a non-negligible overlap with O Bπ ) is similar ÔB X3 x 32 Bπ ÔBπ e E 3t 1 e E(t t 1) t 1 = t ÔB B x Bπ ÔBπ e Et } {{ } ground state contribution Excited states contributions are suppressed by a factor t To be compared with the usual exponential suppression 1 t ÔB X3 x 32 ÔB B x t 1 e (E3 E)t1 R GEVP (t) A + xt

25 Cross-check: box and cross diagrams [Phys Lett B556 (24)] (McNeile et al.) y 1 x 1 R(t) = (v Bπ(t, t ), C connected (t) v Bπ (t, t )) (v Bπ (t, t ), C Bπ Bπ (t) v Bπ (t, t )) = B x2 t 2 + O(t) C connected (t) = 3 2 C box(t) C cross(t). y 2 x 2 y 1 x 1.1 y 2 x β(t) = t R R.4 Previous analysis : ax =.241(1).2 Box + Cross diagrams : ax =.237(8) (t/a) 2 a =.65 fm, m π = 44 MeV Perfect agreement! Antoine Gérardin 22 Pion couplings to the scalar B meson

26 Mass of the scalar B meson Antoine Gérardin 23 Pion couplings to the scalar B meson a m(a, m π ) = Estat(a, s m π ) E ps stat (a, m π) with En eff (t, t ) = a 1 λ n (t, t ) log λ n (t + a, t ) 45 m B (z,m π,a) 45 m B (z,m π,a) y phys y 25 y phys y m B = 385(21) stat (3) syst

27 Local vs Derivative interpolating operator Antoine Gérardin 24 Pion couplings to the scalar B meson J P Local Derivative + Γ = γ Γ = γ i i 1 + Γ = γ 5 γ i Γ = γ 5 i Interpolating operators built from covariant derivatives are beneficial to reduce the contamination from higher excited states Table: Interpolating operators -.5 derivative local -.5 derivative local sgevp R sgevp R t/a E5g : a =.65 fm, m π = 44 MeV t/a F6 : a =.65 fm, m π = 31 MeV