Analyticity and unitarity constraints on form factors

Transcription

1 Analyticity and unitarity constraints on form factors Irinel Caprini IFIN-HH, Bucharest, Romania Based on papers with G. Abbas, B. Ananthanarayan, C. Bourrely, L. Lellouch, I. Sentitemsu Imsong, S. Ramanan

2 Outline 1 Analyticity and unitarity for form factors 2 Method of unitarity bounds Okubo (1971) 3 Meiman problem and generalizations 4 Applications 5 Conclusions

3 Form factors considered Electromagnetic form factors of light pseudoscalar mesons: P = π, K P + (p ) J elm µ P + (p) = (p + p ) µf P (t) Form factors relevant for weak semileptonic transitions: P π, P = K, D, B π + (p ) J weak µ P 0 (p) = (p + p) µf +(t) + (p p ) µf (t) f +(t): vector form factor f 0 (t) = f +(t) + t M P 2 f (t): scalar form factor M2 π In the general discussion F P (t) and f ± (t) shall be denoted generically as F(t)

4 Basic properties Theoretical description of F(t) at low energies: ChPT, lattice, QCD-SR at high t = Q 2 < 0 perturbative QCD (1/t scaling) intermediate region: big uncertainties

5 Basic properties Theoretical description of F(t) unitarity cut inelastic cut ChPT or lattice at low energies: ChPT, lattice, QCD-SR Im t QCD t + t in at high t = Q 2 < 0 perturbative QCD (1/t scaling) 0 intermediate region: big uncertainties Re t

6 Basic properties Theoretical description of F(t) unitarity cut inelastic cut ChPT or lattice at low energies: ChPT, lattice, QCD-SR Im t QCD t + t in at high t = Q 2 < 0 perturbative QCD (1/t scaling) 0 intermediate region: big uncertainties Re t Analyticity and unitarity Causality: F(t) real analytic function, F(t ) = F (t), in the complex t-plane with a cut along the real axis from the lowest unitarity threshold t + to infinity Unitarity: ImF(t + iǫ) = θ(t t +) σ(t)f (t)f(t) + θ(t t in )Σ in (t) σ(t) = p 1 t +/t: two particle phase space Fermi-Watson theorem: for t + t t in, arg[f(t + iǫ)] = δ(t), δ(t): phase-shift of the related scattering amplitude f (t) = e2iδ(t) 1 2iσ(t) Complications: unphysical regions, anomalous thresholds (not encountered)

7 Exploiting analyticity Dispersive representations

8 Exploiting analyticity Dispersive representations Standard dispersion relation (Cauchy integral) F(t) = 1 π R t + ImF(t +iǫ)dt t t Im F(t + iǫ) = σ(t)f (t)f(t) (modulo subtractions) for t < t in

9 Exploiting analyticity Dispersive representations Standard dispersion relation (Cauchy integral) F(t) = 1 π R t + ImF(t +iǫ)dt t t Im F(t + iǫ) = σ(t)f (t)f(t) (modulo subtractions) for t < t in Omnès (phase) representations F(t) = P(t) exp t R π t + dt δ(t ) t (t t) P(t): real polynomial (accounts for zeros: P(t i ) = 0)

10 Exploiting analyticity Dispersive representations Standard dispersion relation (Cauchy integral) F(t) = 1 π R t + ImF(t +iǫ)dt t t Im F(t + iǫ) = σ(t)f (t)f(t) (modulo subtractions) for t < t in Omnès (phase) representations F(t) = P(t) exp t R π t + dt δ(t ) t (t t) P(t): real polynomial (accounts for zeros: P(t i ) = 0) Representation in terms of modulus «t+ t R F(t) = B(t) exp ln F(t ) dt π t + t t +(t t B(t): Blaschke factor ( B(t) = 1 for t > t +, B(t i ) = 0)

11 Exploiting analyticity Dispersive representations Standard dispersion relation (Cauchy integral) F(t) = 1 π R t + ImF(t +iǫ)dt t t Im F(t + iǫ) = σ(t)f (t)f(t) (modulo subtractions) for t < t in Omnès (phase) representations F(t) = P(t) exp t R π t + dt δ(t ) t (t t) P(t): real polynomial (accounts for zeros: P(t i ) = 0) Representation in terms of modulus «t+ t R F(t) = B(t) exp ln F(t ) dt π t + t t +(t t B(t): Blaschke factor ( B(t) = 1 for t > t +, B(t i ) = 0) Analytic parametrizations - little predictive power outside their original range

12 Okubo s approach Method of unitarity bounds Okubo (1971), Micu (1973), Auberson et al (1975), Singh and Raina (1979) Polarization tensor of the relevant current calculated from current algebra at spacelike momenta Dispersion relation for the invariant polarization amplitudes Unitarity and positivity of the spectral functions an upper bound on an integral of the modulus squared of the form factor along the unitarity cut mathematical techniques of complex analysis lead to bounds on the values of the form factor and its derivatives at points inside the analyticity domain

13 Okubo s approach Method of unitarity bounds Okubo (1971), Micu (1973), Auberson et al (1975), Singh and Raina (1979) Polarization tensor of the relevant current calculated from current algebra at spacelike momenta Dispersion relation for the invariant polarization amplitudes Unitarity and positivity of the spectral functions an upper bound on an integral of the modulus squared of the form factor along the unitarity cut mathematical techniques of complex analysis lead to bounds on the values of the form factor and its derivatives at points inside the analyticity domain Modern version: the spacelike correlators are obtained from perturbative QCD and OPE Bourrely, Machet, de Rafael (1981), de Rafael and Taron (1992)

14 Illustration Polarization tensor of a relevant weak current V µ: i R n d 4 x e iq x 0 T V µ (x)v ν (0) o 0 = ( g µν q 2 + q µ q ν )Π 1(q 2 ) + q µ q ν Π 0(q 2 ) Unsubtracted dispersion relations for suitable correlators χ 1(Q 2 ) 1 2 h i Q 2 Π 1( Q 2 ) = 1 Z 2 (Q 2 ) 2 π 0 χ 0(Q 2 ) h i Q 2 Π 0( Q 2 ) = 1 Z Q 2 π 0 dt timπ1(t) (t + Q 2 ) 3 dt timπ0(t) (t + Q 2 ) 2 Unitarity and the positivity of the spectral functions (t ± = (M P ± M π) 2 ) ImΠ 1(t) π [(t t +)(t t )] 3/2 t 3 f +(t) 2 ImΠ 0(t) 3 t + t [(t t + )(t t )] 1/2 2 16π t 3 f 0(t) 2 R 1 π ρ(t, Q 2 ) F(t) 2 dt I(Q 2 ), t + I(Q 2 ) calculated from pqcd and OPE

15 Illustration Polarization tensor of a relevant weak current V µ: i R n d 4 x e iq x 0 T V µ (x)v ν (0) o 0 = ( g µν q 2 + q µ q ν )Π 1(q 2 ) + q µ q ν Π 0(q 2 ) Unsubtracted dispersion relations for suitable correlators χ 1(Q 2 ) 1 2 h i Q 2 Π 1( Q 2 ) = 1 Z 2 (Q 2 ) 2 π 0 χ 0(Q 2 ) h i Q 2 Π 0( Q 2 ) = 1 Z Q 2 π 0 dt timπ1(t) (t + Q 2 ) 3 dt timπ0(t) (t + Q 2 ) 2 Unitarity and the positivity of the spectral functions (t ± = (M P ± M π) 2 ) ImΠ 1(t) π [(t t +)(t t )] 3/2 t 3 f +(t) 2 ImΠ 0(t) 3 t + t [(t t + )(t t )] 1/2 2 16π t 3 f 0(t) 2 R 1 π ρ(t, Q 2 ) F(t) 2 dt I(Q 2 ), t + I(Q 2 ) calculated from pqcd and OPE Q 2 sufficiently large for light mesons; Q 2 = 0 for heavy-heavy or heavy-light form factors R 1 more general relation: π ρ t + i,j(t, Q 2 )F i(t)fj (t)dt I(Q 2 ) (BD ( ) form factors)

16 Comment: connection with stability of analytic continuation Analyticity: its splendour and its dangers Splendour: analytic continuation is unique Dangers: analytic continuation is unstable (ill posed problem in the Hadamard sense) two analytic functions very close along a range Γ may differ arbitrarily outside Γ determination of remote resonances from Breit-Wigner parametrizations!

17 Comment: connection with stability of analytic continuation Analyticity: its splendour and its dangers Splendour: analytic continuation is unique Dangers: analytic continuation is unstable (ill posed problem in the Hadamard sense) two analytic functions very close along a range Γ may differ arbitrarily outside Γ determination of remote resonances from Breit-Wigner parametrizations! Mathematical result (Tikhonov regularization): analytic continuation is stabilized if the class of admissible functions forms a compact set Ciulli et al (1975) Role of the stabilizing condition: Let C be a compact class of analytic functions If F j (t) C and sup Γ F 1 (t) F 2 (t) < ǫ, then for t outside Γ the inequality F 1 (t) F 2 (t) < M(ǫ,t) holds, such that M(ǫ,t) 0 when ǫ 0

18 Comment: connection with stability of analytic continuation Analyticity: its splendour and its dangers Splendour: analytic continuation is unique Dangers: analytic continuation is unstable (ill posed problem in the Hadamard sense) two analytic functions very close along a range Γ may differ arbitrarily outside Γ determination of remote resonances from Breit-Wigner parametrizations! Mathematical result (Tikhonov regularization): analytic continuation is stabilized if the class of admissible functions forms a compact set Ciulli et al (1975) Role of the stabilizing condition: Let C be a compact class of analytic functions If F j (t) C and sup Γ F 1 (t) F 2 (t) < ǫ, then for t outside Γ the inequality F 1 (t) F 2 (t) < M(ǫ,t) holds, such that M(ǫ,t) 0 when ǫ 0 Important remark: The inequality derived from Okubo s approach defines a compact set in the Hardy space H 2 of analytic functions with finite L 2 norm on the boundary this ensures the stability of extrapolation to points inside the holomorphy domain

19 Consequences of the integral condition Problem 1: From the L 2 -norm condition R 1 ρ(t) F(t) 2 dt I π t + find constraints on the values of the values F(t n) and the derivatives F (k) (t j ) at some real or complex points

20 Consequences of the integral condition Problem 1: From the L 2 -norm condition R 1 ρ(t) F(t) 2 dt I π t + find constraints on the values of the values F(t n) and the derivatives F (k) (t j ) at some real or complex points Write the problem in a canonical form: Conformal mapping of the t-plane cut for t > t + onto a unit disk by z z(t, t 0 ), with the inverse t(z, t 0 ) t+ t z(t, t 0 ) = 0 t + t, z(t t+ t 0,t 0 ) = t + t Define an outer function w(z), i.e. analytic and without zeros in z < 1, with modulus squared on z = 1 equal to ρ(t) d t/dz : h w(z) = exp 1 R i 2π 2π 0 dθ eiθ +z e iθ z ln[ρ( t(eiθ )) d t/dz ] the function g(z) = F( t(z,t 0 )) w(z) is analytic in z < 1 and satisfies the inequality R 1 2π 2π 0 g(e iθ ) 2 dθ I

21 Meiman problem (interpolation in L 2 norm) If g(z) analytic in z < 1 and R 1 2π g(e iθ ) 2 dθ I 2π 0» 1 k! d k g(z) dz k = g k, 0 k K 1, g(z n) = ξ n, z n = zn, 1 n N z=0 KX 1 Ī = I gk 2, k=0 KX 1 ξ n = ξ n g k zn k k=0 positivity of the following determinant and of its minors: ξ N Ī ξ1 ξ2 ξn z1 ξ 2K (z 1 z 2 ) K (z 1 z N ) K 1 1 z1 2 1 z 1 z 2 1 z 1 z N (z 1 z 2 ) ξ K (z 2 ) 2K (z 2 z N ) K 2 1 z 1 z 2 1 z2 2 1 z 2 z N..... (z 1 z N ) K 1 z 1 z N (z 2 z N ) K 1 z 2 z N z 2K N 1 z 2 N 0

22 Alternative solution based on analytic interpolation theory Convex domain in the space of parameters, defined by the quadratic inequality: NX m,n=1 A mnξ nξ m + K 1 X j,k=0 B jk g j g k + 2 NX KX 1 C kn g k ξ n I n=1 k=0 Blaschke factors: B j (z) = 1 for z = 1 defined recurrently by B 1 (z) = 1, B n(z) = z z n 1 1 zz n 1 B n 1 (z), 2 n N + 1, 1 d K+l k»» 1 z zn β kl = (K+l k)! dz K+l k, Y n = B N+1 (z) z=0 B N+1 (z) z=z n A mn = YnYm z K n zk m 1 X 1, B jk = 1 z nz m l=l C kn = Yn z K n 1 X l=k K β kl z l+1 n β jl β kl

23 Inclusion of the phase in the elastic region Problem 2: From the conditions R 1 ρ(t) F(t) π dt I t + and arg[f(t + iǫ) = δ(t), t + t t in find constraints on the values of the derivatives F (k) (t j ) and the values F(t n)

24 Inclusion of the phase in the elastic region Problem 2: From the conditions R 1 ρ(t) F(t) π dt I t + and arg[f(t + iǫ) = δ(t), t + t t in find constraints on the values of the derivatives F (k) (t j ) and the values F(t n) Standard techniques of functional optimization solution described by the inequality: NX KX 1 NX KX 1 A mnξ nξ m + B jk g j g k + 2 C kn g k ξ n + 1 dθλ(θ)v(θ) I π m,n=1 j,k=0 n=1 k=0 θ in Zθ in λ(θ): the solution of a Fredholm integral equation θr in λ(θ) 1 dθ 2π λ(θ )K Ψ (θ, θ ) = V(θ), e iθ in = z(t in, t 0 ) θ in K Ψ (θ, θ ) sin[(k 1/2)(θ θ ) Ψ(θ)+Ψ(θ )] sin[(θ θ )/2] Ψ(θ) and V(θ): known functions depending linearly on the input values

25 Modified analytic optimization problem Problem 3: From the conditions R 1 ρ(t) F(t) π dt I t in and arg[f(t + iǫ) = δ(t), t + t t in find constraints on the derivatives F (k) (t j ) and the values F(t n)

26 Modified analytic optimization problem Problem 3: From the conditions R 1 ρ(t) F(t) π dt I t in and arg[f(t + iǫ) = δ(t), t + t t in find constraints on the derivatives F (k) (t j ) and the values F(t n) I determined from Okubo s method and data below t in, or from data above t in

27 Modified analytic optimization problem Problem 3: From the conditions R 1 ρ(t) F(t) π dt I t in and arg[f(t + iǫ) = δ(t), t + t t in find constraints on the derivatives F (k) (t j ) and the values F(t n) I determined from Okubo s method and data below t in, or from data above t in Steps of the proof: Define the Omnès function (for t > t in, δ(t) arbitrary smooth function): Z! t δ(t ) O(t) = exp dt π t + t (t t) Define the function h(t) by F(t) = O(t)h(t), with the properties: h(t) is real below t in, i.e. is analytic in the t-plane cut only for t > t in R 1 ρ(t) O(t) π 2 h(t) 2 dt I t in for h(t) we obtained Problem 1, with two modifications: the t-plane cut for t > t + is replaced by the t-plane cut for t > t in the weight ρ(t) is replaced by ρ(t) O(t) 2

28 Solution of Problem 3 Conformal mapping of the t-plane cut for t > t in onto the unit disc z < 1: z(t, t 0 ) = tin t 0 t in t tin t 0 + t in t, z(t 0,t 0 ) = 0 Outer functions with modulus related to ρ(t) and O(t) :» Z 1 2π w(z) = exp dθ eiθ + z 2π 0 e iθ z ln[ρ( t(eiθ, t 0 )) d t/dz ] p Z! tin t(z,t 0 ) ω(z) = exp dt ln O(t ) π t in t t in (t t(z,t 0 )) the function g(z) defined by: g(z) F( t(z,t 0 )) [O( t(z,t 0 ))] 1 w(z) ω(z) is analytic in z < 1 and satisfies 1 2π Z 2π g(e iθ ) 2 dθ I 0 the standard Meiman problem

29 Properties of the bounds Rigorous properties: are independent of the conformal mapping (the parameter t 0 ) and the arbitrary phase δ(t) for t > t in remain the same if the sign is replaced by the equality sign depend in a monotonous way on I: larger I, weaker constraints

30 Properties of the bounds Rigorous properties: are independent of the conformal mapping (the parameter t 0 ) and the arbitrary phase δ(t) for t > t in remain the same if the sign is replaced by the equality sign depend in a monotonous way on I: larger I, weaker constraints Related problems: Nevanlinna-Pick and Schur-Carathéodory interpolation for bounded functions F L sup t>t + F(t) I The bounds in L -norm are stronger than those based in L 2 -norm By varying ρ(t), we can approach the stronger bounds given by the L -norm hints for a suitable choice of ρ(t): compromise between choices leading to strong bounds need to exploit properly the available knowledge of the modulus

31 Particular consequence: domains without zeros The formalism predicts domains in the t-plane where zeros are excluded Insert the assumption F(t c)=0 in the determinant along with other input values If the consistency inequality is violated, the zero is excluded rigorous description of the domains where zeros are forbidden Example: given the input values F(0), F (0) and F(t 1 ), the domain of points t c = t(z c, t 0 ) where zeros of F(t) are excluded is defined by the inequality I g 2 0 g2 1 g 0 g 1 z c g(z 1 ) g 0 g 1 z 1 g 0 g 1 z c z 4 c 1 z 2 c (z cz 1 ) 2 1 z cz 1 g(z 1 ) g 0 g 1 z 1 (z 0 z 1 ) 2 1 z 0 z 1 (z 1 ) 4 1 z 2 1 < 0 The knowledge of zeros is important for testing symmetry properties and as input in some dispersive representations

32 Applications 1 Constraints on the low energy parameters and zeros BD ( ) form factors (Isgur-Wise function) IC, Lellouch, Neubert (1998) Kπ form factors Bourrely, IC (2005), Abbas, Ananthanarayan, IC, Imsong, Ramanan (2010) (Anant s talk) pion electromagnetic form factor IC (2000), Abbas, Anant, IC, Imsong (2011) Dπ form factors Ananthanarayan, IC, Imsong (2011) 2 Extrapolations to intermediate spacelike energies onset of pqcd for the pion form factor Ananthanarayan, IC, Imsong (2012) 3 Bounds on the modulus on the timelike axis consistency checks on pion form factor data Ananthanarayan, IC, Das, Imsong (work in progress) 4 Analytic parametrizations with unitarity constraints BD ( ) form factors (Isgur-Wise function) IC, Lellouch, Neubert (1998) Bπ vector form factor Bourrely, IC, Lellouch (2009) Dπ form factors, pion electromagnetic form factor (work in progress)

33 Applications 1 Constraints on the low energy parameters and zeros BD ( ) form factors (Isgur-Wise function) IC, Lellouch, Neubert (1998) Kπ form factors Bourrely, IC (2005), Abbas, Ananthanarayan, IC, Imsong, Ramanan (2010) (Anant s talk) pion electromagnetic form factor IC (2000), Abbas, Anant, IC, Imsong (2011) Dπ form factors Ananthanarayan, IC, Imsong (2011) 2 Extrapolations to intermediate spacelike energies onset of pqcd for the pion form factor Ananthanarayan, IC, Imsong (2012) 3 Bounds on the modulus on the timelike axis consistency checks on pion form factor data Ananthanarayan, IC, Das, Imsong (work in progress) 4 Analytic parametrizations with unitarity constraints BD ( ) form factors (Isgur-Wise function) IC, Lellouch, Neubert (1998) Bπ vector form factor Bourrely, IC, Lellouch (2009) Dπ form factors, pion electromagnetic form factor (work in progress) Historical review and references in: G. Abbas, B. Ananthanarayan, IC, I.S. Imsong and S. Ramanan, Eur. Phys. J. A 45, 389 (2010), arxiv: [hep-ph]

34 Application I: Low energy constraints on the Dπ weak form factors Ananthanarayan, IC, Imsong, EPJ A 47, 147 (2011) Of interest for the determination of the element V cd of the CKM matrix Input: f +(0) = 0.67 ± 0.10, from LCSR Khodjamirian et al (2009) and lattice HPQCD (2011) low-energy soft-pion theorem (Callan-Treiman): f 0 (M 2 D M2 π ) = f D/f π Dominguez et al (1990) phase at low energies from dominant resonances D and D 0 Okubo s approach: derivatives χ (n) k of a polarization function at Q 2 = 0 R 1 π t + ρ (n) k (t) f k(t) 2 dt χ (n) k, k = +,0 χ (n) k = χ (n)pt k + χ (n)np k : pqcd to two loops Chetyrkin et al (2001) n χ (n)pt + χ (n)np + χ (n) + χ (n)pt 0 χ (n)np 0 χ (n)

35 Application I: Low energy constraints on the Dπ weak form factors Ananthanarayan, IC, Imsong, EPJ A 47, 147 (2011) Of interest for the determination of the element V cd of the CKM matrix Input: f +(0) = 0.67 ± 0.10, from LCSR Khodjamirian et al (2009) and lattice HPQCD (2011) low-energy soft-pion theorem (Callan-Treiman): f 0 (M 2 D M2 π ) = f D/f π Dominguez et al (1990) phase at low energies from dominant resonances D and D 0 Okubo s approach: derivatives χ (n) k of a polarization function at Q 2 = 0 R 1 π t + ρ (n) k (t) f k(t) 2 dt χ (n) k, k = +,0 χ (n) k = χ (n)pt k + χ (n)np k : pqcd to two loops Chetyrkin et al (2001) n χ (n)pt + χ (n)np + χ (n) + χ (n)pt 0 χ (n)np 0 χ (n) Taylor expansion at t = 0: f k (t) = f k (0) 1 + λ t k Mπ λ k t 2 Mπ 4 +, k = +,0

36 Dπ form factors: constraints on slope and curvature Scalar form factor, moment χ (0) 0 4 standard bound standard bound + CT phase + CT pole fit 2 x 10 5 λ 0 // / 2 λ 0 x 10

37 Dπ form factors: constraints on slope and curvature Scalar form factor, moment χ (0) 0 Constraints from various moments 4 2 standard bound standard bound + CT phase + CT pole fit χ 0 (0) χ 0 (1) χ 0 (2) pole fit x 10 5 λ 0 // 0 x 10 5 λ 0 // phase + CT / 2 λ 0 x / 2 λ 0 x 10 Intersection small allowed domain Pole ansatz from Becirevic, Kaidalov (1999) excluded by imposing all the constraints

38 Dπ form factor: domains where zeros are excluded Vector form factor, moment χ (0) 1

39 Dπ form factor: domains where zeros are excluded Vector form factor, moment χ (0) 1 Scalar form factor, moment χ (0) 0 larger region using CT theorem

40 Application I: Low-energy constraints on the pion form factor Ananthanarayan, IC, Imsong, Phys Rev D83, (2011) Input: δ(t) = δ 1 1 (t) for t t in = (M π + M ω) 2 = (0.917 GeV) 2 from Roy equations for ππ amplitude Ananthanarayan et al (2001), Garcia-Martin et al (2011) Recent measurements of the modulus up to high energies BaBar (2009) Precise measurements at spacelike points Horn et al (2008), Huber et al (2008) t Value [ GeV 2 ] F(t) t ± t ±

41 Application I: Low-energy constraints on the pion form factor Ananthanarayan, IC, Imsong, Phys Rev D83, (2011) Input: δ(t) = δ 1 1 (t) for t t in = (M π + M ω) 2 = (0.917 GeV) 2 from Roy equations for ππ amplitude Ananthanarayan et al (2001), Garcia-Martin et al (2011) Recent measurements of the modulus up to high energies BaBar (2009) Precise measurements at spacelike points Horn et al (2008), Huber et al (2008) t Value [ GeV 2 ] F(t) t ± t ± Best results obtained from Problem 3: Example: suitable choice of ρ(t): R 1 ρ(t) F(t) π t 2 dt I in ρ(t) = ρ µ(t) = α2 M 2 µ 12π (t t + ) 3/2 t 7/2 K(t), K(t) = R 1 0 du (1 u)u 2 1 u+m µ 2 u2 /t I â ππ µ = Davier et al (2010), Malaescu (private communication)

42 Application I: Low-energy constraints on the pion form factor Ananthanarayan, IC, Imsong, Phys Rev D83, (2011) Input: δ(t) = δ 1 1 (t) for t t in = (M π + M ω) 2 = (0.917 GeV) 2 from Roy equations for ππ amplitude Ananthanarayan et al (2001), Garcia-Martin et al (2011) Recent measurements of the modulus up to high energies BaBar (2009) Precise measurements at spacelike points Horn et al (2008), Huber et al (2008) t Value [ GeV 2 ] F(t) t ± t ± Best results obtained from Problem 3: Example: suitable choice of ρ(t): R 1 ρ(t) F(t) π t 2 dt I in ρ(t) = ρ µ(t) = α2 M 2 µ 12π (t t + ) 3/2 t 7/2 K(t), K(t) = R 1 0 du (1 u)u 2 1 u+m µ 2 u2 /t I â ππ µ = Davier et al (2010), Malaescu (private communication) Taylor expansion: F(t) = r2 π t + ct2 + dt 3 + r 2 π = 0.43 ± 0.01 fm2

43 Low-energy constraints: allowed domain in the c-d plane 10.5 <r π 2 > = 0.43 fm 2 <r π 2> = 0.44 fm 2 d [GeV -6 ] c [GeV -4 ]

44 Low-energy constraints: allowed domain in the c-d plane 2 2 no spacelike datum 10.5 <r π> = 0.43 fm F(t 1 ) (cent) <r π 2> = 0.44 fm 2 F(t 1 ) (max) F(t 1 ) (min) 10 d [GeV -6 ] 10 d [GeV -6 ] <r π 2 > = 0.43 fm c [GeV -4 ] c [GeV -4 ]

45 Low-energy constraints: allowed domain in the c-d plane 2 2 no spacelike datum 10.5 <r π> = 0.43 fm F(t 1 ) (cent) <r π 2> = 0.44 fm 2 F(t 1 ) (max) F(t 1 ) (min) 10 d [GeV -6 ] 10 d [GeV -6 ] <r π 2 > = 0.43 fm c [GeV -4 ] c [GeV -4 ] By varying all the input parameters, r 2 π, F(t 1 ),â ππ µ, δ 1 1 (t): 3.75 GeV 4 c 3.98 GeV 4, 9.91 GeV 6 d GeV 6 with a strong correlation between the values of c and d

46 Pion form factor: domains without zeros rπ 2 = 0.43 ± 0.01fm2 and F(t 1 ) = simple zeros on the real axis are excluded in the range 4.46 GeV 2 t GeV 2

47 Pion form factor: domains without zeros rπ 2 = 0.43 ± 0.01fm2 and F(t 1 ) = simple zeros on the real axis are excluded in the range 4.46 GeV 2 t GeV 2 Complex zeros, no spacelike input r 2 π = 0.43 fm2 (smaller domain) r 2 π = 0.44 fm2 (bigger domain)

48 Pion form factor: domains without zeros rπ 2 = 0.43 ± 0.01fm2 and F(t 1 ) = simple zeros on the real axis are excluded in the range 4.46 GeV 2 t GeV 2 Complex zeros, no spacelike input Complex zeros, with spacelike input: rπ 2 = 0.43 fm2 (smaller domain) F(t 1 ) = 0.234, rπ 2 = 0.43 fm2 r 2 π = 0.44 fm2 (bigger domain)

49 Application II: Bounds on the pion form factor on the spacelike axis Ananthanarayan, IC, Imsong, Phys Rev D85, (2012) Input: F(0) = 1, r 2 π = 0.43 ± 0.01 fm 2, F( 2.45 GeV 2 ) = ± arg[f(t + iǫ)] = δ 1 1 (t), 4M2 π t t in, t in = (M ω + M π) 2 R 1 ρ(t) F(t) 2 dt I, π t in for suitable choices of the weight ρ(t) Direct evaluation of I : BaBar data Aubert et al (2009) up to 3 GeV very conservative estimate above 3 GeV, imposing the decrease F(t) 1/t above 20 GeV small sensitivity to the high-energy assumptions for weights that decrease as 1/ t or faster

50 Application II: Bounds on the pion form factor on the spacelike axis Ananthanarayan, IC, Imsong, Phys Rev D85, (2012) Input: F(0) = 1, r 2 π = 0.43 ± 0.01 fm 2, F( 2.45 GeV 2 ) = ± arg[f(t + iǫ)] = δ 1 1 (t), 4M2 π t t in, t in = (M ω + M π) 2 R 1 ρ(t) F(t) 2 dt I, π t in for suitable choices of the weight ρ(t) Direct evaluation of I : BaBar data Aubert et al (2009) up to 3 GeV very conservative estimate above 3 GeV, imposing the decrease F(t) 1/t above 20 GeV small sensitivity to the high-energy assumptions for weights that decrease as 1/ t or faster ρ(t) I ± / t ± /t ± /t ± 0.017

51 Choice of the best weight ρ(t) = 1/ t, inclusion of F(t 2 ) 1 without spacelike datum with spacelike datum 0.8 Q 2 F π (-Q 2 ) [GeV 2 ] Q 2 [GeV 2 ]

52 Choice of the best weight ρ(t) = 1/ t, inclusion of F(t 2 ) Comparison of various weights Q 2 F π (-Q 2 ) [GeV 2 ] without spacelike datum with spacelike datum Q 2 F π (-Q 2 ) [GeV 2 ] /t 1/2 1/2 1/t 1/t 1/t 2 1/t Q 2 [GeV 2 ] Q 2 [GeV 2 ] weights that decrease too fast have a weak constraining power at large energies the weight related to the standard Okubo approach and ρ µ, which decrease like 1/t 2, are not useful for extrapolation to large Q 2 weights that decrease too slowly are sensitive to the asymptotic tail suitable choice: ρ(t) = 1/ t

53 Effect of the uncertainty of the input Including the errors: vary separately each input and combine the resulting errors in quadrature vary simultaneously all the input quantities inside their error intervals lead to comparable results

54 Effect of the uncertainty of the input Including the errors: vary separately each input and combine the resulting errors in quadrature vary simultaneously all the input quantities inside their error intervals lead to comparable results 0.6 Q 2 F π (-Q 2 ) [GeV 2 ] Q 2 [GeV 2 ] Bounds for the optimal choice ρ = 1 t white band: allowed domain for central values of the input variables grey bands: enlarged domain when the input is varied inside the error bars

55 Comparison with experimental data Low energy data Q 2 F π (-Q 2 ) [GeV 2 ] Our prediction Ackermann Amendolia Bebek Brauel Brown Horn Volmer Q 2 [GeV 2 ] A few points (Amendolia, Bebek) in conflict with the bounds

56 Comparison with experimental data Low energy data High energy data Q 2 F π (-Q 2 ) [GeV 2 ] Our prediction Ackermann Amendolia Bebek Brauel Brown Horn Volmer Q 2 F π (-Q 2 ) [GeV 2 ] our prediction Ackermann Amendolia Bebek Brauel Brown Horn Volmer Q 2 [GeV 2 ] Q 2 [GeV 2 ] A few points (Amendolia, Bebek) in conflict with the bounds

57 Comparison with pqcd and nonperturbative theoretical models F LO QCD ( Q2 ) = 8πf 2 π αs(µ2 ) Q 2 FQCD NLO( Q2 ) = 8f π 2 h i α2 s (µ2 ) β0 Q 2 4 ln µ2 Q The asymptotic regime is known to set in quite slowly Various nonperturbative models for the intermediate regions Definite conclusions difficult due to lack of data at high energy

58 Comparison with pqcd and nonperturbative theoretical models F LO QCD ( Q2 ) = 8πf 2 π αs(µ2 ) Q 2 FQCD NLO( Q2 ) = 8f π 2 h i α2 s (µ2 ) β0 Q 2 4 ln µ2 Q The asymptotic regime is known to set in quite slowly Various nonperturbative models for the intermediate regions Definite conclusions difficult due to lack of data at high energy Q 2 F π (-Q 2 ) [GeV 2 ] Our prediction LO LO + NLO Braun, Khodjamirian, Maul (2000) Radyushkin (2001) Bakulev, Passek-Kumericki, Schroers, Stefanis (2004) Grigoryan, Radyushkin (2007) Brodsky, de Teramond (2008) Bakulev, Pimikov, Stefanis (2009) Q 2 [GeV 2 ]

59 Application III: Bounds on the pion form factor in the timelike region Ananthanarayan, IC, Das, Imsong, preliminary results The formalism applied up to now (Problem 3) does not use as input data on the modulus F(t) for t < t in On the other hand, the formalism allows to find bounds on this quantity The function is analytic and real for t < t in g(z) F( t(z, t 0 ))[O( t(z, t 0 ))] 1 w(z) ω(z) Derive upper and lower bounds on g(z) from Meiman condition They lead to bounds on the modulus F(t), calculated as O(t) = exp F(t) = O(t) g( z(t 0,t)) w( z(t 0,t)) t π PV Z! δ(t ) dt t + t (t t)

60 Bounds on the pion form factor in the timelike region Isospin breaking by ω-ρ interference in e + e annihilation:» δ1 1 (t) δ1 1 (t) + arg 1 + ǫt, t ω = (M ω i/2γ ω) 2, ǫ t ω t Leutwyler (2002), Hanhart (2012)

61 Bounds on the pion form factor in the timelike region Isospin breaking by ω-ρ interference in e + e annihilation:» δ1 1 (t) δ1 1 (t) + arg 1 + ǫt, t ω = (M ω i/2γ ω) 2, ǫ t ω t Methods of including the uncertainties: Leutwyler (2002), Hanhart (2012) vary separately each input and combine the errors in quadrature vary simultaneously all the input variables inside their error intervals central inputs errors method_1 errors method_2 F(t) ρ(t) = 1/t t 1/2 [GeV]

62 Bounds on the pion form factor in the timelike region Isospin breaking by ω-ρ interference in e + e annihilation:» δ1 1 (t) δ1 1 (t) + arg 1 + ǫt, t ω = (M ω i/2γ ω) 2, ǫ t ω t Leutwyler (2002), Hanhart (2012) ρ(t) = 1/t Belle ρ(t) = 1/t, isospin breaking BaBaR CMD-2 KLOE F(t) F(t) t 1/2 [GeV] t 1/2 [GeV] accurate recent data: Belle (2008), BaBar (2009), KLOE (2011), CMD-2 (2007)

63 Bounds on the pion form factor in the timelike region Isospin breaking by ω-ρ interference in e + e annihilation:» δ1 1 (t) δ1 1 (t) + arg 1 + ǫt, t ω = (M ω i/2γ ω) 2, ǫ t ω t Leutwyler (2002), Hanhart (2012) ρ(t) = 1/t, isospin breaking BaBaR KLOE 3 F(t) t 1/2 [GeV] The bounds describe more precisely the modulus at low energies than the data A few experimental points (BaBar) in conflict with the bounds

64 Application IV: Parametrization of the Bπ vector form factor Bourrely, IC, Lellouch, arxiv: , Phys Rev D79, (2009) Of interest for the determination of the element V ub of the CKM matrix dγ dq 2 ( B 0 π + l ν l ) = G2 F V ub 2 192π 3 mb 3 λ 3/2 (q 2 ) f +(q 2 ) 2 Physical range of semileptonic decays: 0 q 2 t (m B 0 m π +) 2 = GeV 2 Basic properties: f +(q 2 ) analytic in the q 2 -plane cut for q 2 t +, with t + = (m B 0 + m π +) 2 except for a pole at q 2 = M 2 B Threshold behaviour: Imf +(q 2 ) (q 2 t +) 3/2 Unitarity constraint (Okubo): χ 1 (0) = 3[ αs( m b)] 32π 2 m 2 b R 1 ρ(t) f π +(t) 2 dt χ 1 (0) t + m b ūu m 6 b αsg2 12πm 6 b Generalis (1990), Lellouch (1996), Arnesen et al (2005)

65 New parametrization of f + (q 2 ) K f +(q 2 1 X 1 ) = 1 q 2 /mb 2 b k (t 0 )»z k ( 1) k K kk zk, z = z(q 2,t 0 ) k=0 Unitarity constraint: KP B jk (t 0 )b j (t 0 )b k (t 0 ) 1 j,k=0 t 0(GeV 2 ) B 00 B 01 B 02 B 04 B 04 B t opt t B j(j+k) = B 0k, B jk = B kj Optimal choice of t 0 : t opt (m B + m π)( m B m π) 2 = GeV Im z Re z

66 Fit of theoretical and experimental data Theoretical and experimental input: f +(0) = 0.26 ± 0.03, from LCSR Khodjamirian et al (1007, Ball (2007) lattice calculations at eight q 2 -points FNAL-MILC and HPQCD experimental data on the partial branching fractions over bins in q 2 CLEO BaBar, Belle,

67 Fit of theoretical and experimental data Theoretical and experimental input: f +(0) = 0.26 ± 0.03, from LCSR Khodjamirian et al (1007, Ball (2007) lattice calculations at eight q 2 -points FNAL-MILC and HPQCD experimental data on the partial branching fractions over bins in q 2 CLEO BaBar, Belle, Combined fit of experimental and theoretical points, with unitarity constraint: 0 1 KX L(b j, V ub ) = χ 2 (b j, V ub ) + B jk b j b k 1A j,k=0 χ 2 (b k, V ub ) = χ 2 th + χ2 exp χ 2 th = P 8 [fj in f +(q 2 1 j )]Cjk [f in k f+(q2 k )] + (f+(0) f LCSR) 2 /(δf LCSR ) 2 j,k=1 χ 2 P exp = 22 [Bj in B j (f +)]C 1 B jk [Bin k B k(f +)] j,k=1

68 Systematic error and fitting strategy Systematic error: In this approach the systematic error is the truncation error of the expansion Due to the unitarity constraint, the higher order coefficients cannot grow arbitrarily the truncation error can be controlled let b max K+1 be the maximum value of b K+1, allowed by the unitarity constraint, for fixed values of b k, k K, obtained from the fit realistic prescription for the error: δf +(q 2 ) syst = bmax K+1 zk+1 1 q 2 /m 2 B

69 Systematic error and fitting strategy Systematic error: In this approach the systematic error is the truncation error of the expansion Due to the unitarity constraint, the higher order coefficients cannot grow arbitrarily the truncation error can be controlled let b max K+1 be the maximum value of b K+1, allowed by the unitarity constraint, for fixed values of b k, k K, obtained from the fit realistic prescription for the error: δf +(q 2 ) syst = bmax K+1 zk+1 1 q 2 /m 2 B Fitting strategy: Increase the number of parameters until the systematic error becomes negligible compared to the statistical error along the whole semileptonic region

70 Optimal fit j b j 0.42 ± ± ± ± 4.1 Also: V ub = (3.54 ± 0.30) 10 3 (remarkably close to expectations from global CKM fits) total LCSR LQCD Belle CLEO Babar-t BaBar-u n data χ

71 Optimal fit j b j 0.42 ± ± ± ± 4.1 Also: V ub = (3.54 ± 0.30) 10 3 (remarkably close to expectations from global CKM fits) total LCSR LQCD Belle CLEO Babar-t BaBar-u n data χ f + (q 2 ) LCSR FNAL-MILC HPQCD f + (q 2 ) (1-q 2 2 / m * B ) q 2 [GeV 2 ] q 2 [GeV 2 ] Errors found by standard χ 2 analysis, not the linear approximation in the error propagation

72 Summary and conclusions The availability of a reliable bound on an integral involving the square of the modulus of a form factor on the unitarity cut allows one to: constrain the form factor shape parameters at low energy isolate domains in the complex plane where zeros are excluded find bounds at intermediate spacelike regions and test the onset of perturbative QCD control the truncation error of analytic parametrizations The knowledge of the phase along a part of the unitarity cut considerably improves the results Precise values at points inside the analyticity domain (from ChPT, lattice, experiment) crucial for improving the predictions