[PPT] - Light mesons from tau decays Sergi Gonzlez-Sols 1 Indiana University PowerPoint Presentation

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Light mesons from tau decays

Sergi Gonzàlez-Solís1

Indiana University Center for Exploration of Energy and Matter based on: Escribano, Gonzàlez-Solís, Jamin, Roig JHEP 1409 (2014), Escribano, Gonzàlez-Solís, Roig PRD 94 (2016), 034008, Gonzàlez-Solís, Roig 1902.02273 [hep-ph]

International Workshop on e+e− collisions from Phi to Psi

Budker INP , Novosibirsk, march 1, 2019

1sgonzal@iu.edu

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Test of QCD and ElectroWeak Interactions Inclusive decays: τ − → (¯ ud, ¯ us)ντ Full hadron spectra (precision physics) Fundamental SM parameters: αs(mτ),ms,∣Vus∣ Exclusive decays: τ − → (PP,PPP,...)ντ specific hadron spectrum (approximate physics) Hadronization of QCD currents, study of Form Factors, resonance parameters (MR,ΓR)

S.Gonzàlez-Solís Phi to Psi 2019 march 1, 2019 2 / 33

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τ decays into two mesons

hadronization

τ − ντ W − ¯ u d′ = Vudd + Vuss P − P ′0

dΓ(τ − → P −P 0ντ) ds = G2

F ∣Vui∣2m3 τ

768π3 Shad

EW C2 P P ′ (1 −

s M 2

τ

)

2

{(1 + 2s m2

τ

)λ3/2

P −P 0(s)∣F P −P 0 V

(s)∣2 + 3∆2

P −P 0

s2 λ1/2

P −P 0(s)∣F P −P 0 S

(s)∣2}

τ − → π−π0ντ: Pion vector form factor, ρ(770),ρ(1450),ρ(1700) τ − → K−KSντ: Kaon vector form factor, ρ(770),ρ(1450),ρ(1700) τ − → KSπ−ντ: Kπ form factor, K∗(892),K∗(1410), Kℓ3, Vus (Passemar) τ − → K−η(′)ντ: K∗(1410), Vus τ − → π−η(′)ντ: isospin-violating decays

S.Gonzàlez-Solís Phi to Psi 2019 march 1, 2019 3 / 33

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The pion vector form factor: Motivation Enters the description of many physical processes

π− π0

F π

V (s) ∝

see talk by Colangelo

Belle measurement of the pion vector form factor (0805.3773)

high-statistics data until de τ mass
sensitive to ρ(1450) and ρ(1700)
our aim: to improve the description
f the ρ(1450) and ρ(1700) region

S.Gonzàlez-Solís Phi to Psi 2019 march 1, 2019 4 / 33

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Dispersive representation of the pion vector form factor F π

V (s) = exp[α1s + α2

2 s2 + s3 π ∫

scut 4m2

π

ds′ δ1

1(s′)

(s′)3(s′ − s − i0)] , Form Factor phase δ1

1(s)

4m2

π < s < 1 GeV: ππ phase from Roy

0.5 1.0 1.5 2.0 50 100 150 s GeV ΠΠ Phase

(García-Martín et.al PRD 83, 074004 (2011))

1 < s < m2

τ: "Pheno" phase shift

m2

τ < s: phase guided smoothly to π

Low-energy observables

F π

V (s)

= 1 + 1 6⟨r2⟩π

V s + cπ V s2 + dπ V s3 + ⋯.

⟨r2⟩π

V

= 6α1 , cπ

V = 1

2 (α2 + α2

1) . S.Gonzàlez-Solís Phi to Psi 2019 march 1, 2019 5 / 33

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ChPT with resonances + Omnès: Exponential representation Get a model for the (Pheno) phase F π

V (s) = Pn(s)exp

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ sn π ∫

∞ 4m2

π

ds′ (s′)n δ1

1(s′)

s′ − s − i0 ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ , ππ → ππ scattering at O(p2) T(s) = s − m2

π

F 2

π

→ T 1

1 (s) = sσ2 π(s)

96πF 2

π

→ δ1

1(s) = σπ(s)T 1 1 (s) = sσ3 π(s)

96πF 2

π

, Exponential Omnès representation of the form factor F π

V (s)

= M2

ρ

M2

ρ − s − iMρΓρ(s) exp

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ − s 96π2F 2

π

Re[Aπ(s,µ2) + 1 2AK(s,µ2)] ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ Γρ(s) = − Mρs 96π2F 2

π

Im[Aπ(s) + 1 2AK(s)] = Mρs [ ( )3 ( −

2) +

( )3 ( −

2 )]

S.Gonzàlez-Solís Phi to Psi 2019 march 1, 2019 6 / 33

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Incorporation of the ρ′ ≡ ρ(1450),ρ′′ ≡ ρ(1700) F π

V (s)

= M2

ρ + s(γeiφ1 + δeiφ2)

M2

ρ − s − iMρΓρ(s)

exp ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ Re ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ − s 96π2F 2

π

(Aπ(s) + 1 2AK(s)) ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ −γ seiφ1 M2

ρ′ − s − iMρ′Γρ′(s) exp

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ − sΓρ′(M2

ρ′)

πM3

ρ′σ3 π(M2 ρ′)ReAπ(s)

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ −δ seiφ2 M2

ρ′′ − s − iMρ′′Γρ′′(s) exp

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ − sΓρ′′(M2

ρ′′)

πM3

ρ′′σ3 π(M2 ρ′′)ReAπ(s)

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ , Γρ′,ρ′′(s) = Γρ′,ρ′′ s M2

ρ′,ρ′′

σ3

π(s)

σ3

π(M2 ρ′,ρ′′)θ(s − 4m2 π).

tanδ1

1(s) = ImF π V (s)

ReF π

V (s)

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Dispersive Fits to the Pion Vector Form Factor Fits for different values of scut and matching at 1 GeV

Parameter scut [GeV2] Fits m2

τ

4 (reference fit) 10 ∞ Fit 1 α1 [GeV−2] 1.87(1) 1.88(1) 1.89(1) 1.89(1) α2 [GeV−4] 4.40(1) 4.34(1) 4.32(1) 4.32(1) mρ [MeV] = 773.6(9) = 773.6(9) = 773.6(9) = 773.6(9) Mρ [MeV] = mρ = mρ = mρ = mρ Mρ′ [MeV] 1365(15) 1376(6) 1313(15) 1311(5) Γρ′ [MeV] 562(55) 603(22) 700(6) 701(28) Mρ′′[MeV] 1727(12) 1718(4) 1660(9) 1658(1) Γρ′′ [MeV] 278(1) 465(9) 601(39) 602(3) γ 0.12(2) 0.15(1) 0.16(1) 0.16(1) φ1 −0.69(1) −0.66(1) −1.36(10) −1.39(1) δ −0.09(1) −0.13(1) −0.16(1) −0.17(1) φ2 −0.17(5) −0.44(3) −1.01(5) −1.03(2) χ2/d.o.f 1.47 0.70 0.64 0.64

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Form Factor phase shift for different values of scut

––– –––

–––

–––

0.0

0.5 1.0 1.5 2.0 50 100 150 200 s GeV ∆1

1s degrees Roy scutmΤ

2

scut4 GeV2 scut10 GeV2 scut

The results can be found in tables provided as ancillary material in

1902.02273 [hep-ph]

S.Gonzàlez-Solís Phi to Psi 2019 march 1, 2019 9 / 33

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Modulus squared of the pion vector form factor

–––

–––

0.0

0.5 1.0 1.5 2.0 2.5 3.0 0.001 0.01 0.1 1 10 s GeV2 FV

Π 2 Belle data 2008 This work scut4 GeV2 This work scut

The results can be found in tables provided as ancillary material in

1902.02273 [hep-ph]

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Variant (I) Fits for different matching point and with scut = 4 GeV

Parameter Matching point [GeV] Fits 0.85 0.9 0.95 1 (reference fit) Fit I α1 [GeV−2] 1.88(1) 1.88(1) 1.88(1) 1.88(1) α2 [GeV−4] 4.35(1) 4.35(1) 4.34(1) 4.34(1) mρ [MeV] = 773.6(9) = 773.6(9) = 773.6(9) = 773.6(9) Mρ [MeV] = mρ = mρ = mρ = mρ Mρ′ [MeV] 1394(6) 1374(8) 1351(5) 1376(6) Γρ′ [MeV] 592(19) 583(27) 592(2) 603(22) Mρ′′[MeV] 1733(9) 1715(1) 1697(3) 1718(4) Γρ′′ [MeV] 562(3) 541(45) 486(7) 465(9) γ 0.12(1) 0.12(1) 0.13(1) 0.15(1) φ1 −0.44(3) −0.60(1) −0.80(1) −0.66(1) δ −0.13(1) −0.13(1) −0.13(1) −0.13(1) φ2 −0.38(3) −0.51(2) −0.62(1) −0.44(3) χ2/d.o.f 0.75 0.74 0.68 0.70

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Variant (II): Inclusion of intermediate states other than ππ Fit A: ρ′ → K ¯ K and ρ′′ → K ¯ K Fit B: ρ′ → K ¯ K + ρ′ → ωπ

Parameter scut = 4 GeV2 Fit A Fit B reference fit α1 [GeV−2] 1.87(1) 1.88(1) 1.88(1) α2 [GeV−4] 4.37(1) 4.35(1) 4.34(1) mρ [MeV] = 773.6(9) = 773.6(9) = 773.6(9) Mρ [MeV] = mρ = mρ = mρ Mρ′ [MeV] 1373(5) 1441(3) 1376(6) Γρ′ [MeV] 462(14) 576(33) 603(22) Mρ′′[MeV] 1775(1) 1733(9) 1718(4) Γρ′′ [MeV] 412(27) 349(52) 465(9) γ 0.13(1) 0.15(3) 0.15(1) φ1 −0.80(1) −0.53(5) −0.66(1) δ −0.14(1) −0.14(1) −0.13(1) φ2 −0.44(2) −0.46(3) −0.44(3) χ2/d.o.f 0.93 0.70 0.70

S.Gonzàlez-Solís Phi to Psi 2019 march 1, 2019 12 / 33

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Variant (III) Dispersive representation of the pion vector form factor F π

V (s) = exp[ s

π ∫

scut 4m2

π

ds′ δ1

1(s′)

(s′)(s′ − s − i0) + s π ∫

∞ scut

ds′ δeff(s′) (s′)(s′ − s − i0)]Σ(s) Properties for δeff(s)

δeff(scut) = δ1

1(scut) and δeff(s) → π for large s to recover 1/s fall-off

δeff(s) = π + (δ1

1(scut) − π) scut

s Integrating the piece with δeff(s) F π

V (s)

= e1−

δ1 1(scut) π

(1 − s scut )

(1−

δ1 1(scut) π

) scut

s

(1 − s scut )

−1

× exp[ s π ∫

scut 4m2

π

ds′ δ1

1(s′)

(s′)(s′ − s − i0)]Σ(s) Σ(s) =

∞

∑

i=0

aiωi(s), ω(s) = √scut − √scut − s √scut + √scut − s

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The resulting fit parameters are found to be a1 = 2.99(12), Mρ′ = 1261(7)MeV , Γρ′ = 855(15)MeV , Mρ′′ = 1600(1)MeV , Γρ′′ = 486(26)MeV , γ = 0.25(2), φ1 = −1.90(6), δ = −0.15(1), φ2 = −1.60(4), with a χ2/d.o.f = 32.3/53 ∼ 0.61 for the one-parameter fit, and a1 = 3.03(20), a2 = 1.04(2.10), Mρ′ = 1303(19)MeV , Γρ′ = 839(102)MeV , Mρ′′ = 1624(1)MeV , Γρ′′ = 570(99)MeV γ = 0.22(10), φ1 = −1.65(4), δ = −0.18(1), φ2 = −1.34(14), with a χ2/d.o.f = 35.6/52 ∼ 0.63 for the two-parameter fit.

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Form Factor phase shift for different parametrizations

––– –––

0.0

0.5 1.0 1.5 2.0 50 100 150 200 s GeV ∆1

1s degrees Fit 1 reference fit Fit 1–Ρ Fit I Fit A Fit singularities

The results can be found in tables provided as ancillary material in

1902.02273 [hep-ph]

S.Gonzàlez-Solís Phi to Psi 2019 march 1, 2019 15 / 33

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Modulus squared of the pion vector form factor

–––

–––

0.0

0.5 1.0 1.5 2.0 2.5 3.0 0.001 0.01 0.1 1 10 s GeV2 FV

Π 2 Belle data 2008 Fit 1 reference fit Fit 1–Ρ Fit I Fit A Fit singularities

The results can be found in tables provided as ancillary material in

1902.02273 [hep-ph]

S.Gonzàlez-Solís Phi to Psi 2019 march 1, 2019 16 / 33

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Central results Fit results (central value ± statistical fit error ± systematic th. error) α1 = 1.88 ± 0.01 ± 0.01 GeV−2, α2 = 4.34 ± 0.01 ± 0.03 GeV−4, Mρ ≐ 773.6 ± 0.9 ± 0.3 MeV , Mρ′ = 1376 ± 6+18

−73 MeV ,

Γρ′ = 603 ± 22+236

−141 MeV ,

Mρ′′ = 1718 ± 4+57

−94 MeV ,

Γρ′′ = 465 ± 9+137

−53 MeV ,

γ = 0.15 ± 0.01+0.07

−0.03 ,

φ1 = −0.66 ± 0.01+0.22

−0.99 ,

δ = −0.13 ± 0.01+0.00

−0.05 ,

φ2 = −0.44 ± 0.03+0.06

−0.90 ,

Physical pole mass and width Mpole

ρ

= 760.6 ± 0.8 MeV , Γpole

ρ

= 142.0 ± 0.4 MeV , Mpole

ρ′

= 1289 ± 8+52

−143 MeV ,

Γpole

ρ′

= 540 ± 16+151

−111 MeV ,

Mpole

ρ′′

= 1673 ± 4+68

−125 MeV ,

Γpole

ρ′′

= 445 ± 8+117

−49 MeV ,

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Determination of the ρ(1450) and ρ(1700) resonance parameters

Reference Model parameters Pole parameters Data Mρ′, Γρ′ [MeV] Mpole

ρ′

, Γpole

ρ′

[MeV] ALEPH 1328 ± 15, 468 ± 41 1268 ± 19, 429 ± 31 τ ALEPH 1409 ± 12, 501 ± 37 1345 ± 15, 459 ± 28 τ & e+e− Belle (fixed ∣F π

V (0)∣2)

1446 ± 7 ± 28, 434 ± 16 ± 60 1398 ± 8 ± 31, 408 ± 13 ± 50 τ Belle (all free) 1428 ± 15 ± 26, 413 ± 12 ± 57 1384 ± 16 ± 29, 390 ± 10 ± 48 τ Dumm et. al. — 1440 ± 80, 320 ± 80 τ Celis et. al. 1497 ± 7, 785 ± 51 1278 ± 18, 525 ± 16 τ Bartos et. al. — 1342 ± 47, 492 ± 138 e+e− Bartos et. al. — 1374 ± 11, 341 ± 24 τ This work 1376 ± 6+18

−73, 603 ± 22+236 −141

1289 ± 8+52

−143, 540 ± 16+151 −111

τ Reference Model parameters Pole parameters Data (Mρ′′, Γρ′′) [MeV] (Mpole

ρ′′

, Γpole

ρ′′ ) [MeV]

ALEPH = 1713, = 235 1700, 232 τ ALEPH 1740 ± 20, = 235 1728 ± 20, 232 τ & e+e− Belle (fixed ∣F π

V (0)∣2)

1728 ± 17 ± 89, 164 ± 21+89

−26

1722 ± 18, 163 ± 21+88

−27

τ Belle (all free) 1694 ± 41, 135 ± 36+50

−26

1690 ± 94, 134 ± 36+49

−28

τ Dumm et. al. — 1720 ± 90, 180 ± 90 τ Celis et. al. 1685 ± 30, 800 ± 31 1494 ± 37, 600 ± 17 τ Bartos et. al. — 1719 ± 65, 490 ± 17 e+e− Bartos et. al. — 1767 ± 52, 415 ± 120 τ This work 1718 ± 4+57

−94, 465 ± 9+137 −53

1673 ± 4+68

−125, 445 ± 8+117 −49

τ

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Determination of resonance parameters To look for a zero of the propagator in the complex plane

M pole

ρ

= 760.6 ± 0.8 MeV , Γpole

ρ

= 142.0 ± 0.4 MeV , M pole

ρ′

= 1289 ± 8+52

−143 MeV ,

Γpole

ρ′

= 540 ± 16+151

−111 MeV ,

M pole

ρ′′

= 1673 ± 4+68

−125 MeV ,

Γpole

ρ′′

= 445 ± 8+117

−49 MeV ,

S.Gonzàlez-Solís Phi to Psi 2019 march 1, 2019 19 / 33

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Kaon vector Form Factor dΓ(τ − → K−K0ντ) d√s = G2

F ∣Vud∣2

768π3 M3

τ (1 −

s M2

τ

)

2

(1 + 2s M2

τ

)σ3

K(s)∣F K V (s)∣2 ,

Chiral Perturbation Theory O(p4) FK+K−(s) = 1 + 2Lr

9

F 2

π

− s 192π2F 2

π

[Aπ(s,µ2) + 2AK(s,µ2)] , FK0 ¯

K0(s)

= − s 192π2F 2

π

[Aπ(s,µ2) − AK(s,µ2)] . Extract the I = 1 component F K

V (s) = 1 + 2Lr 9

F 2

π

− s 96π2F 2

π

[Aπ(s,µ2) + 1 2AK(s,µ2)] . At O(p4), the pion and kaon vector form factor are the same Assumption: we consider that both are also the same at higher energies

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Kaon vector form factor Omnès exponential representation Different resonance mixing contribution than F π

V (s)

F K

V (s)

= M2

ρ + s(˜

γei˜

φ1 + ˜

δei˜

φ2)

M2

ρ − s − iMρΓρ(s)

exp ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ Re ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ − s 96π2F 2

π

(Aπ(s) + 1 2AK(s)) ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ −˜ γ sei˜

φ1

M2

ρ′ − s − iMρ′Γρ′(s) exp

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ − sΓρ′(M2

ρ′)

πM3

ρ′σ3 π(M2 ρ′)ReAπ(s)

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ −˜ δ sei˜

φ2

M2

ρ′′ − s − iMρ′′Γρ′′(s) exp

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ − sΓρ′′(M2

ρ′′)

πM3

ρ′′σ3 π(M2 ρ′′)ReAπ(s)

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ , Γρ′,ρ′′(s) = Γρ′,ρ′′ s M2

ρ′,ρ′′

σ3

π(s)

σ3

π(M2 ρ′,ρ′′)θ(s − 4m2 π).

Extract the phase tanφKK(s) = ImF K

V (s)/ReF K V (s)

Use a three-times subtracted dispersion relation

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Fit results to BaBar τ − → K−KSντ data

Parameter scut = 4 [GeV2] Fit i) Fit ii) Fit iii) Fit iv) ˜ α1 = 1.88(1) = 1.84 = 1.88(1) — ˜ α2 = 4.34(1) = 4.34 = 4.34(1) — Mρ′ [MeV] 1467(24) 1538(32) 1489(25) 1411(12) Γρ′ [MeV] 415(48) 604(83) 297(36) 394(35) ˜ γ 0.10(2) 0.36(11) 0.10(2) 0.09(1) ˜ φ1 −1.19(16) −1.48(13) −1.10(15) −1.88(9) χ2/d.o.f. 2.9 1.9 2.9 3.3

—

1.0 1.2 1.4 1.6 20 40 60 80 100 mKKS GeV 1N dN dmKKS 103

BaBar data 2018 Our prediction Our fit exponential Our fit dispersive

1902.02273 [hep-ph]

S.Gonzàlez-Solís Phi to Psi 2019 march 1, 2019 22 / 33

SLIDE 23

Combined analysis of F π

V (s) and τ − → K−KSντ Parameter scut = 4 [GeV2] Fit a Fit b Fit c α1 1.88(1) 1.89(1) 1.87(1) α2 4.34(2) 4.31(2) 4.38(3) ˜ α1 = α1 = α1 1.88(24) ˜ α2 = α2 = α2 4.38(29) mρ [MeV] = 773.6(9) = 773.6(9) = 773.6(9) Mρ [MeV] = mρ = mρ = mρ Mρ′ [MeV] 1396(19) 1453(19) 1406(61) Γρ′ [MeV] 507(31) 499(51) 524(149) Mρ′′[MeV] 1724(41) 1712(32) 1746(1) Γρ′′ [MeV] 399(126) 284(72) 413(362) γ 0.12(3) 0.15(3) 0.11(11) ˜ γ 0.11(2) = γ 0.11(5) φ1 −0.23(26) 0.29(21) −0.27(42) ˜ φ1 −1.83(14) −1.48(13) −1.90(67) δ −0.09(2) −0.07(2) −0.10(5) ˜ δ = 0 = 0 −0.01(4) φ2 −0.20(31) 0.27(29) −1.15(71) ˜ φ2 = 0 = 0 0.40(3) χ2/d.o.f 1.52 1.19 1.25

—

0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.001 0.01 0.1 1 10 s GeV2 FV

Π 2 Belle data 2008 Fit a Fit b Fit c

—

1.0 1.2 1.4 1.6 20 40 60 80 100 mKKS GeV 1NdN dq 103

BaBar 2018 Fit a Fit b Fit c

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SLIDE 24

Combined analysis of τ − → KSπ−ντ and τ − → K−ηντ decays RχT with two resonances: K∗(892) and K∗(1410)

figure courtesy of

D. Boito

̃ F Kπ

V

(s) = m2

K⋆ − κK⋆ ̃

HKπ(0) + γs D(mK⋆, γK⋆) − γs D(mK⋆′, γK⋆′) , D(mn,γn) ≡ m2

n − s − κnRe[HKπ(s)] − imnΓn(s),

κn = 192πFKFπ σKπ(m2

K∗)

γK∗ mK∗ , Γn(s) = Γn s m2

n

σ3

Kπ(s)

σ3

Kπ(m2 n)

We then have a phase with two resonances δKπ(s) = tan−1 [ImF Kπ

V

(s) ReF Kπ

V

(s)]

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SLIDE 25

Vector Form Factor: Dispersive representation Three subtractions: helps the convergence of the form factor and suppresses the the high-energy region of the integral F Kπ

V

(s) = P(s)exp[α1 s m2

π−

+ 1 2α2 s2 m4

π−

+ s3 π ∫

scut sKπ

ds′ δKπ(s′) (s′)3(s′ − s − i0)] α1 = λ

′

+ and α2 1 + α2 = λ

′′

+ low energies parameters

F Kπ

V

(t) = 1 + λ

′

+

M 2

π−

t + 1 2 λ

′′

+

M 4

π−

t2

scut ∶ cut-off to check stability Parameters to Fit: λ′

+ ,λ′′ +, mK∗,γK∗,mK∗′,γK∗′

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SLIDE 26

Results of the combined τ − → KSπ−ντ and τ − → K−ηντ analysis

ËËË ËËËËËËËËË ËË Ë Ë Ë ËËË Ë Ë ËËËËËËËËËËËËËËËËËËËËËËËËËËËËËËËËËËËËËËËËËËË ËË Ë Ë ËËËËËËË Ë ËË ËË Ë Ë Ë Ë Ë È ÈÈÈ ÈÈÈ È È È È È È † † † † † † † † † † † † † † † † † † † † † † † † † † † † „ „ „

0.6 0.8 1.0 1.2 1.4 1.6 1.8 0.1 1 10 100 1000 104 s HGeVL Eventsêbin

t-ØK-hnt excluded fit points 'Unfolded' t-ØK-hnt Belle data t-ØKSp-nt excluded fit points Unfolded t-ØKSp-nt Belle data Fit to t-ØK-hnt Fit to t-ØKSp-nt

Scalar contributions

Escribano, Gonzàlez-Solís, Jamin, Roig JHEP 1409 (2014) 042 S.Gonzàlez-Solís Phi to Psi 2019 march 1, 2019 26 / 33

SLIDE 27

Results of the combined τ − → KSπ−ντ and τ − → K−ηντ analysis

Different choices regarding linear slopes and resonance mixing parameters (scut = 4 GeV2)

Fitted value Reference Fit Fit A Fit B Fit C ¯ BKπ(%) 0.404 ± 0.012 0.400 ± 0.012 0.404 ± 0.012 0.397 ± 0.012 (Bth

Kπ)(%)

(0.402) (0.394) (0.400) (0.394) MK∗ 892.03 ± 0.19 892.04 ± 0.19 892.03 ± 0.19 892.07 ± 0.19 ΓK∗ 46.18 ± 0.42 46.11 ± 0.42 46.15 ± 0.42 46.13 ± 0.42 MK∗′ 1305+15

−18

1308+16

−19

1305+15

−18

1310+14

−17

ΓK∗′ 168+52

−44

212+66

−54

174+58

−47

184+56

−46

γKπ × 102 = γKη −3.6+1.1

−1.5

−3.3+1.0

−1.3

= γKη λ′

Kπ × 103

23.9 ± 0.7 23.6 ± 0.7 23.8 ± 0.7 23.6 ± 0.7 λ′′

Kπ × 104

11.8 ± 0.2 11.7 ± 0.2 11.7 ± 0.2 11.6 ± 0.2 ¯ BKη × 104 1.58 ± 0.10 1.62 ± 0.10 1.57 ± 0.10 1.66 ± 0.09 (Bth

Kη) × 104

(1.45) (1.51) (1.44) (1.58) γKη × 102 −3.4+1.0

−1.3

−5.4+1.8

−2.6

−3.9+1.4

−2.1

−3.7+1.0

−1.4

λ′

Kη × 103

20.9 ± 1.5 = λ′

Kπ

21.2 ± 1.7 = λ′

Kπ

λ′′

Kη × 104

11.1 ± 0.4 11.7 ± 0.2 11.1 ± 0.4 11.8 ± 0.2 χ2/n.d.f. 108.1/105 ∼ 1.03 109.9/105 ∼ 1.05 107.8/104 ∼ 1.04 111.9/106 ∼ 1.06

S.Gonzàlez-Solís Phi to Psi 2019 march 1, 2019 27 / 33

SLIDE 28

Results of the combined τ − → KSπ−ντ and τ − → K−ηντ analysis

Reference fit results obtained for different values of scut

❤❤❤❤❤❤❤❤❤❤❤ ❤

Parameter scut(GeV2) 3.24 4 9 ∞ ¯ BKπ(%) 0.402 ± 0.013 0.404 ± 0.012 0.405 ± 0.012 0.405 ± 0.012 (Bth

Kπ)(%)

(0.399) (0.402) (0.403) (0.403) MK∗ 892.01 ± 0.19 892.03 ± 0.19 892.05 ± 0.19 892.05 ± 0.19 ΓK∗ 46.04 ± 0.43 46.18 ± 0.42 46.27 ± 0.42 46.27 ± 0.41 MK∗′ 1301+17

−22

1305+15

−18

1306+14

−17

1306+14

−17

ΓK∗′ 207+73

−58

168+52

−44

155+48

−41

155+47

−40

γKπ = γKη = γKη = γKη = γKη λ′

Kπ × 103

23.3 ± 0.8 23.9 ± 0.7 24.3 ± 0.7 24.3 ± 0.7 λ′′

Kπ × 104

11.8 ± 0.2 11.8 ± 0.2 11.7 ± 0.2 11.7 ± 0.2 ¯ BKη × 104 1.57 ± 0.10 1.58 ± 0.10 1.58 ± 0.10 1.58 ± 0.10 (Bth

Kη) × 104

(1.43) (1.45) (1.46) (1.46) γKη × 102 −4.0+1.3

−1.9

−3.4+1.0

−1.3

−3.2+0.9

−1.1

−3.2+0.9

−1.1

λ′

Kη × 103

18.6 ± 1.7 20.9 ± 1.5 22.1 ± 1.4 22.1 ± 1.4 λ′′

Kη × 104

10.8 ± 0.3 11.1 ± 0.4 11.2 ± 0.4 11.2 ± 0.4 χ2/n.d.f. 105.8/105 108.1/105 111.0/105 111.1/105

S.Gonzàlez-Solís Phi to Psi 2019 march 1, 2019 28 / 33

SLIDE 29

Results of the combined τ − → KSπ−ντ and τ − → K−ηντ analysis

Central results including the largest variation of scut

1200

1300 1400 1500 1600

mK' 1410 MeV

Boito et al. '1 0 Boito et al. '0 9 This work Es cribano et al.'1 3

ΤKSΠΝΤ ΤKSΠΝΤKl3 ΤKΗΝΤ

100

200 300 400 500

K' 1410 MeV

Boito et al. '1 0 Boito et al. '0 9 This work Es cribano et al. '1 3

ΤKSΠΝΤ ΤKSΠΝΤKl3 ΤKΗΝΤ

MK∗−(892) = 892.03 ± 0.19 MeV ΓK∗−(892) = 46.18 ± 0.44 MeV MK∗−(1410) = 1305+16

−18 MeV

ΓK∗−(1410) = 168+65

−59 MeV

γKπ = γKη = −3.4+1.2

−1.4 ⋅ 10−2

¯ BKπ = (0.0404 ± 0.012)% ¯ BKη = (1.58 ± 0.10) ⋅ 10−4

χ2/d.o.f = 108.1/105 = 1.03 ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ no gain ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ improvement

S.Gonzàlez-Solís Phi to Psi 2019 march 1, 2019 29 / 33

SLIDE 30

Results of the combined τ − → KSπ−ντ and τ − → K−ηντ analysis

20

22 24 26 28 30 32 Λ

' 103

FLAVIANET '1 0 KTeV '1 0 KLOE '0 7 NA4 8 '0 4 ISTRA '0 4 Boito et al. '1 0 Boito et al. '0 9 Jamin et al. '0 8 Mous s allam et al. '0 8 This work KSΠ This work KΗ Bernard '1 3 Antonelli et al. '1 3

Kl 3 Τ ΤKl 3

λ′

Kπ = (23.9 ± 0.9) ⋅ 10−3

λ′

Kη = (20.9 ± 2.7) ⋅ 10−3

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

isospin violation? spectra for τ − → K−π0ντ needed

S.Gonzàlez-Solís Phi to Psi 2019 march 1, 2019 30 / 33

SLIDE 31

Results of the combined τ − → KSπ−ντ and τ − → K−ηντ analysis

‡

‡ Ê Ê Ê

Ï Ï Ï

Ú Ú Ú Ú Ú

4 8 12 16 20 24 28 4 8 12 16 20 24 28 l+

'' ◊104

FLAVIANET '10 KTeV '10 KLOE '07

NA48 '04

ISTRA+ '04 Boito et al. '10 Boito et al. '09 Jamin et al. '08 Moussallam et al. '08

This work @KSp-D This work @K -hD

Bernard '13

Antonelli et al. '13

Kl 3 t t+Kl 3

λ′′

Kπ = (11.8 ± 0.2) ⋅ 10−4

λ′′

Kη = (11.1 ± 0.5) ⋅ 10−4 S.Gonzàlez-Solís Phi to Psi 2019 march 1, 2019 31 / 33

SLIDE 32

τ − → π−η(′)ντ: Invariant mass distribution and Branching Ratio

Escribano, Gonzàlez-Solís, Roig PRD 94 (2016), 0.6 0.8 1.0 1.2 1.4 1.6 1.8 10-5 10-4 0.001 0.01 0.1 1 10 s HGeVL 1016ÿ dGêd s

Vector Full: vector + B. Wigner H2 resL Full: vector + 3 coupled channels Full: vector + elastic

1.0 1.2 1.4 1.6 1.8 2.0 10-10 10-8 10-6 10-4 0.01 1 s HGeVL 1017ÿ dGêd s

Vector Full: vector + B. Wigner H2 resL Full: vector + 3 coupled channels Full: vector + elastic scalar

see talk by P . Rados

Challenging for Belle II

τ − → π−ηντ Theory predictions: BR ∼ 1 × 10−5 (Escribano’16, Moussallam’14) BaBar: BR < 9.9 ⋅ 10−5 95% CL , Belle: BR < 7.3 ⋅ 10−5 90% CL τ − → π−η′ντ Theory predictions: BR ∼ [10−7,10−6] (Escribano’16) BaBar: BR < 4 ⋅ 10−6 90% CL

S.Gonzàlez-Solís Phi to Psi 2019 march 1, 2019 32 / 33

SLIDE 33

Outlook Tau physics is a very rich field to test QCD and EW Important experimental activities: Belle (II), BaBar, LHCb, BESIII τ decays into two mesons are a privileged laboratory to access the non-perturbative regime of QCD Form Factors from dispersion relations with subtractions

Extraction of the K∗(892) parameters from a fit to τ → KSπ−ντ Extraction of the K∗(1410) from τ − → KSπ−ντ and τ − → K−ηντ F π

V (s): important for testing QCD dynamics and the SM and NP

τ − → KSK−ντ: extraction of the ρ(1450) and ρ(1700) parameters

A lot of interesting physics to be done in the tau sector

S.Gonzàlez-Solís Phi to Psi 2019 march 1, 2019 33 / 33

SLIDE 34

Hadronic Tau decays

¡

h a d r

n

i z a t i

n

τ − ντ W − ¯ u d′ = Vudd + Vuss P − P ′0

Test of QCD and ElectroWeak Interactions

S.Gonzàlez-Solís Phi to Psi 2019 march 1, 2019 34 / 33

SLIDE 35

Pion vector form factor: Chiral Perturbation Theory O(p4)

= π− π0 π0 π− + π− π0 π, K π, K + π0 π− + π0 π−

F V

π (s)

= 1 + 2Lr

9(µ)

F 2

π

s − s 96π2F 2

π

(Aπ(s,µ2) + 1 2AK(s,µ2)) AP (s,µ2) = log m2

P

µ2 + 8m2

P

s − 5 3 + σ3

P (s)log (σP (s) + 1

σP (s) − 1) ,σP (s) = √ 1 − 4m2

P

s

S.Gonzàlez-Solís Phi to Psi 2019 march 1, 2019 35 / 33

SLIDE 36

Pion vector form factor: Chiral Perturbation Theory O(p4)

0.0

0.5 1.0 1.5 2.0 2.5 3.0 0.001 0.01 0.1 1 10 s GeV2 FV

Π 2

Belle 2008 ChPT at p4

S.Gonzàlez-Solís Phi to Psi 2019 march 1, 2019 36 / 33

SLIDE 37

Pion vector Form Factor: ChPT with resonances Resonace Chiral Theory F π

V (s) = 1 + FV GV

F 2

π

s M2

ρ − s FV GV =F 2

π

⇒

M2

ρ

M2

ρ − s ,

Expansion in s and comparing ChPT and RχT F V

π (s)

= 1 + 2Lr

9(µ)

F 2

π

s − s 96π2F 2

π

(Aπ(s,µ2) + 1 2AK(s,µ2)) F V

π (s)

= 1 + ( s M2

ρ

) + ( s M2

ρ

)

2

+ ⋅ Chiral coupling estimate: Lr

9(Mρ) = FV GV 2M2

ρ

=

F 2

π

2M2

ρ ∼ 7.2 × 10−3

Combining ChPT and RχT F π

V (s) =

M2

ρ

M2

ρ − s −

s 96π2F 2

π

[Aπ(s,µ2) + 1 2AK(s,µ2)] ,

S.Gonzàlez-Solís Phi to Psi 2019 march 1, 2019 37 / 33

SLIDE 38

ChPT with resonances + Omnès: Exponential representation Incorporate the (off-shell) ρ width Γρ(s) = − Mρs 96π2F 2

π

Im[Aπ(s) + 1 2AK(s)] = Mρs 96πF 2

π

[σπ(s)3θ(s − 4m2

π) + σK(s)3θ(s − 4m2 K)] .

F π

V (s) =

M2

ρ

M2

ρ − s − iMρΓρ(s) exp

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ − s 96π2F 2

π

Re[Aπ(s,µ2) + 1 2AK(s,µ2)] ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ .

S.Gonzàlez-Solís Phi to Psi 2019 march 1, 2019 38 / 33

SLIDE 39

ChPT with resonances + Omnès+ρ′,ρ′′: Exponential representation

—

0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.001 0.01 0.1 1 10 s GeV2 FV

Π 2

Belle 2008 ChPT at p4 Our fit: exponential respresentation

Mρ = 775.2(4) MeV , γ = 0.15(4) , φ1 = −0.36(24) , Mρ′ = 1438(39) MeV , Γρ′ = 535(63) MeV , δ = −0.12(4) , φ2 , = −0.02(45) , Mρ′′ = 1754(91) MeV , Γρ′′ = 412(102) MeV , χ2

dof = 0.92 S.Gonzàlez-Solís Phi to Psi 2019 march 1, 2019 39 / 33

SLIDE 40

ChPT with resonances + Omnès+ρ′,ρ′′: Exponential representation

—

0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.001 0.01 0.1 1 10 s GeV2 FV

Π 2

Belle 2008 ChPT at p4 Our fit: exponential respresentation

M pole

ρ

= 762.0(3) MeV , Γpole

ρ

= 143.0(2) MeV , M pole

ρ′

= 1366(38) MeV , Γpole

ρ′

= 488(48) MeV , M pole

ρ′′

= 1718(82) MeV , Γpole

ρ′′

= 397(88) MeV ,

S.Gonzàlez-Solís Phi to Psi 2019 march 1, 2019 40 / 33

SLIDE 41

Variant (I) Fits for different values of scut and allowing the ρ-mass to float

Parameter scut [GeV2] Fits m2

τ

4 (reference fit) 10 ∞ Fit 1-ρ α1 [GeV−2] 1.88(1) 1.88(1) 1.89(1) 1.88(1) α2 [GeV−4] 4.37(3) 4.34(1) 4.31(3) 4.34(1) mρ [MeV] 773.9(3) 773.8(3) 773.9(3) 773.9(3) Mρ [MeV] = mρ = mρ = mρ = mρ Mρ′ [MeV] 1382(71) 1375(11) 1316(9) 1312(8) Γρ′ [MeV] 516(165) 608(35) 728(92) 726(26) Mρ′′[MeV] 1723(1) 1715(22) 1655(1) 1656(8) Γρ′′ [MeV] 315(271) 455(16) 569(160) 571(13) γ 0.12(13) 0.16(1) 0.18(2) 0.17(1) φ1 −0.56(35) −0.69(1) −1.40(19) −1.41(8) δ −0.09(3) −0.13(1) −0.17(4) −0.17(3) φ2 −0.19(69) −0.45(12) −1.06(10) −1.05(11) χ2/d.o.f 1.09 0.70 0.63 0.66

S.Gonzàlez-Solís Phi to Psi 2019 march 1, 2019 41 / 33

SLIDE 42

Dispersive representation: on the singularities at s = scut Modulus squared of the pion form factor for scut = mτ,4 GeV2

—

—

1 2 3 4 0.001 0.01 0.1 1 10 s GeV2 FΠΠ 2

Belle data 2008 scutmΤ

2 GeV2

scut4 GeV2

S.Gonzàlez-Solís Phi to Psi 2019 march 1, 2019 42 / 33

SLIDE 43

Form Factor phase shift including systematic th. uncertainties

––– –––

0.0

0.5 1.0 1.5 2.0 50 100 150 200 250 s GeV ∆1

1s degrees Fit 1 reference fit Fit 1–Ρ Fit I Fit A Fit singularities

The results can be found as ancillary material in 1902.02273

[hep-ph]

S.Gonzàlez-Solís Phi to Psi 2019 march 1, 2019 43 / 33

SLIDE 44

Form Factor including systematic th. uncertainties

–––

–––

0.0

0.5 1.0 1.5 2.0 2.5 3.0 0.001 0.01 0.1 1 10 s GeV2 FV

Π 2 Belle data 2008 Fit 1 reference fit Fit 1–Ρ Fit I Fit A Fit singularities

The results can be found as ancillary material in 1902.02273

[hep-ph]

S.Gonzàlez-Solís Phi to Psi 2019 march 1, 2019 44 / 33

SLIDE 45

τ − → ντ+strange Tau partial width to strange ∼ 3% τ → (Kπ)−ντ and τ → K−η(′)ντ → this talk

S.Gonzàlez-Solís Phi to Psi 2019 march 1, 2019 45 / 33