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

Light mesons from tau decays Sergi Gonzàlez-Solís 1 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 1 sgonzal@iu.edu

Test of QCD and ElectroWeak Interactions Inclusive decays: τ − → ( ¯ us ) ν τ ud, ¯ Full hadron spectra (precision physics) Fundamental SM parameters: α s ( m τ ) ,m s , ∣ V us ∣ Exclusive decays: τ − → ( PP,PPP,... ) ν τ specific hadron spectrum (approximate physics) Hadronization of QCD currents, study of Form Factors, resonance parameters ( M R , Γ R ) S.Gonzàlez-Solís Phi to Psi 2019 march 1, 2019 2 / 33

τ − ν τ τ decays into two mesons W − d ′ = V ud d + V us s d Γ ( τ − → P − P 0 ν τ ) F ∣ V ui ∣ 2 m 3 2 hadronization = G 2 u P P ′ ( 1 − ) ¯ s τ S had EW C 2 P − 768 π 3 M 2 ds τ ( s )∣ 2 + 3∆ 2 ) λ 3 / 2 λ 1 / 2 {( 1 + 2 s P − P 0 ( s )∣ F P − P 0 P − P 0 ( s )∣ F P − P 0 ( s )∣ 2 } P ′ 0 P − P 0 V S m 2 s 2 τ τ − → π − π 0 ν τ : Pion vector form factor, ρ ( 770 ) ,ρ ( 1450 ) ,ρ ( 1700 ) τ − → K − K S ν τ : Kaon vector form factor, ρ ( 770 ) ,ρ ( 1450 ) ,ρ ( 1700 ) τ − → K S π − ν τ : Kπ form factor, K ∗ ( 892 ) ,K ∗ ( 1410 ) , K ℓ 3 , V us (Passemar) τ − → K − η (′) ν τ : K ∗ ( 1410 ) , V us τ − → π − η (′) ν τ : isospin-violating decays S.Gonzàlez-Solís Phi to Psi 2019 march 1, 2019 3 / 33

The pion vector form factor: Motivation Enters the description of many physical processes π 0 V ( s ) ∝ F π π − 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 of the ρ ( 1450 ) and ρ ( 1700 ) region S.Gonzàlez-Solís Phi to Psi 2019 march 1, 2019 4 / 33

Dispersive representation of the pion vector form factor 1 ( s ′ ) 2 s 2 + s 3 δ 1 V ( s ) = exp [ α 1 s + α 2 ( s ′ ) 3 ( s ′ − s − i 0 )] , s cut π ∫ ds ′ F π 4 m 2 π 1 ( s ) Form Factor phase δ 1 π < s < 1 GeV: ππ phase from Roy 150 4 m 2 ΠΠ Phase 100 (García-Martín et.al PRD 83, 074004 (2011)) 1 < s < m 2 τ : "Pheno" phase shift 50 τ < s : phase guided smoothly to π m 2 0 0.5 1.0 1.5 2.0 s � GeV � Low-energy observables 1 + 1 V s 2 + d π V s 3 + ⋯ . 6 ⟨ r 2 ⟩ π F π V ( s ) = V s + c π V = 1 ⟨ r 2 ⟩ π c π 2 ( α 2 + α 2 = 1 ) . 6 α 1 , V S.Gonzàlez-Solís Phi to Psi 2019 march 1, 2019 5 / 33

ChPT with resonances + Omnès: Exponential representation Get a model for the (Pheno) phase ⎧ ⎫ ⎪ ⎪ 1 ( s ′ ) ⎪ ⎪ s n ds ′ δ 1 V ( s ) = P n ( s ) exp ⎨ ∞ ⎬ π ∫ s ′ − s − i 0 F π ⎪ ⎪ ( s ′ ) n ⎪ ⎪ , ⎩ ⎭ 4 m 2 π ππ → ππ scattering at O( p 2 ) T ( s ) = s − m 2 π ( s ) π ( s ) 1 ( s ) = sσ 2 1 ( s ) = sσ 3 � → T 1 � → δ 1 1 ( s ) = σ π ( s ) T 1 π , F 2 96 πF 2 96 πF 2 π π π Exponential Omnès representation of the form factor ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ M 2 V ( s ) ⎨ − Re [ A π ( s,µ 2 ) + 1 2 A K ( s,µ 2 )] ⎬ = s ρ F π ⎪ ⎪ ρ − s − iM ρ Γ ρ ( s ) exp ⎪ ⎪ ⎩ ⎭ M 2 96 π 2 F 2 π Γ ρ ( s ) − M ρ s Im [ A π ( s ) + 1 2 A K ( s )] = 96 π 2 F 2 π M ρ s [ ( ) 3 ( − ( ) 3 ( − 2 )] S.Gonzàlez-Solís Phi to Psi 2019 2 ) + march 1, 2019 6 / 33 =

Incorporation of the ρ ′ ≡ ρ ( 1450 ) ,ρ ′′ ≡ ρ ( 1700 ) ⎧ ⎡ ⎤ ⎫ ρ + s ( γe iφ 1 + δe iφ 2 ) ⎪ ⎢ ⎥ ⎪ ⎪ ⎪ ⎢ ⎥ M 2 V ( s ) = ⎨ − ( A π ( s ) + 1 2 A K ( s )) ⎬ s ⎢ ⎥ F π ⎪ ⎪ ρ − s − iM ρ Γ ρ ( s ) exp Re ⎢ ⎥ ⎪ ⎪ ⎩ ⎣ ⎦ ⎭ M 2 96 π 2 F 2 π ⎧ ⎫ ⎪ s Γ ρ ′ ( M 2 ρ ′ ) ⎪ ⎪ ⎪ se iφ 1 − γ ⎨ − ρ ′ ) Re A π ( s ) ⎬ ⎪ ⎪ ρ ′ − s − iM ρ ′ Γ ρ ′ ( s ) exp π ( M 2 ⎪ ⎪ ⎩ ⎭ M 2 πM 3 ρ ′ σ 3 ⎧ ⎫ ⎪ s Γ ρ ′′ ( M 2 ρ ′′ ) ⎪ ⎪ ⎪ − δ se iφ 2 ⎨ − ρ ′′ ) Re A π ( s ) ⎬ ⎪ ⎪ ρ ′′ − s − iM ρ ′′ Γ ρ ′′ ( s ) exp π ( M 2 , ⎪ ⎪ ⎩ ⎭ M 2 πM 3 ρ ′′ σ 3 π ( s ) Γ ρ ′ ,ρ ′′ ( s ) = Γ ρ ′ ,ρ ′′ σ 3 ρ ′ ,ρ ′′ ) θ ( s − 4 m 2 π ) . s π ( M 2 M 2 σ 3 ρ ′ ,ρ ′′ V ( s ) 1 ( s ) = Im F π tan δ 1 V ( s ) Re F π S.Gonzàlez-Solís Phi to Psi 2019 march 1, 2019 7 / 33

Dispersive Fits to the Pion Vector Form Factor Fits for different values of s cut and matching at 1 GeV s cut [GeV 2 ] ∞ Parameter m 2 Fits 4 (reference fit) 10 α 1 [GeV − 2 ] 1 . 87 ( 1 ) τ 1 . 88 ( 1 ) 1 . 89 ( 1 ) 1 . 89 ( 1 ) Fit 1 α 2 [GeV − 4 ] 4 . 40 ( 1 ) 4 . 34 ( 1 ) 4 . 32 ( 1 ) 4 . 32 ( 1 ) = 773 . 6 ( 9 ) = 773 . 6 ( 9 ) = 773 . 6 ( 9 ) = 773 . 6 ( 9 ) m ρ [MeV] = m ρ = m ρ = m ρ = m ρ M ρ [MeV] 1365 ( 15 ) 1376 ( 6 ) 1313 ( 15 ) 1311 ( 5 ) M ρ ′ [MeV] 562 ( 55 ) 603 ( 22 ) 700 ( 6 ) 701 ( 28 ) Γ ρ ′ [MeV] 1727 ( 12 ) 1718 ( 4 ) 1660 ( 9 ) 1658 ( 1 ) M ρ ′′ [MeV] 278 ( 1 ) 465 ( 9 ) 601 ( 39 ) 602 ( 3 ) Γ ρ ′′ [MeV] 0 . 12 ( 2 ) 0 . 15 ( 1 ) 0 . 16 ( 1 ) 0 . 16 ( 1 ) γ − 0 . 69 ( 1 ) − 0 . 66 ( 1 ) − 1 . 36 ( 10 ) − 1 . 39 ( 1 ) φ 1 − 0 . 09 ( 1 ) − 0 . 13 ( 1 ) − 0 . 16 ( 1 ) − 0 . 17 ( 1 ) δ − 0 . 17 ( 5 ) − 0 . 44 ( 3 ) − 1 . 01 ( 5 ) − 1 . 03 ( 2 ) φ 2 χ 2 /d.o.f 1 . 47 0 . 70 0 . 64 0 . 64 S.Gonzàlez-Solís Phi to Psi 2019 march 1, 2019 8 / 33

Form Factor phase shift for different values of s cut 200 ––– ––– Roy ��� ��� 2 s cut � m Τ 150 ––– ––– s cut � 4 GeV 2 1 � s � � degrees � �� �� s cut � 10 GeV 2 s cut �� 100 � ∆ 1 50 0 0.0 0.5 1.0 1.5 2.0 s � GeV � 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

Modulus squared of the pion vector form factor ��� Belle data � 2008 � � � � � � ––– ––– � This work � s cut � 4 GeV 2 � 10 ������������������������������ � ������ This work � s cut �� � � 1 Π � 2 ����������� � F V 0.1 � ������ 0.01 0.001 0.0 0.5 1.0 1.5 2.0 2.5 3.0 s � GeV 2 � 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 10 / 33

Variant (I) Fits for different matching point and with s cut = 4 GeV Matching point [GeV] Parameter Fits 0 . 85 0 . 9 0 . 95 1 (reference fit) 1 . 88 ( 1 ) 1 . 88 ( 1 ) 1 . 88 ( 1 ) 1 . 88 ( 1 ) α 1 [GeV − 2 ] Fit I 4 . 35 ( 1 ) 4 . 35 ( 1 ) 4 . 34 ( 1 ) 4 . 34 ( 1 ) α 2 [GeV − 4 ] = 773 . 6 ( 9 ) = 773 . 6 ( 9 ) = 773 . 6 ( 9 ) = 773 . 6 ( 9 ) m ρ [MeV] = m ρ = m ρ = m ρ = m ρ M ρ [MeV] 1394 ( 6 ) 1374 ( 8 ) 1351 ( 5 ) 1376 ( 6 ) M ρ ′ [MeV] 592 ( 19 ) 583 ( 27 ) 592 ( 2 ) 603 ( 22 ) Γ ρ ′ [MeV] 1733 ( 9 ) 1715 ( 1 ) 1697 ( 3 ) 1718 ( 4 ) M ρ ′′ [MeV] 562 ( 3 ) 541 ( 45 ) 486 ( 7 ) 465 ( 9 ) Γ ρ ′′ [MeV] 0 . 12 ( 1 ) 0 . 12 ( 1 ) 0 . 13 ( 1 ) 0 . 15 ( 1 ) γ − 0 . 44 ( 3 ) − 0 . 60 ( 1 ) − 0 . 80 ( 1 ) − 0 . 66 ( 1 ) φ 1 − 0 . 13 ( 1 ) − 0 . 13 ( 1 ) − 0 . 13 ( 1 ) − 0 . 13 ( 1 ) δ − 0 . 38 ( 3 ) − 0 . 51 ( 2 ) − 0 . 62 ( 1 ) − 0 . 44 ( 3 ) φ 2 χ 2 /d.o.f 0 . 75 0 . 74 0 . 68 0 . 70 S.Gonzàlez-Solís Phi to Psi 2019 march 1, 2019 11 / 33

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 s cut = 4 GeV 2 Parameter Fit A Fit B reference fit 1 . 87 ( 1 ) 1 . 88 ( 1 ) 1 . 88 ( 1 ) α 1 [GeV − 2 ] 4 . 37 ( 1 ) 4 . 34 ( 1 ) α 2 [GeV − 4 ] 4.35(1) = 773 . 6 ( 9 ) = 773 . 6 ( 9 ) = 773 . 6 ( 9 ) m ρ [MeV] = m ρ = m ρ = m ρ M ρ [MeV] 1373 ( 5 ) 1441 ( 3 ) 1376 ( 6 ) M ρ ′ [MeV] 462 ( 14 ) 576 ( 33 ) 603 ( 22 ) Γ ρ ′ [MeV] 1775 ( 1 ) 1733 ( 9 ) 1718 ( 4 ) M ρ ′′ [MeV] 412 ( 27 ) 349 ( 52 ) 465 ( 9 ) Γ ρ ′′ [MeV] 0 . 13 ( 1 ) 0 . 15 ( 3 ) 0 . 15 ( 1 ) γ − 0 . 80 ( 1 ) − 0 . 53 ( 5 ) − 0 . 66 ( 1 ) φ 1 − 0 . 14 ( 1 ) − 0 . 14 ( 1 ) − 0 . 13 ( 1 ) δ − 0 . 44 ( 2 ) − 0 . 46 ( 3 ) − 0 . 44 ( 3 ) φ 2 χ 2 /d.o.f 0 . 93 0 . 70 0 . 70 S.Gonzàlez-Solís Phi to Psi 2019 march 1, 2019 12 / 33

Variant (III) Dispersive representation of the pion vector form factor 1 ( s ′ ) δ eff ( s ′ ) δ 1 V ( s ) = exp [ s ( s ′ )( s ′ − s − i 0 ) + s ∞ ( s ′ )( s ′ − s − i 0 )] Σ ( s ) s cut π ∫ π ∫ F π ds ′ ds ′ 4 m 2 s cut π Properties for δ eff ( s ) δ eff ( s cut ) = δ 1 1 ( s cut ) and δ eff ( s ) → π for large s to recover 1 / s fall-off δ eff ( s ) = π + ( δ 1 1 ( s cut ) − π ) s cut s Integrating the piece with δ eff ( s ) ( 1 − δ 1 ) s cut − 1 1 ( s cut ) V ( s ) = e 1 − ( 1 − ) ( 1 − ) δ 1 s s 1 ( s cut ) π s F π π s cut s cut 1 ( s ′ ) δ 1 × exp [ s ds ′ ( s ′ )( s ′ − s − i 0 )] Σ ( s ) s cut π ∫ 4 m 2 π √ s cut − √ s cut − s ∞ Σ ( s ) = a i ω i ( s ) , ω ( s ) = √ s cut + √ s cut − s ∑ i = 0 S.Gonzàlez-Solís Phi to Psi 2019 march 1, 2019 13 / 33