Chapter on Analytic approaches to HLbL Status report Gilberto - PowerPoint PPT Presentation

Intro HLbL to ( g − 2 ) µ Exp. input Conclusions Chapter on “Analytic approaches to HLbL ” Status report Gilberto Colangelo ( g − 2 ) µ Theory Initiative Seattle, 9.9.2019

Intro HLbL to ( g − 2 ) µ Exp. input Conclusions List of authors Johan Bijnens, GC, Francesca Curciarello, Henryk Czy˙ z, Igor Danilkin, Franziska Hagelstein, Martin Hoferichter, Bastian Kubis, Andreas Nyffeler, Vladimir Pascalutsa, Elena Perez del Rio, Massimiliano Procura, Christoph Florian Redmer, Pablo Sanchez-Puertas, Peter Stoffer, Marc Vanderhaeghen

Intro HLbL to ( g − 2 ) µ Exp. input Conclusions Outline Introduction: structure of the chapter Hadronic light-by-light contribution to ( g − 2 ) µ PS-pole contribution Two-pion contributions Higher hadronic intermediate states Short-distance constraints Summary Experimental input Conclusions

Intro HLbL to ( g − 2 ) µ Exp. input Conclusions Table of content Talk by Pauk Talk by Hagelstein Session after this talk Gasparyan, Redmer Talk by Czyz

Intro HLbL to ( g − 2 ) µ Exp. input Conclusions Table of content Fischer Talks by Holz & Nyffeler Talks by Danilkin & Stoffer Talks by Kampf & Hoferichter Talks by Bijnens, Hoferichter and Laub

Intro HLbL to ( g − 2 ) µ Exp. input Conclusions Table of content

Intro HLbL to ( g − 2 ) µ Exp. input Conclusions Dispersive approaches ◮ model independent ◮ unambiguous definition of the various contributions ◮ makes a data-driven evaluation possible (in principle) ◮ if data not available: use theoretical calculations of subamplitudes, short-distance constraints etc. ◮ First attempts: GC, Hoferichter, Procura, Stoffer (14) Pauk, Vanderhaeghen (14) ◮ similar philosophy, with a different implementation: Schwinger sum rule Hagelstein, Pascalutsa (17)

Intro HLbL to ( g − 2 ) µ Exp. input Conclusions PS -pole 2 π Higher hadrons SDC Summary The HLbL tensor HLbL tensor: � � � Π µνλσ = i 3 � � dz e − i ( x · q 1 + y · q 2 + z · q 3 ) � 0 | T j µ ( x ) j ν ( y ) j λ ( z ) j σ ( 0 ) dx dy | 0 � k 2 = 0 q 4 = k = q 1 + q 2 + q 3 General Lorentz-invariant decomposition: � Π µνλσ = g µν g λσ Π 1 + g µλ g νσ Π 2 + g µσ g νλ Π 3 + q µ l Π 4 i q ν j q λ k q σ ijkl + . . . i , j , k , l consists of 138 scalar functions { Π 1 , Π 2 , . . . } , but in d = 4 only 136 are linearly independent Eichmann et al. (14) Constraints due to gauge invariance? (see also Eichmann, Fischer, Heupel (2015)) ⇒ Apply the Bardeen-Tung (68) method + Tarrach (75) addition

Intro HLbL to ( g − 2 ) µ Exp. input Conclusions PS -pole 2 π Higher hadrons SDC Summary Gauge-invariant hadronic light-by-light tensor Applying the Bardeen-Tung-Tarrach method to Π µνλσ one ends up with: GC, Hoferichter, Procura, Stoffer (2015) ◮ 43 basis tensors (BT) in d = 4: 41=no. of helicity amplitudes ◮ 11 additional ones (T) to guarantee basis completeness everywhere ◮ of these 54 only 7 are distinct structures ◮ all remaining 47 can be obtained by crossing transformations of these 7: manifest crossing symmetry ◮ the dynamical calculation needed to fully determine the LbL tensor concerns these 7 scalar amplitudes 54 � Π µνλσ = T µνλσ Π i i i = 1

Intro HLbL to ( g − 2 ) µ Exp. input Conclusions PS -pole 2 π Higher hadrons SDC Summary Master Formula � 12 � i = 1 ˆ T i ( q 1 , q 2 ; p )ˆ d 4 q 1 d 4 q 2 Π i ( q 1 , q 2 , − q 1 − q 2 ) a HLbL = − e 6 2 ( q 1 + q 2 ) 2 [( p + q 1 ) 2 − m 2 µ ][( p − q 2 ) 2 − m 2 µ ( 2 π ) 4 ( 2 π ) 4 q 2 1 q 2 µ ] ◮ ˆ T i : known kernel functions ◮ ˆ Π i : linear combinations of the Π i ◮ the Π i are amenable to a dispersive treatment: their imaginary parts are related to measurable subprocesses ◮ 5 integrals can be performed with Gegenbauer polynomial techniques GC, Hoferichter, Procura, Stoffer (2015)

Intro HLbL to ( g − 2 ) µ Exp. input Conclusions PS -pole 2 π Higher hadrons SDC Summary Master Formula After performing the 5 integrations: � 1 � ∞ � ∞ 12 = 2 α 3 � � T i ( Q 1 , Q 2 , τ )¯ a HLbL dQ 4 dQ 4 1 − τ 2 d τ Π i ( Q 1 , Q 2 , τ ) µ 1 2 48 π 2 0 0 − 1 i = 1 where Q µ i are the Wick-rotated four-momenta and τ the four-dimensional angle between Euclidean momenta: Q 1 · Q 2 = | Q 1 || Q 2 | τ The integration variables Q 1 := | Q 1 | , Q 2 := | Q 2 | . GC, Hoferichter, Procura, Stoffer (2015)

Intro HLbL to ( g − 2 ) µ Exp. input Conclusions PS -pole 2 π Higher hadrons SDC Summary Setting up the dispersive calculation The HLbL tensor is split as follows: Π µνλσ = Π π -pole µνλσ + ¯ µνλσ + Π π -box Π µνλσ + · · · Last diagrams = all partial waves ⇔ scalars and tensors etc. 3 π states are in . . . ⇒ axial vector resonances

Intro HLbL to ( g − 2 ) µ Exp. input Conclusions PS -pole 2 π Higher hadrons SDC Summary Outline Introduction: structure of the chapter Hadronic light-by-light contribution to ( g − 2 ) µ PS-pole contribution Two-pion contributions Higher hadronic intermediate states Short-distance constraints Summary Experimental input Conclusions

Intro HLbL to ( g − 2 ) µ Exp. input Conclusions PS -pole 2 π Higher hadrons SDC Summary Pion-pole contribution ◮ The pion transition form factor completely fixes this contribution Knecht-Nyffeler (01) Π 1 = F π 0 γ ∗ γ ∗ ( q 2 1 , q 2 2 ) F π 0 γ ∗ γ ∗ ( q 2 3 , 0 ) ¯ q 2 3 − M 2 π 0 ◮ Both transition form factors (TFF) must be included: [dropping one bc short-distance not correct Melnikov-Vainshtein (04) ] ◮ data on singly-virtual TFF available CELLO, CLEO, BaBar, Belle, BESIII ◮ several calculations of the transition form factors in the literature Masjuan & Sanchez-Puertas (17), Eichmann et al. (17), Guevara et al. (18) ◮ dispersive approach works here too Hoferichter et al. (18) ◮ quantity where lattice calculations can have a significant impact Gèrardin, Meyer, Nyffeler (16,19)

Intro HLbL to ( g − 2 ) µ Exp. input Conclusions PS -pole 2 π Higher hadrons SDC Summary PS-pole contributions B. Kubis and P . Sanchez Puertas Philosophy adopted in the section: The calculations must be model-independent and data-driven to as large an extent as possible (...) Three criteria must be fulfilled: 1. TFF normalization given by the real-photon decay widths, and high-energy constraints must be fulfilled; 2. at least the space-like experimental data for the singly-virtual TFF must be reproduced; 3. systematic uncertainties must be assessed with a reasonable procedure.

Intro HLbL to ( g − 2 ) µ Exp. input Conclusions PS -pole 2 π Higher hadrons SDC Summary Results above the bar ◮ Dispersive calculation of the pion TFF Hoferichter et al. (18) a π 0 µ = 63 . 0 + 2 . 7 − 2 . 1 × 10 − 11 ◮ Padé-Canterbury approximants Masjuan & Sanchez-Puertas (17) a π 0 µ = 63 . 6 ( 2 . 7 ) × 10 − 11 ◮ Lattice Gérardin, Meyer, Nyffeler (19) a π 0 µ = 62 . 3 ( 2 . 3 ) × 10 − 11

Intro HLbL to ( g − 2 ) µ Exp. input Conclusions PS -pole 2 π Higher hadrons SDC Summary Results above the bar

Intro HLbL to ( g − 2 ) µ Exp. input Conclusions PS -pole 2 π Higher hadrons SDC Summary Results above the bar 0.07 Q 2 F π 0 γ ∗ γ ∗ ( − Q 2 , − Q 2 ) [GeV] 0.06 0.05 0.04 0.03 dispersive 0.02 Canterbury lattice 0.01 OPE limit 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Q 2 � GeV 2 �

Intro HLbL to ( g − 2 ) µ Exp. input Conclusions PS -pole 2 π Higher hadrons SDC Summary η - and η ′ -pole contribution ◮ Dispersive calculation not yet available → talk by S. Holz ( η - η ′ mixing, different isospin structure etc.) ◮ Less data (BaBar) ◮ Canterbury approach: µ = 16 . 3 ( 1 . 0 ) stat ( 0 . 5 ) a P ; 1 , 1 ( 0 . 9 ) sys × 10 − 11 → 16 . 3 ( 1 . 4 ) × 10 − 11 a η µ = 14 . 5 ( 0 . 7 ) stat ( 0 . 4 ) a P ; 1 , 1 ( 1 . 7 ) sys × 10 − 11 → 14 . 5 ( 1 . 9 ) × 10 − 11 a η ′

Intro HLbL to ( g − 2 ) µ Exp. input Conclusions PS -pole 2 π Higher hadrons SDC Summary η - and η ′ -pole contribution ( GeV - 1 ) 15 2 )× 10 - 3 10 2 , - Q 2 F ηγ * γ * (- Q 1 5 0 ( 6.5,6.5 ) ( 16.9,16.9 ) ( 14.8,4.3 ) ( 38.1,15.0 ) ( 45.6,45.6 ) 2 ,Q 2 2 ) ( GeV 2 ) ( Q 1 Data points: BaBar. Blue band: Canterbury representation.

Intro HLbL to ( g − 2 ) µ Exp. input Conclusions PS -pole 2 π Higher hadrons SDC Summary η - and η ′ -pole contribution Q 2 F η ' γ * γ * (- Q 2 , - Q 2 ) ( GeV ) 0.10 0.08 0.06 0.04 0.02 0.00 0 1 2 3 4 5 6 7 Q 2 ( GeV 2 ) Data points: BaBar. Blue band: Canterbury representation.

Intro HLbL to ( g − 2 ) µ Exp. input Conclusions PS -pole 2 π Higher hadrons SDC Summary PS-poles: conclusion Dispersive ( π 0 ) + Canterbury ( η , η ′ ): a π 0 + η + η ′ = 93 . 8 + 4 . 0 − 3 . 6 × 10 − 11 µ Canterbury: a π 0 + η + η ′ = 94 . 3 ( 5 . 3 ) × 10 − 11 µ Outlook: Dispersive evaluation of the η, η ′ contributions will give two fully independent evaluations ⇒ better control over systematics

Intro HLbL to ( g − 2 ) µ Exp. input Conclusions PS -pole 2 π Higher hadrons SDC Summary Outline Introduction: structure of the chapter Hadronic light-by-light contribution to ( g − 2 ) µ PS-pole contribution Two-pion contributions Higher hadronic intermediate states Short-distance constraints Summary Experimental input Conclusions