Dispersion relations for 2 and 2 Bachir Moussallam work with: - PowerPoint PPT Presentation

GDR Light and Shadow ***Montpellier, Nov. 16-17 2016 Dispersion relations for γγ → 2 π and γγ ∗ → 2 π Bachir Moussallam work with: R. Garc´ ıa-Mart´ ın

Introduction γγ → hadrons at e + e − colliders: σ ∼ α 4 ln 4 E ln E m e m π [ Brodsky, Kinoshita, Terazawa (1970) ] 2 γ couplings of hadrons ( J PC = 0 ++ , 0 −+ , 2 ++ , 2 −+ , 3 ++ , · · · ) π 0 , η → 2 γ : measurement via Primakov [ Browman (1974) ] was not correct ! σ , f 0 ( 980 ) , a 0 ( 980 ) → 2 γ Analyticity based extraction: [ Mennessier, Z.Phys. C16 (1983) 241 ] 2/40

LBL hadronic contributions to muon g − 2 Largest contribution: one-pion pole Large N c approach, next largest: η , σ , f 0 , · · · poles More general Bern dispersive approach: � [ γγ ∗ → ππ ] J × [ γ ∗ γ ∗ → ππ ] J Ingredients: J 3/40

γγ → ππ : analyticity 4/40

Independent amplitudes Amplitude γ ( ∗ ) ( q 1 ) γ ( ∗ ) ( q 2 ) → π ( p 1 ) π ( p 2 ) derived from the matrix element � e 2 W µν ( q i , p i ) = i d 4 xe − iq 1 x � π ( p 1 ) π ( p 2 ) | T ( j µ ( x ) j ν ( 0 )) | 0 � Ward identities: q µ 1 W µν = q ν 2 W µν = 0. Expand on basis T n µν ( q i , p i ) , n = 1 · · · 5 W µν = A ( s , t , q 2 i ) T 1 µν + B ( s , t , q 2 i ) T 2 µν + C ( s , t , q 2 i ) T 3 µν + · · · Helicity amplitudes: H ++ ( s , θ ) , H +− ( s , θ ) , H + 0 ( s , θ ) 5/40

Analyticity of scattering amplitudes Starting point: Mandelstam analyticity conjecture [ PR 112 (1958) 1344 ] � � ρ st ( s ′ , t ′ ) ds ′ dt ′ A ( s , t , u ) = ( s ′ − s )( t ′ − t ) + D st � � � � ρ su ( t ′ , u ′ ) ρ su ( u ′ , s ′ ) dt ′ du ′ du ′ ds ′ ( t ′ − t )( u ′ − u ) + ( u ′ − u )( s ′ − s ) D tu D us Partial-wave amplitudes s-plane are analytic functions of s 0 4 m 2 π γγ → ππ : h I J ( s ) = h I J , L ( s ) + h I J , R ( s ) 6/40

Discontinuity across RHC (in elastic region 4 m 2 π � s � s in ) � � ∗ 1 t I 2 i disc [ h I J ( s )] = Im [ h I � �� � h I J ( s )] = σ π ( s ) J ( s ) J ( s ) ππ amplitude FSI theory [ Omn` es (1958) ] : � s � � ∞ ds ′ Ω I s ′ ( s ′ − s ) φ I J ( s ′ ) J ( s ) = exp π 4 m 2 π with: J ( s ′ ) , s ′ � s in (Fermi-Watson) φ I = δ I J ( s ′ ) J ( s ′ )] , s ′ > s in = Phase [ h I Application to 2 γ → 2 π : [ Gourdin,Martin Nuov.Com.17 (1960) 224 ] 7/40

Omn` es function removes right-hand cut � h I � J ( s ) s > 4 m 2 Im = 0, π Ω I J ( s ) General representation: h I J ( s ) = h I J , L ( s ) � � � ∞ P n − 1 ( s ) + s n ds ′ h I J , L ( s ′ ) sin ( φ I J ( s ′ )) + Ω I J ( s ) ( s ′ ) n ( s ′ − s ) | Ω I π J ( s ′ ) | 4 m 2 π Exact, but needs HE information In practice: in finite energy region efficient approximation, few parameters, Fix (partly) from chiral constraints 8/40

Phase φ I J in inelastic region ? ππ J = 0, I = 0 : main contrib. to inelasticity from KK Coupled channel extension: Ω I J is 2 × 2 matrix: must be computed numerically from ππ , KK 2 × 2 T-matrix Prediction for φ 0 0 = Phase [ h 0 0 ] 350 φ 0 0 δ 0 300 0 0 ], Phase-shift 250 200 150 Phase[ h 0 100 50 0 0.2 0.4 0.6 0.8 1 1.2 1.4 √ s GeV Phase [ h 0 0 ] displays sharp fall off 9/40

Left-hand cut Leading contribution at small s: from pion pole in γπ + → γπ + , computed from sQED Lagrangian called Born term Soft photon theorem [ F. Low, PR 110(1958)974 ] h I J ( s ) → h I , Born ( s ) + O ( s ) J ⇒ P n − 1 ( 0 ) = 0. 10/40

Beyond pion pole: two options 1) Start from Mandelstam based DR’s (e.g. family with ( t − a )( u − a ) = b ) [ Hoferichter et al. EPJ C71 (2011)1743 ] . Then, project on PW’s: − → h J , L ( s ) in terms of Im [ γπ → γπ ] J (but no detailed exp. inputs) 2) Less rigorous (large N c ): resonance contributions to γπ → γπ , from Lagrangian : ρ , ω , a 1 , b 1 , a 2 ... Extension to γ ∗ rather simple 11/40

Chiral expansion results and constraints 12/40

Historically: γγ → π 0 π 0 : [ Bijnens, Cornet NP B296 (1988) 557, Donoghue, Holstein,Lin PR D37 (1988)2423 ] . One-loop calc. in ChPT: finite, no LEC’s � � ++ ( s ) = m 2 1 + m 2 π − s s log 2 σ π ( s ) − 1 H n π 8 π 2 F 2 σ π ( s ) + 1 π Measured at DESY [ Crystal Ball, PR D41 (1990) 3324 ] γγ → π 0 π 0 20 ChPT p 4 18 Crystal Ball (1990) 16 σ ( | cos( θ ) | < 0 . 8) (nb) 14 12 10 8 6 4 2 Two-loop calc. [ Bellucci et 0 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 al.(1990) ] improves energy dep. √ s (GeV) 13/40

Matching DR’s and ChPT Pion polarisabilities electric ( α i ) and magnetic ( β i ) e 2 π ) = s ( α 1 − β 1 ) + s 2 2 π m π H ++ ( s , t = m 2 12 ( α 2 − β 2 ) + · · · − e 2 π ) = s ( α 1 + β 1 ) + s 2 2 π m π H +− ( s , t = m 2 12 ( α 2 + β 2 ) + · · · Polarisabilities in ChPT [ Gasser, Ivanov, Sainio NP B728 (2005) 31,B745(2006)84 ] ( α − β ) [ 10 − 4 fm 3 ] p 4 , p 6 Couplings π 0 one-loop − 1.0 None two-loops −( 1.9 ± 0.2 ) c 29 , c 30 , c 31 , c 32 , c 33 , c 34 π + ¯ l 5 − ¯ one-loop 6.0 ± 0.6 l 6 two-loops 5.7 ± 1.0 · · · + c 6 , c 35 , c 44 , c 46 , c 47 , c 50 , c 51 Compass(2014): ( α − β ) π + = ( 4.0 ± 1.2 ± 1.4 ) · 10 − 4 fm 3 14/40

Use four constraints from ChPT: 1) For π + : [ Gasser et al., NP B745 (2006)84 ] ( α 1 − β 1 ) = [ 4.7 − 5.7 ] 10 − 4 fm 3 2) For π 0 : relation between dipole and quadrupole polarisabilities in terms of one p 6 coupling π ( α 2 − β 2 ) π 0 =( 6.20 + 10 5 c r 6 ( α 1 − β 1 ) π 0 + m 2 34 ) 10 − 4 fm 3 c r 34 can be estimated from sum rule: ( τ decays inputs) 10 5 c r 34 = ( 1.18 ± 0.31 ) : [ D¨ urr,Kambor PR D61 (2000) ] 10 5 c r 34 = ( 1.37 ± 0.16 ) : [ Golterman et al.,PR D89 (2014) ] 3) For K + , K 0 : ChPT p 4 2 e 2 ( α 1 − β 1 ) K 0 = 0, ( α 1 − β 1 ) K + = ( L r 9 + L r 10 ) π m K F 2 15/40 K

180 σ (cos θ ≤ 0 . 8) (nb) 160 Five polyn.parameters fitted Belle 140 Crystal Ball 120 100 80 π 0 π 0 data : [ γγ → π 0 π 0 ] 60 Integrated cross sections: 40 20 agreement is fair 0 0.4 0.6 0.8 1 1.2 E (GeV) 18 18 18 16 E=0.61 16 E=0.81 16 E=0.87 14 14 14 dσ/d cos θ (nb) 12 12 12 10 10 10 8 8 8 6 6 6 4 4 4 2 2 2 0 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Differential cross sections 50 60 350 [ γγ → π 0 π 0 ] E=0.97 55 E=1.07 E=1.29 300 45 50 larger angular coverage 45 250 40 40 200 35 35 needed 150 30 30 25 100 20 25 50 15 20 10 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 16/40

σ (cos θ ≤ 0 . 6) (nb) π + π − data 300 Belle Mark II CELLO 250 200 Integrated cross sections: 150 [ γγ → π + π − ] some tension w. Mark II 100 50 0.4 0.6 0.8 1 1.2 E (GeV) 250 250 250 dσ/d cos θ (nb) E=0.97 E=0.80 E=0.90 Differential cross sections: 200 200 200 150 150 150 100 Needed: 100 100 50 Better precision 50 50 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 [ γγ → π + π − ] 300 550 600 Larger angular 500 550 E=1.29 E=1.17 250 E=1.07 450 500 400 450 200 350 400 coverage 300 350 150 250 300 200 250 100 150 200 100 150 50 50 100 0 0 50 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 17/40

σ → 2 γ couplings Dai Pennington (2014) Present work Hoferichter (2011) Mennessier (2011) Mao (2009) Mennessier (2008) Bernabeu (2008) Oller (2008) Pennington (2008) Oller (2008) Pennington (2006) 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Γ( σ → 2 γ ) (KeV) 18/40

From γγ to γγ ∗ ( q 2 ) 19/40

Goal is to extend the same formalism to γγ ∗ ( q 2 ) → ππ (not seemed to have been considered previously) Two physically accessible situations: e − e + → γππ ( q 2 > 4 m 2 π ) ( q 2 < 0) e − γ → e − ππ Issues to address 1) “left-hand” cut ( q 2 > 4 m 2 π ) 2) Form factors 20/40

Left-hand cut when q 2 � = 0 Definition of Born term: compute diagrams with sQED and q 2 � = 0 Influence of pion form factor � π + ( p ) | j µ ( 0 ) | π + ( p ′ ) � = ( p + p ′ ) µ F v π (( p − p ′ ) 2 ) First two diagrams mult. by F v π ( q 2 ) Gauge invariance: ⇒ mult. also third diagram 21/40

Consider J = 0 partial wave projection � 4 m 2 � ( s , q 2 ) = F v π ( q 2 ) σ π ( s ) log 1 + σ π ( s ) h Born π 1 − σ π ( s ) − 2 q 2 0 s − q 2 � with σ π ( s ) = 1 − 4 m 2 π / s Cut is on the negative real axis π pole at s = q 2 in the physical region But: if q 2 > 4 m 2 Note: s = q 2 corresponds to soft photon 22/40

Omn` es dispersive integral: � ∞ ds ′ sin δ ( s ′ ) ( 4 m 2 π L π ( s ′ ) − 2 q 2 ) I ( s , q 2 ) = 1 ( s ′ − s )( s ′ − q 2 ) | Ω ( s ′ ) | π 4 m 2 π well defined: both s , q 2 energy variables, I ( s , q 2 ) defined with lim ǫ → 0 s + i ǫ , q 2 + i ǫ Note: Fermi-Watson breaks down [ Creutz, Einhorn PR D1 (1970) 2537 ] Unitarity: Im � γγ ∗ | ππ � = � γγ ∗ | ππ �� ππ | ππ � + � γ ∗ | ππ �� γππ | ππ � 23/40

Further contributions to left-hand cut 8 Cut : 6 4 Im ( s ) /m 2 π 2 0 -2 -4 -6 -8 -20 -10 0 10 20 30 40 50 Re ( s ) /m 2 π Overlaps with positive real axis Problem with Omn` es ? 24/40