Meson Form Factors and the BaBar Puzzle

Transcription

1 Meson Form Factors and the BaBar Puzzle Nils Offen Universität Regensburg Séminaire Particule Orsay 17.November 211 Nils Offen (Universität Regensburg) The BaBar Puzzle Orsay / 3

2 1 Introduction: The BaBar Puzzle 2 Collinear Factorisation 3 Proposed Solutions 4 Light Cone Sum Rules 5 Reprise: η ( ) γγ -Transitions 6 Conclusions/Summary Nils Offen (Universität Regensburg) The BaBar Puzzle Orsay / 3

3 Introduction: The BaBar Puzzle π(η ( ) ) γ γ-transition form factors Z d 4 x e iq 1 x P(p) T {j µ(x) j ν()} = i e 2 ǫ µναβ q α 1 q β 2 F γ γ P(q 2 1, q 2 2) related to axial anomaly for q 2 1 = q 2 2 = F(, ) = 1 4pi 2 f π theoretically cleanest case: both photons virtual q 2 1, q 2 2 e? e? e? e + e + e + experimentally easier: one real photon q 2 2 =, q 2 1 = Q 2 < Nils Offen (Universität Regensburg) The BaBar Puzzle Orsay / 3

4 Introduction: The BaBar Puzzle Good Old Times.22.2 Q 2 F γ γπ (Q 2 ) [GeV] Q 2 [GeV 2 ] asymptotic limit from handbag diagram π lim Q 2 F πγγ (Q 2 ) = 2 f π Q 2 collinear QCD seemed to describe main part of FF asymptotic regime reached for Q 2 few GeV 2? Brodsky, Lepage Nils Offen (Universität Regensburg) The BaBar Puzzle Orsay / 3

5 Introduction: The BaBar Puzzle The BaBar-Puzzle Q 2 F(Q 2 ) (GeV).3.2 CELLO CLEO BABAR CZ.1 BMS ASY Q 2 (GeV 2 ) BaBar-Puzzle part I: experimental results exceed asymptotic limit for the π form factor Nils Offen (Universität Regensburg) The BaBar Puzzle Orsay / 3

6 Introduction: The BaBar Puzzle The BaBar-Puzzle Q 2 F(Q 2 ) (GeV) CELLO CLEO BABAR BMS CZ ASY Q 2 (GeV 2 ) (3/5)Q 2 F n (Q 2 ) (GeV) BABAR (γ γ π ) CLEO* (γ γ η,η / ) CLEO (e + e - γη,γη / ) BABAR (γ γ η,η / ) BABAR (e + e - γη,γη / ) Q 2 (GeV 2 ) BaBar-Puzzle part I: experimental results exceed asymptotic limit for the π form factor BaBar-Puzzle part II: η ( ) form factors behave as expected assume flavour mixing scheme n = 1 2 ( ūu + dd ), s = ss η = cos φ n sin φ s, η = sin φ n + cos φ s for similar DAs the difference between F πγ γ and F n γ γ factor 3 5 Nils Offen (Universität Regensburg) The BaBar Puzzle Orsay / 3

7 Introduction: The BaBar Puzzle The BaBar puzzle II Fixed-order NLO QCD calculation with µ 2 = Q 2 does not work:.3 a 2, a 4 Q 2 F(Q 2 ) [GeV].2.1 a 2 AS fit BABAR CLEO Input parameters at 1 GeV: magenta: a = 1, blue: a = 1, a 2 =.39, black: a = 1, a 2 =.39, Q 2 [GeV 2 ] Figure: The fixed-order NLO QCD calculation Changing pion distribution amplitude does not help at all? Power-suppressed effects 1/Q p?? Nils Offen (Universität Regensburg) The BaBar Puzzle Orsay / 3

8 Collinear Factorisation The general picture Ì À Ì ¼ À Ë µ µ µ Figure: Schematic structure of the QCD factorization for the F γ γ π (Q2 ) formfactor. A : hard subgraph that includes both photon vertices B : real photon is emitted at large distances C : Feynman Mechanism: soft quark spectator 1 Q Q Q Q 4 Contributions of regions A, B, C are additive All other possibilities lead to exponentially small corrections exp[ Q 2 ] not seen in OPE Nils Offen (Universität Regensburg) The BaBar Puzzle Orsay / 3

9 Collinear Factorisation Region A: 1 Q 2 -Terms leading term of OPE from T{ j µ(x) j ν()} can be written in factorised form: Z F γγ π(q 2 2fπ ) = 3 dx T H (x, Q 2, µ, α s(µ))φ π(x, µ) T H known to NLO in MS scheme and NNLO in conformal scheme φ π: leading twist distribution amplitude Z 1 2 fπ p µ dx e ixp z φ(x, µ) = π(p) ū(z)γ µ[z, ]u() z 2 = ER-BL evolution implies expansion in Gegenbauer-polynomials φ π(x, µ) = 6x(1 x) X n= αs(µ) () γ n /2β an(µ )C 3/2 n (2x 1), a (µ) = 1 α s(µ ) expect 1 = a > a 2 > a 4 > a 6 > a 2 [1 GeV] =.3 ±.15, a 4 [1 GeV] j.1 B-decays.1 NLC SR [BMS-model], a n>4 [1 GeV] unconstrained Nils Offen (Universität Regensburg) The BaBar Puzzle Orsay / 3

10 Collinear Factorisation Region A: Twist 4 terms twist 4 term from OPE of T{ j µ(x) j ν()} µ µ µ µ Figure: Twist-4 corrections to the pion transition form factor involves twist-4 quark-gluon pion distribution amplitudes Z «2fπ 1 dx 8 δ F γ γ π (Q2 π 2 ) = φπ(x) Q 2 3 x 27 Q 2 δ 2 π.2 GeV 2 Might be significant at Q GeV 2 but does not change high Q 2 behaviour Nils Offen (Universität Regensburg) The BaBar Puzzle Orsay / 3

11 Collinear Factorisation Region B: Photon Emission From Large Distances hard scattering kernel convoluted with twist three pion and photon DA π γ π γ results in F (B) γ γ π (Q 2 ) = 2fπ 3 16πα sχ qq 2 9f 2 πq 4 infrared divergent overlap with region C regularised result Z 1 dx φp 3;π (x) x Z 1 dy φγ(y) ȳ 2 F γ γ π (Q2 ) = 2fπ Q 2 Z 1 3 «dx.2 GeV2 φπ(x) + ln 2 Q 2 x Q 2 µ 2 IR might be significant up to Q 2 5 GeV 2 Nils Offen (Universität Regensburg) The BaBar Puzzle Orsay / 3

12 Collinear Factorisation Region C: Feynman Mechanism truly non-perturbative one quark carries almost all momentum overlap integral of wave functions use e.g. Drell-Yan representation as convolution of light-cone WFs (Brodsky-Lepage) (ε q )F qq γ γ π (Q2 ) = f π 4π 3 3 Z 1 Z dx d 2 k (ε (xq + k )) (xq + k ) 2 iǫ Ψ qq (x, k ) has to be calculated in some model Needs approaches that go geyond this picture. Nils Offen (Universität Regensburg) The BaBar Puzzle Orsay / 3

13 Proposed Solutions Sudakov Suppression Kroll; Li, Sterman; Botts, Sterman general idea: keep k dependence in hard scattering kernel 1 xq 2 1 xq 2 + k 2... and in wave function double logs from collinear and soft regions exponentiate F γ γ π (Q2 ) = Z 2fπ 3 Z dx d 2 b 2π e T H (x, Q 2, b, µ, α s(µ)) e S φ π(x, b /b) Sudakov factor suppresses region of large b soft contributions modelled by wave function hard scattering kernel and Sudakov factor suppress higher Gegenbauer moments Q 2 [GeV 2 ] Nils Offen (Universität Regensburg) The BaBar Puzzle Orsay / C n /C n

14 Proposed Solutions State-of-the-art calculations in k factorization k factorization fit ) [GeV] 2 (Q F πγ 2 Q Q [GeV ] Asymptotic DA, LO Asymptotic DA, NLO Flat DA, LO Flat DA, NLO Q 2 F(Q 2 ) [GeV].2.1 BaBar CLEO Q 2 [GeV 2 ] Li, Mishima, arxiv: P. Kroll, arxiv: flat: a 2 =.39, a 4 =.24 fit: a 2 =.25, a 4 =.7 R dx R d 2 k t Ψ qq(x, k ) 2 = needs separate fit for η γγ Nils Offen (Universität Regensburg) The BaBar Puzzle Orsay / 3

15 Proposed Solutions Quark Models Dorokhov π simple interal over quark loop γ5 with constituent quark mass M q 135 MeV reproduces BaBar data for F γγ π caveat: f π in same model divergent regularised model does not reproduce data... no Pion in QCD, no γ 5 -vertex, wrong chiral limit M q, m π similar model from PCAC by Pham, Pham quark models imply flat DA for light quarks, more peaked DAs for heavier quarks introducing dynamical constituent mass m(k 2 Λ k2 ) = M q e nonlocal quark photon vertex Γ µ = ie q(γ µ + Γ µ) pion quark vertex LCWF γ 5 F(k1,k 2 2) 2 can describe γγ π, η ( ), η c data with three different M q physical justification for m(k 2 )? models soft contribution... Nils Offen (Universität Regensburg) The BaBar Puzzle Orsay / 3

16 Proposed Solutions Musatov-Radyushkin Model I Use Drell-Yan representation as convolution of light-cone WFs (Brodsky-Lepage) (ε q )F qq γ γ π (Q 2 ) = with a model wave function f π 4π 3 3 Z 1 Z dx d 2 k (ε (xq + k )) (xq + k ) 2 iǫ Ψ qq (x, k ) to get Ψ qq(x, k ) = 4π2 σ φ π(x) exp k «2 6 x x 2σx x Z 1» «F MR 2fπ dx φ π(x) γ γ π (Q2 ) = 1 exp xq2 3 xq 2 2 xσ using σ =.53 GeV 2 and flat pion DA φ π(x) = 1 can fit the BABAR data! caveat: R R dx d 2 k t Ψ qq(x, k ) 2 =,?! F γ γ π() R 1 φπ(x) dx =,?! x Nils Offen (Universität Regensburg) The BaBar Puzzle Orsay / 3

17 Proposed Solutions Musatov-Radyuskin Model II correction in Musatov-Radyushkin model is exponentially suppressed Z 1» «F MR 2fπ dx φ π(x) γ γ π (Q2 ) = 1 exp xq2 3 xq 2 2 xσ absent in OPE for flat DA and large Q 2 numerically very similar to 2fπ 3 Z 1 φ π(x) dx xq 2 + M, 2 2 M.6GeV 2 σ.53 average k, 2 k 2 = σ = (.42 GeV)2 3 q close to folklore value k 2 3 MeV flat DA does not evolve for Photon-Pion form factor Nils Offen (Universität Regensburg) The BaBar Puzzle Orsay / 3

18 Proposed Solutions Flat Distribution Amplitude? flat distribution amplitude would force us to reconsider pqcd predictions e.g. as (pqcd) Fπ (Q 2 ) = 8παs 9Q 2 Z 1 dx Z 1 φπ(x) φπ(y) dy x y Q 2 flat DA really necessary in MR-model? Answer: BaBar flat n=4 n= n=2 n= Q x alternatively, check how much is contributed by each successive Gegenbauer polynomial: F MR flat (Q 2 = 2) = = n = n = 2 n = 4 n = 6 Nils Offen (Universität Regensburg) The BaBar Puzzle Orsay / 3

19 Proposed Solutions First Summary shape of Pion distribution amplitude The Gegenbauer expansion for the form factor calculated with flat DA converges very fast At Q 2 < 1 2 GeV 2 using n = 4 truncation is sufficient End-point behavior of a true pion DA is irrelevant soft corrections are modelled by different approaches physical (QCD) interpretation not always clear... Systematic calculation of soft effects possible? Dispersion relations. Nils Offen (Universität Regensburg) The BaBar Puzzle Orsay / 3

20 Light Cone Sum Rules The Method I The QCD result satisfies an unsubtracted dispersion relation F QCD γ γ π (Q 2, q 2 ) = 1 π hadronic sum looks like F γ γ π (Q2, q 2 2fρ F γ ) = ρ π (Q2 ) mρ q2 π ds ImF QCD γ γ π (Q 2, s) s + q 2. s Khodjamirian ds ImF γ γ π(q2, s) s + q 2. Duality: assume that above a certain threshold ImF γ γ π (Q2, s) = ImF QCD γ γ π (Q 2, s) for s > s Asymptotic freedom: QCD expression must be correct at q 2, therefore 2fρ F γ ρ π (Q2 ) = 1 π s ds ImF QCD γ γ π (Q 2, s). Duality sum rules: use this result to correct the QCD calculation Nils Offen (Universität Regensburg) The BaBar Puzzle Orsay / 3

21 Light Cone Sum Rules The Method II correlation function F(q1, 2 q2) 2 = i d 4 xe iq1x π(p) T { jµ(x)jν()} hadronic dispersion relation operator-product-expansion F (had) (Q 2, q 2 ) = 2 fρ f ρπ (Q 2 ) m 2 ρ + + q2 s ds ρ(s) s + q 2 F (OPE) (Q 2, q 2 ) = 1 π ds Im F (OPE) (Q 2, s) s + q 2 s quark-hadron-duality ds ρ(s) s + q 2 1 π s ds Im F (OPE) (Q 2, s) s + q 2 2fρ f ρπ (Q 2 ) (m 2 ρ + q2 ) = 1 π s ds Im F (OPE) (Q 2, s) s + q 2 main uncertainties universal input Borel-transformation F γπ (Q 2 ) = 1 π m 2 ρ s sum rule for lim q 2 ds Im F (OPE) (s)e (m2 ρ s)/m2 + 1 π ds Im F (OPE) (s) s s Nils Offen (Universität Regensburg) The BaBar Puzzle Orsay / 3

22 Light Cone Sum Rules The Method II correlation function F(q1, 2 q2) 2 = i d 4 xe iq1x π(p) T { jµ(x)jν()} hadronic dispersion relation operator-product-expansion F (had) (Q 2, q 2 ) = 2 fρ f ρπ (Q 2 ) m 2 ρ + + q2 s ds ρ(s) s + q 2 F (OPE) (Q 2, q 2 ) = 1 π ds Im F (OPE) (Q 2, s) s + q 2 s quark-hadron-duality ds ρ(s) s + q 2 1 π s ds Im F (OPE) (Q 2, s) s + q 2 2fρ f ρπ (Q 2 ) (m 2 ρ + q2 ) = 1 π Borel-transformation s ds Im F (OPE) (Q 2, s) s + q 2 main uncertainties universal input truncated OPE F γπ (Q 2 ) = 1 π m 2 ρ s sum rule for lim q 2 ds Im F (OPE) (s)e (m2 ρ s)/m2 + 1 π ds Im F (OPE) (s) s s Nils Offen (Universität Regensburg) The BaBar Puzzle Orsay / 3

23 Light Cone Sum Rules The Method II correlation function F(q1, 2 q2) 2 = i d 4 xe iq1x π(p) T { jµ(x)jν()} hadronic dispersion relation operator-product-expansion F (had) (Q 2, q 2 ) = 2 fρ f ρπ (Q 2 ) m 2 ρ + + q2 s ds ρ(s) s + q 2 F (OPE) (Q 2, q 2 ) = 1 π ds Im F (OPE) (Q 2, s) s + q 2 s quark-hadron-duality ds ρ(s) s + q 2 1 π s ds Im F (OPE) (Q 2, s) s + q 2 2fρ f ρπ (Q 2 ) (m 2 ρ + q2 ) = 1 π Borel-transformation s ds Im F (OPE) (Q 2, s) s + q 2 main uncertainties universal input truncated OPE quark-hadron duality F γπ (Q 2 ) = 1 π m 2 ρ s sum rule for lim q 2 ds Im F (OPE) (s)e (m2 ρ s)/m2 + 1 π ds Im F (OPE) (s) s s Nils Offen (Universität Regensburg) The BaBar Puzzle Orsay / 3

24 Light Cone Sum Rules Leading order example QCD calculation F QCD γ γ π (Q 2, q 2 ) = 2fπ 3 Z 1 dx φ π(x) xq 2 + xq 2. LCSR F LCSR γ γ π (Q2 ) = 1 Im s xq 2 xs π x δ s Q2 x x ( Z 1 2fπ 3 x Z dx φ x π(x) + xq 2 ) dx φ π(x), x xm ρ 2 = s s + Q 2 Cn/C.6 The difference is a soft correction that suppresses higher Gegenbauer-moments qualitative picture stays the same after inclusion of NLO corrections Q 2 [GeV 2 ] Nils Offen (Universität Regensburg) The BaBar Puzzle Orsay / 3

25 Light Cone Sum Rules Results I Agaev et al. Three models with a n>4 a 2 < a 4 a 2 [1 GeV].15, a 4 [1GeV].2 Nils Offen (Universität Regensburg) The BaBar Puzzle Orsay / 3

26 Light Cone Sum Rules Results II comparison of soft and hard contributions in LCSRs Agaev et al Q 2 F π γ γ(q 2 ) soft+hard hard soft Q 2 soft part still 25% at Q 2 4 GeV 2 asymptotic regime starts later than assumed?! Nils Offen (Universität Regensburg) The BaBar Puzzle Orsay / 3

27 Light Cone Sum Rules Results III The same models describe pion form EM factor and B πlν l width Agaev et al. NLO LCSRs including twist up to 6 and up to 4, respectively What about η, η? Nils Offen (Universität Regensburg) The BaBar Puzzle Orsay / 3

28 Reprise: η ( ) γγ -Transitions η η -mixing singlet-octet scheme J i µ5 P(p) = i f i P p µ (i = 1, 8; P = η, η ) f 8 η = f 8 cos θ 8, f 8 η = f 8 sin θ 8, f 1 η = f 1 sin θ f 1 η = f 1 cos θ flavour scheme J q µ5 = 1 2 (ūγ µγ 5u + dγ µγ 5d), J s µ5 = sγ µγ 5s J r µ5 P(p) = i f r P p µ (r = q, s) f q η = f q cos φ q, f q η = f q sin φ q, f s η = f s sin φ s f s η = fs cos φs neglect difference φ q φ s φ q φ s 41 include η c and G into mixing? Nils Offen (Universität Regensburg) The BaBar Puzzle Orsay / 3

29 Reprise: η ( ) γγ -Transitions BaBar-Measurement Q 2 F(Q 2 ) (GeV).25.2 CLEO* (γ γ η) CLEO (e + e - ηγ) (a) Q 2 F(Q 2 ) (GeV).3.25 CLEO (γ γ η / ) CLEO (e + e - η / γ) (b) BABAR (γ γ η) BABAR (e + e - ηγ) Q 2 (GeV 2 ).15 BABAR (γ γ η / ) BABAR (e + e - η / γ) Q 2 (GeV 2 ) BaBar measured η ( ) γγ and e + e η ( ) γ form factors Nils Offen (Universität Regensburg) The BaBar Puzzle Orsay / 3

30 Reprise: η ( ) γγ -Transitions BaBar-Measurement (3/5)Q 2 F n (Q 2 ) (GeV).3.2 Q 2 F s (Q 2 ) (GeV).15.1 CLEO* (γ γ η,η / ) CLEO (e + e - γη,γη / ) BABAR (γ γ η,η / ) BABAR (e + e - γη,γη / ).1 BABAR (γ γ π ) CLEO* (γ γ η,η / ) CLEO (e + e - γη,γη / ) BABAR (γ γ η,η / ) BABAR (e + e - γη,γη / ) Q 2 (GeV 2 ) Q 2 (GeV 2 ) BaBar measured η ( ) γγ and e + e η ( ) γ form factors used flavour scheme to translate to light quark and strange quark content n FF does not rise as pion FF s FF falls short even of prediction with asymptotic DA Nils Offen (Universität Regensburg) The BaBar Puzzle Orsay / 3

31 Reprise: η ( ) γγ -Transitions BaBar-Measurement (3/5)Q 2 F n (Q 2 ) (GeV).3.2 Q 2 F s (Q 2 ) (GeV).15.1 CLEO* (γ γ η,η / ) CLEO (e + e - γη,γη / ) BABAR (γ γ η,η / ) BABAR (e + e - γη,γη / ).1 BABAR (γ γ π ) CLEO* (γ γ η,η / ) CLEO (e + e - γη,γη / ) BABAR (γ γ η,η / ) BABAR (e + e - γη,γη / ) Q 2 (GeV 2 ) Q 2 (GeV 2 ) BaBar measured η ( ) γγ and e + e η ( ) γ form factors used flavour scheme to translate to light quark and strange quark content n FF does not rise as pion FF s FF falls short even of prediction with asymptotic DA Can additional contributions solve this? Nils Offen (Universität Regensburg) The BaBar Puzzle Orsay / 3

32 Reprise: η ( ) γγ -Transitions Additional Corrections for the η ( ) form factors gluonic content Agaev et al. work in progress include three Gluon or Twist 4 contribution? η ( ) η ( ) η ( ) mass corrections due to m 2 η ( ) SU(3)-breaking due to additional twist 3 corrections for massive strange quark ms µ ( η ( ( ) 2f ) Z 1 σ η dx dφ (x) Z x η ( ),3 dx dφ σ + (x) ) η ( ),3 x Q 2 3 xq 2 dx xm ρ 2 = s + ms 2 dx s + Q 2 x even though m 2 n > m 2 π larger effect if η c is taken into account unlikely to cure discrepancy of F γγ π F γγ n Nils Offen (Universität Regensburg) The BaBar Puzzle Orsay / 3

33 Conclusions/Summary Conclusions/Summary picture still rather confusing some important issues LCSR fits generally prefer a small value a 2 (1 GeV) compared to a 2 (1 GeV).35 ±.15 from lattice calculations higher precision lattice data needed BES data? BABAR data in the Q 2 = 1 2 GeV 2 range require large a 4 (1 GeV).25; older data/other reactions not sensitive because of lower effective Q 2 No natural explanation for the difference γ γ π and γ γ η more experimental data needed (KEK?) Nils Offen (Universität Regensburg) The BaBar Puzzle Orsay / 3

34 Method µ = 1 GeV µ = 2 GeV Reference LO QCDSR, CZ model CZ 1981 QCDSR Khodjamirian et al. 24 QCDSR.28 ±.8.19 ±.5 Ball et al. 26 QCDSR, NLC.19 ±.6.13 ±.4 BMS 91, 98, 1 F πγγ, LCSR.19 ±.5.12 ±.3 (µ = 2.4) Schmedding, Yakovlev 99 F πγγ, LCSR.32.2(µ = 2.4) BMS 2 F πγγ, LCSR, R.44.3 BMS 5 F πγγ, LCSR, R Agaev 5 Fπ em,lcsr.24 ±.14 ±.8.16 ±.9 ±.5 Braun 99, Bijnens 2 Fπ em,lcsr, R.2 ±.3.13 ±.2 Agaev 5 F B πlν, LCSR.19 ± ±.13 Ball 5 F B πlν, LCSR.16.1 Duplancic 8 LQCD, N f = 2, CW.329 ± ±.114 QCDSF/UKQCD 6 LQCD,N f = 2+1, DWF.382 ± ±.88 RBS/UKQCD 7 Nils Offen (Universität Regensburg) The BaBar Puzzle Orsay / 3

35 Region C: LCSR vs. MR model Separate contributions of different Gegenbauer polynomials Q 2 F γ γ π (Q2 ) = o 2f π nf (Q 2 ) + a 2 f 2 (Q 2 ) + a 4 f 4 (Q 2 ) and compare the coefficients f n(q 2 ) n= n=2 n= n= n=2 n= Q Q2 A qualitative agreement MR Model LCSR Convincing evidence for strong suppression of end-point regions alias contributions of higher Gegenbauer polynomials in pion DA Nils Offen (Universität Regensburg) The BaBar Puzzle Orsay / 3