Theory of hadronic B decays

Theory of hadronic B decays Dan Pirjol MIT BEAUTY 2005, 10th International Conference on B physics Assisi, Italy - 20-24 June 2005

Outline Hadronic B decays - probes of the flavor structure of the Standard model The challenge of the strong interactions Factorization methods (QCDF, pqcd) Progress from the Soft-Collinear Effective Theory (SCET) Factorization in B decays into two light mesons Phenomenological study of corrections to factorization Factorization in B D ( ) π - implications for 2β + γ B ππ Conclusions

The goal 1.5 excluded area has CL < 0.05 1 sin 2b Dm d V ubv ud γ α V tbv td β 0 1 h -1.5-1 -0.5 0 0.5 1 1.5 2 Independent measurements of the sides and angles of the UT should overconstrain the CKM matrix Test consistency with the SM picture, or find signals for new physics 0.5 0-0.5-1 e K V ub /V cb C K M f i t t e r ICHEP 2004 a g a r b a Dm s & Dm d e K

Electroweak Hamiltonian Integrate out the top and W H W = G F 2 i Tree operators λ (i) CKM C i(µ)o i O u 1 = (ūb) V A ( du) V A W O u 2 = (ū i b j ) V A ( d j u i ) V A b W t d Penguin operators O 3 6 = ( db) V A ( qq) V ±A q O 7 10 = ( db) V A e q ( qq) V ±A q + new physics contributions... Now known at NNLO Gorbahn, Haisch - hep-ph/0411071 Gorbahn, Haisch, Misiak - hep-ph/0504194

Strong-interaction effects Weak interactions of quarks take place inside hadrons -> need to account for strong interaction nonperturbative effects M 1 M 2 H EW B =? Controlling these effects is a central part of SM physics Lattice QCD Exploit symmetries of QCD: flavor SU(3) chiral symmetry heavy quark symmetry Factorization theorems of hard QCD Effective field theories Recent progress from Soft-Collinear Effective Theory

Flavor symmetry - SU(3) Gronau, Hernandez, London,Rosner The B decay amplitude can be given as a sum over graphical amplitudes 6 graphical amplitudes = 5 (equivalent to SU(3) Wigner-Eckart) reduced matrix elements W p B b u p B b d p B p p p Tree (T) Color-suppressed (C) Penguin (P) + EWPs p p p B W B B p p p W-exchange (E) Weak annihilation (A) Penguin annihilation (PA)

Counting amplitudes and strong phases B P 8 P 8 B P 8 V 8 B V 8 V 8 B Mη 1 5 amps 4 phases 10 amps 4 phases 5 amps 4 phases 4 amps In certain lucky cases the strong amplitudes can be completely eliminated using experimental input Isospin analysis in B ππ Gronau, London W 1 +- - 2 A 2q 2 P sin a P 1 ~ +- A ~00 2 A A 00 More often than not, additional dynamical information is required - usually assuming negligible rescattering or weak annihilation A E P A 0 Powerful approach, if used carefully A dynamical approach needed, providing a guide to the hierarchy of amps A + 0 X P 2a Z Y see talk of M. Gronau

Theoretical approaches - factorization Schwinger; Bauer,Stech,Wirbel Bardeen, Buras, Gerard Large Nc QCD Blok, Shifman Neubert,Stech Buras,Silvestrini; Ali, Lu Naive/`improved factorization QCD factorization Beneke,Buchalla,Neubert,Sachrajda Du, Lu, Sun, Yang, Zhu Ciuchini et al. pqcd - kt factorization SCET Keum, Li, Sanda Bauer, Fleming, DP, Rothstein,Stewart Chay, Kim Beneke,Chapovsky, Diehl,Feldmann Hill, Becher,Neubert,Lee

QCD factorization Example: B M 1 M 2 M 1,2 = π, K, η, Λ/m b O(α s ) At leading order in and the hadronic matrix elements of simplify H W Beneke,Buchalla,Neubert,Sachrajda A(B M 1 M 2 ) = C {F BM 1 T I Φ M2 + Φ B T II Φ M1 Φ M2 } ``form-factor term ``hard-scattering term T I and T II are calculable kernels. Strong phases introduced through the Bander,Silverman,Soni mechanism and hard loop corrections -> generically small φ O(α s, Λ m b )

Theoretical issues All-order (in α s ) proof? Factorization scale separation Identify all relevant scales How well do we know the hadronic input parameters? Form-factors, wave functions, decay constants - nonperturbative QCD Factorization for power correction terms? Can be addressed in the Soft-Collinear Effective Theory

Soft-Collinear Effective Theory Systematic power counting in hadrons Λ/m b for B processes involving energetic Introduce distinct fields for the relevant degrees of freedom: soft, collinear, hard-collinear, etc. Construct effective Lagrangians for strong and weak interactions describing the dynamics of the relevant degrees of freedom Guiding principles - new symmetries: collinear/soft gauge invariance, reparameterization invariance

Example: Semileptonic B πlν decay at large recoil E π m B /2 b W u B 0 d Relevant modes: - Soft - Collinear p 2 s Λ 2 p 2 c Λ 2 - Hard-collinear p 2 hc QΛ p µ s Λ n p c Q n p hc Q

SCET - degrees of freedom Introduce fields for each degree of freedom, with well-defined momentum scaling and power counting mode field momentum p µ (+,, ) p 2 hard hard-collinear collinear A n,q, ξ n,p A n,q, ξ n,p (Q, Q, Q) (Λ, Q, QΛ) (Λ 2 /Q, Q, Λ) Q 2 QΛ soft/usoft A s, q, b v (Λ, Λ, Λ) 2-step matching: QCD SCET I SCET II µ 2 Q 2 µ 2 QΛ

Factorization In the large recoil region, the heavy-to-light form factors satisfy a factorization theorem Beneke, Feldman; Bauer, DP, Stewart f i (E) = C i (E, µ)ζ(e, µ) + nonfactorizable 1 0 dxdk + B i (E, µ, z)j(x, z, k + )φ + B (k +)φ π (x) factorizable p 2 ~ Q 2 B M p 2 ~ Λ2 p 2 ~ QΛ p 2 ~ 2 Λ

In the large recoil region, the heavy-to-light form factors satisfy a factorization theorem Beneke, Feldman; Bauer, DP, Stewart f i (E) = C i (E, µ)ζ(e, µ) + nonfactorizable 1 0 dxdk + B i (E, µ, z)j(x, z, k + )φ + B (k +)φ π (x) factorizable Ingredients Wilson coefficients - hard momenta C 1 (ω) = 1 + O(α s (Q)) Jet function - hard-collinear momenta J = πα s( QΛ)C F N c 1 xk + + O(α 2 s) term also known (Becher, Hill, Lee, Neubert) Soft form factor - matrix element in SCET-II ζ(e, µ) Collinear matrix elements - light-cone wave functions

Nonleptonic B decays 1. Recent theory progress from SCET SCET operator analysis and factorization in 2-body nonleptonic B decays Universality of the jet function in form factors and nonleptonic B decays Comparison with QCDF, pqcd Chay, Kim Bauer, DP, Rothstein, Stewart 2. Phenomenology of power corrections Grossman, Hoecker, Ligeti, DP

SCET proof of factorization in B M 1 M 2 QCD SCET I matching Bauer, Rothstein, DP, Stewart d u d n u n + b u b v u n (ūb)( du) c i (ω i, µ)q (0) i + b i (ω i, µ)q (1) i + [W id/ W ] n The LO Wilson coefficients O(λ 0 ) c i (ω i, µ) O(λ) known to 1-loop order λ 2 = Λ m b b i (ω i, µ) known only at tree level BBNS

SCET I SCET II matching Nonfactorizable terms T {Q (0) i, il (1,2) qξ } ζ(e, µ) Factorizable terms (soft form factor) Add the interaction with the spectator quark A n q ξ n il qξ Beneke, Chapovsky,Diehl,Feldmann T {Q (1) i, il (1) qξ } J (q nb v ) ( q n q n ) ( q n q n ) (convolutions of light-cone B and light mesons w.f. s)

Factorization in B M 1 M 2 After matching onto SCET-II, scale separation is achieved hard hard-collinear collinear } soft p 2 Q 2 p 2 QΛ p 2 Λ 2 B p 2 ~ Λ2 p 2 ~ Q 2 M p 2 ~ QΛ p 2 ~ 2 Λ M p 2 ~ 2 Λ + LO Factorization relation A( B M 1 M 2 ) = Q cc + f M2 ζ BM 1 1 0 1 0 dut 2 (u)φ M2 (u) + (1 2) dxdzdudk + J(x, z, k + )[T 2J (u, z)φ M1 (x)φ M2 (u)+t 1J (u, z)φ M1 (u)φ M2 (x)]φ + B (k +) Matrix element of Q cc = ( sc)( cb) left in unfactorized form

Jet universality 1. The hard-collinear contribution is contained into a unique jet function (same for all mesons) 2. The jet function is identical to that appearing in the form factors - great computational simplification p p B p B p J(x, k + ) = πα s( QΛ)C F N C 1 xk + + O(α 2 s) The O(α 2 s) terms also known Hill, Becher, Lee, Neubert

The expansion in α s ( ΛQ) might not be perturbative: Introduce the nonperturbative function ΛQ 1.4GeV ζj Bπ (z) = f Bf π m 2 B dxdk + J(x, z, k + )φ π (x)φ + B (k +) and allow as a logical possibility ζ Bπ J (z) ζ Bπ (SCET power counting)

Decay amplitudes All B -> MM amplitudes (and B -> M form factors) can be written in terms of ζ BM, ζj BM (z) The result has a simple form in terms of graphical amplitudes: A(π + π ) T P A(K π + ) T P + P C EW P P EW 2A(π + π 0 ) T C 2A(K π 0 ) T C P + P T EW + P C EW P P EW A(π 0 π 0 ) C + P A( K 0 π ) P + 1 2 P C EW P P EW 2A( K0 π 0 ) C + P + P T EW + 1 2 P C EW + P P EW Relation to SCET parameters: Implications: T Kπ = N 0 f K [ c 1u K ζ Bπ + b 1u ζ Bπ J K ] C Kπ = N 0 f K [ c 2u π ζ BK + b 2u ζ BK π ] Small strong phases of T,C amplitudes A E P A O(Λ/m b ) δ T C O(α s (m b ), Weak annihilation amplitudes A, E, PA power suppressed etc. Λ m b )

Comparison with QCDF and pqcd factorization approaches 1. At lowest order in perturbation theory, the QCDF results are reproduced 2. SCET allows a more general hierarchy of the nonfactorizable vs. factorizable contributions QCDF: pqcd: SCET: ζ J ζ ζ 0 ζ ζ J (Λ/m b ) 3/2 3. The SCET result is a strict implementation of the power counting; QCDF and pqcd include subsets of power suppressed terms A( B M 1 M 2 ) = A 0 + Λ m b A 1 +

Phenomenology Determining α from an isospin analysis in B ππ Introducing dynamical input The B ππ puzzle - possible explanations Study of penguins and power corrections in B ππ Grossman, Hoecker, Ligeti, DP

B π + π and Standard method for determining sin 2α In the absence of penguin contamination β + γ S ππ = sin 2α C ππ = 0 BABAR (227m) Belle (275m) Average S pp 0.30 ± 0.17 ± 0.03 0.67 ± 0.16 ± 0.06 0.50 ± 0.12 C pp 0.09 ± 0.15 ± 0.04 0.56 ± 0.12 ± 0.06 0.37 ± 0.10 BABAR, hep-ex/0501071 Belle, hep-ex/0502035 The penguins are significant: isospin analysis required Gronau-London Br(10 6 ) B + π + π 0 5.5 ± 0.6 B 0 π + π 4.6 ± 0.4 B 0 π 0 π 0 1.51 ± 0.28 S π+ π 0.50 ± 0.12 C π+ π 0.37 ± 0.10 0 0.28 ± 0.40 C π0 π A + = λ (d) u T c (1 + r c e iδ c e iγ ) 2A00 = λ (d) u T n (1 + r n e iδ n e iγ ) 2A 0 = λ (d) u (T n + T c ) 5 hadronic parameters + α -> can be determined using 6 data points

Isospin analysis 1.2 1 C K M f i t t e r Moriond 05 B pp (S/C + from BABAR) B pp (S/A + from Belle) Combined no C/A 00 Confidence level 1.2 1 0.8 0.6 0.4 C K M f i t t e r 1000 fb 1 B 00 = 1.9 B 00 = 1.2 B 00 = 0.66 1 CL 0.8 0.6 0.2 0 0 20 40 60 80 100 120 140 160 180 0.4 a (deg) 0.2 CKM fit no a meas. in fit 0 0 20 40 60 80 100 120 140 160 180 a (deg) Additional dynamical input required _(S + )= _(C + )~0.01

1 CL SCET determination of Bauer, Rothstein, Stewart At leading order in and,the strong phases among tree amplitudes vanish Arg(T n /T c ) O(α s (m b ), Eliminates one hadronic parameter Can be used to make predictions for 1.2 1 0.8 0.6 C K M f i t t e r CKM 2005 α α s (m b ) Λ/m b A+ = λ (d) u T c (1 + r c e iδ c e iγ ) Λ ) 2A00 = λ (d) u T n (1 + r n e iδ n e iγ ) m b 2A 0 = λ (d) u (T n + T c ) IA w/o C 00 and t (t) = 0 IA w/o C 00 and t (c) = 0 full IA IA w/o C 00 α, C π 0 π 0 1 CL 1 0.75 0.5 t-convention c-convention C exp π 0 π 0 = 0.28 ± 0.40 0.4 0.2 CKM fit no a in fit 0.25 0 0 20 40 60 80 100 120 140 160 180 a (deg) 0-1 -0.5 0 0.5 1 C 00

The B ππ `puzzle Determine the tree amplitudes Graphical representation of the isospin relation 1.5 1 0.5 0-0.5-1 1 t t n -1.5 0 0.5 1 1.5 2 γ = 54, 64, 74 T c, T n, T 0 1 + t n = t Large strong phases Arg(T n /T c ) 40 50 Significant color-suppressed amplitudes t n O(1) arg( T C / T + ) (deg) 150 100 50 0-50 -100 T c + T n = T 0 t = T 0 T c t n = T n T c Color suppression in B pp C K M f i t t e r ICHEP 2004 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2-150 0.1 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 T C / T + γ CKMfit = 59 +6.4 4.9 Ali, Lunghi, Parkhomenko Buras, Fleischer,... Bauer, Rothstein,DP,Stewart

Possible explanations Factorizable contributions larger than previous estimates (w/ input from QCD sum rules) Bauer, Rothstein, DP, Stewart Large penguin/power correction terms LO SCET analysis large factorizable contributions can overcome color suppression The tree amplitudes contain penguins! Grossman,Hoecker, Ligeti, DP see also Feldmann, Hurth A(B π + π ) = λ (d) u (T + P u ) + λ (d) c P c + λ (d) t P t = λ u (T + P u P t ) + λ c (P c P t ) c-convention = λ u (T + P u P c ) + λ t (P t P c ) t-convention New physics

B ππ 2A(B π π 0 ) = N π 1 in SCET @ LO Bauer, Rothstein, DP, Stewart Working to leading order in matching at µ = m b 3 λ(d) u (C 1 + C 2 )(4ζ Bπ + 7ζ Bπ J ) + O( Λ m b ) A( B 0 π + π ) = N π λ (d) u {(C 1 + C 2 3 )ζbπ + [C 1 + (1 + ū 1 π ) C 2 3 ]ζbπ J } +λ (d) c Q c c + λ (d) t { (C 4 + C 3 3 )ζbπ [C 4 + (1 + ū 1 π ) C 3 3 ]ζbπ J } + O( Λ ) m b Use the data to extract the SCET parameters ζ Bπ, ζj Bπ ( 0.0039 ) ζ Bπ γ=64 = (0.08 ± 0.03) V ub ( 0.0039 ) ζj Bπ γ=64 = (0.10 ± 0.02) V ub Comparable factorizable vs. nonfactorizable parameters!

Predictions Predict the semileptonic B πlν and rare B πl + l (only exp. errors shown) ( 0.0039 f + (q 2 = 0) = ζ Bπ + ζj Bπ = (0.18 ± 0.02) V ub f T (q 2 = 0) = ζ Bπ ζ Bπ J = (0.02 ± 0.05) form factors ) ( 0.0039 ) V ub Different from the usual factorization prediction (e.g. Luo, Rosner), because of large hard scattering amplitude ζj Bπ 2A(B π π 0 ) = λ (d) u N 0 { 4 3 (C 1 + C 2 )f+ Bπ (q 2 = 0) + (C 1 + C 2 )ζj Bπ )} Smaller results than in QCD sum rule calculations Ball, Zwicky - hep-ph/0406232 f+ Bπ (0) = 0.26 ± 0.03 ft Bπ (0) = 0.25 ± 0.03

Testing penguins/power suppressed terms in B ππ Grossman, Hoecker, Ligeti, DP

The penguin hierarchy problem

The penguin hierarchy problem A(B π + π ) = λ (d) u Theory issues: Use unitarity of the CKM matrix (T + P u ) + λ (d) c P c + λ (d) t P t Pt could contain large chirally enhanced terms = subleading but large corrections - included in QCDF Pc could contain nonperturbative contributions - charming penguins Pu could be enhanced by rescattering Ciuchini et al., Bauer et al. What can we say about their relative magnitude using data? PHP = λ u (T + P u P t ) + λ c (P c P t ) c-convention = λ u (T + P u P c ) + λ t (P t P c ) t-convention Compare the shapes of the tree triangles in different conventions

T + P uc P uc C P uc T + P ut P ut C P ut T + C c-convention T + C t-convention Pc enters tree amplitudes only in the c-convention Pt enters tree amplitudes only in the t-convention Pu enters tree amplitudes in both conventions Compare the strong phases τ (c) Arg[(T + P uc )/(T + C)] vs. τ (t) Arg[(T + P ut )/(T + C)]

Results 1.2 1 t-convention c-convention t-convention fit without using C 00 1.2 1 t-convention c-convention 1 CL 0.8 0.6 1 CL 0.8 0.6 0.4 0.4 0.2 0.2 0-60 -40-20 0 20 40 60 0-60 -40-20 0 20 40 60 B ππ t (deg) B ρρ arg( T +0 / T + ) B ππ Large Pu amplitude, smaller Pc, Pt significant weak annihilation? B ρρ Small penguins SCET constraint on α may help

Factorization in B D ( ) π and implications for 2β + γ

Interference of tree-mediated decays Aleksan et al., Dunietz, Rosner d π u c D + u b c B 0 B 0 D + b u π B 0 D ( )+ π V cb V ud B 0 D ( )+ π V ubv cd B 0 B 0 mixing induces a time-dependent CP asymmetry Final state = non-cp eigenstate The asymmetry depends on hadronic parameters

Time-dependent decay rates Γ(B(t) f) e t /τ 2τ Tagging meson ( B 0 B 0 ) (1 S ζ sin( Mt) ηc ζ cos( Mt)) (η, ζ) = +( ) D π + (D + π ) The asymmetry parameters depend on hadronic physics A(B 0 D + π ) A( B 0 D + π ) = reiδ+γ r λ V ub O(1) 0.02 V cb as S ± = 2r color-allowed 1 + r 2 sin(2β + γ ± δ) C = 1 r2 1 + r 2

BABAR and BELLE measured fully and partially reconstructed D ( )± π BABAR includes also D ± ρ Measurements expressed in terms of 2 parameters a D ( ) π = 2r D ( ) π sin(2β + γ) cos δ D ( ) π c D ( ) π = 2r D ( ) π cos(2β + γ) sin δ D ( ) π WA Dπ D π HFAG a c 0.045 ± 0.027 0.035 ± 0.035 0.027 ± 0.013 0.001 ± 0.018 Extract r from Combine a and c to determine r 2 = Br(B0 D + π ) Br(B 0 D π + ) 3 10 4 2β + γ (up to a 4-fold ambiguity)

Issues The Cabibbo suppressed mode B 0 D + π is very hard to measure Additional theory input is required for extracting r Factorization SU(3) based methods (+ dynamical assumptions) Tagging using nonleptonic modes introduces dependence on the hadronic parameters of the tag mode

Factorization What does factorization tell us about r and? The CKM allowed mode leading order in 1/m b A(B 0 D + π ) A( B 0 D + π ) = reiδ+γ b cdū δ is calculable in factorization at BBNS; Bauer, DP, Stewart d π u A( B 0 D + π ) a 1 F B D f π + O(α s (m b ), Λ m b ) B 0 b c D + small strong phases

No proof of factorization exists so far for the CKM suppressed ( wrongcharm decay) modes b ud c Computations in pqcd C.D.Lu The only known limit of QCD where factorization holds for DCS modes is the large Nc limit Only valid for color-allowed modes! B 0 b c D + u u π D + B 0 O( 1 N c ) π (C 1 + C 2 3 )F B πf D + O( 1 N c )

Large Nc factorization r Dπ = λ V ub V cb R Dπ 0.01 R = ratio of hadronic matrix elements R Dπ = (m2 B m2 π)f0 B π (m 2 D )f D (m 2 B 0.4 m2 D )F B D 0 (m 2 π)f π Corrections ~ 30-50% Small strong phase δ predicted

Consistency check for the large Nc estimate for r What does data tell us about the strong phases? Assume β = 23, γ = 60 re iδ r e iδ B Dπ Significant strong phases allowed! B D π The large Nc estimate can be trusted if future data rule out large strong phases

SU(3) flavor Exact SU(3) relations A(B 0 D ( )+ π ) = λ[a(b 0 D ( )+ s π ) + A(B s D ( )+ π )] = λa(b s D ( )+ s K ) B s + D π B s D ( )+ π is pure W exchange Assuming that it can be neglected, data on BABAR hep-ex/0504035 r = λ B 0 D + s π B(B 0 Ds + π B(B 0 D π + ) can determine r f D f D s = 0.015 +0.006 0.004

Error estimate Sources of theory error in this r determination SU(3) breaking ~ 30% Neglect of the W-exchange amplitude E The W-exchange amplitude can be measured in: S = 1 B s decays : B s D + π, B s D 0 π 0 S = 0 B d decays : B 0 D + s K (but not B 0 D + s K!) In the absence of such data, the error on r could be as large as 100%

Conclusions A model-independent theory of nonleptonic B decays is now available, based on an effective field theory with a well-defined power counting in Λ/m b SCET separates the contributions of the physics on different scales: factorization SCET yields a surprising connection between semileptonic, rare and nonleptonic B decays: at LO they are parameterized in terms of a small set of common parameters A combined fit to all these (measured) modes can extract these parameters and allows predictions for new modes Much work remains to be done to explore power corrections to these results. Phenomenological studies reveal an intriguing pattern of corrections in B ππ