Inclusive B decay Spectra by Dressed Gluon Exponentiation

Transcription

1 Inclusive B decay Spectra by Dressed Gluon Exponentiation Plan of the tal Einan Gardi (Cambridge) Inclusive B decay spectra motivation Strategy of the theoretical study Very short introduction to Sudaov logs and renormalons Kinematics, the endpoint region and the shape function approach Factorization, Sudaov resummation with NNLL accuracy and the divergence of perturbation theory Dressed Gluon Exponentiation: renormalon resummation in the Sudaov exponent The quar distribution function in a meson and in an on-shell heavy quar cancellation of the leading renormalon ambiguity Numerical results for inclusive B-decay spectra by DGE; comparison to data Einan Gardi (University of Cambridge) KEK, October 5 1

2 Inclusive B decay Spectra by Dressed Gluon Exponentiation References Inclusive spectra in charmless semileptonic B decays by DGE, J.R. Andersen, E. Gardi, [hep-ph/5936]. Taming the B X s γ spectrum by Dressed Gluon Exponentiation, J.R. Andersen, E. Gardi, JHEP 56 3 (5) [hep-ph/5159]. On the quar distribution in an on-shell heavy quar and its all-order relations with the perturbative fragmentation function, E. Gardi, JHEP 5, 53 (5) [hep-ph/5157]. Radiative and semi-leptonic B-meson decay spectra: Sudaov resummation beyond logarithmic accuracy and the pole mass, E. Gardi, JHEP 44, 49 (4) [hep-ph/4349]. Einan Gardi (University of Cambridge) KEK, October 5

3 Inclusive B decay Spectra Weights / 1 MeV 4 radiative decay: B Xs γ Data Spectator Model Events / 1 MeV/c semi-leptonic decay: B Xu l ν l signal region (a) E (GeV) CLEO M x (GeV/c ) The distribution peas close to the endpoint (E γ M B /; small M X ) BELLE Example: extracting V ub from the semi-leptonic decay Precise measurements are restricted to the small M X region (charm bacground) Determination of V ub relies on calculation of the spectrum. Einan Gardi (University of Cambridge) KEK, October 5 3

4 Strategy of the theoretical study Decay spectra are complicated quantities. They depend on The underlying decay mechanism The structure of the B meson The jet structure and hadronization in the final state. The latter two involve confinement; they go beyond perturbation theory. To study the applicability of perturbation theory one can Disentangle effects of different characteristic scales; apply factorization. Identify sources of large corrections and resum them. Study the infrared sensitivity: renormalon analysis. Einan Gardi (University of Cambridge) KEK, October 5 4

5 Very short introduction to Sudaov Logs and Renormalons hierarchy of scales logs Soft and collinear gluon radiation (nearly on-shell partons) Running coupling logs in loops d 4 d 4 Sudaov logs Renormalons Einan Gardi (University of Cambridge) KEK, October 5 5

6 Sudaov Logs Incomplete cancellation between real and virtual corrections The quar propagator (for = and p = ) 1 = 1 (p+) p = 1 E g E q (1 cosθ qg ) m r p+ is singular at E g = (soft) and at θ qg = (collinear) For infrared and collinear safe observables, the singularity itself cancels but the coefficients contain residual logarithms. Each gluon emission generates up to two large logarithms = multiple emission is important! m r p 1 3 p 4 Factorization properties of QCD matrix elements (and of the phase space) = Exponentiation Einan Gardi (University of Cambridge) KEK, October 5 6

7 Renormalons IR renormalons: the large order behavior as a probe of large distance effects Example: vacuum polarization D(Q ) = C F α s n = C F For small momenta (IR) φ(ǫ) ǫ d d φ( /Q ) φ( /Q ) α s( ) π β α s π ln( /Q n ) Q For large momenta (UV) φ(ǫ) ln ǫ/ǫ Q A n d Q p ln /Q n = A n n! Minimal term at n n m = p/a. Ambiguity n m!n n m m At large orders perturbation theory is factorially divergent. p n+1 This is dictated by contributions of extreme momenta, UV or IR. A β α s π exp( n m ) = (Λ /Q ) p The contribution from the IR region is non summable and generates ambiguous power terms. Einan Gardi (University of Cambridge) KEK, October 5 7

8 The propagator: Example: renormalon ambiguity in the pole mass i p/ m MS Σ(p, m MS ) Computed in the large N f limit Off shell Σ(p, m MS ) has no renormalons But applying the on shell condition (inverse propagator vanishes at p = m ): m m MS = 1 + C F β du ( Λ m MS ) u [ 3e 5 3u (1 u)γ(1 + u)γ( u) Γ(3 u) + 3 ] 4u R Σ 1 (u). Beyond PT the pole mass is ambiguous... and so is Λ = M m. Benee & Braun; Bigi, Shifman, Uraltsev & Vainshtein (94) Einan Gardi (University of Cambridge) KEK, October 5 8

9 Kinematics in B X s γ In the B meson the b quar is close to its mass shell. Therefore, perturbation theory (with an on-shell quar initial state) applies (up to power corrections...). photon q s quar jet m X = ( P B q) x E γ m b ; 1 Γ tot dγ Perturbative endpoint: x = 1 Physical endpoint: x = M B /m b > 1 dx LO = δ(1 x) b quar p In the endpoint region the distribution is smeared by radiation and by the primordial motion of the quar = conventional approach: leading power NP shape function. P B Neubert; Bigi, Shifman, Uraltsev & Vainshtein (93) Distinguish: Additional energy available in the meson Λ = M B m b Dynamical structure of the meson Einan Gardi (University of Cambridge) KEK, October 5 9

10 Large x factorization in inclusive B decays Hard Jet Hard The spectrum can be computed in PT: infrared and collinear safe Dominated by Sudaov logs, ln(1 x) Quar distribution scales: P Hard: m Jet: m X = (P b q) m (1 x) = m /N Soft: m(1 x) = m/n Spectral moments: Γ PT N B meson (in PT: on shell b quar) 1 dxx N 1 1 dγ PT Γ PT tot dx = H(m)J(m /N; µ)s PT (m/n; µ) + O(1/N) H(m) Sud(N, m) + O(1/N) Korchemsy & Sterman (94) P Einan Gardi (University of Cambridge) KEK, October 5 1

11 n Coefficients in the Sudaov exponent { Sud(N, m) = exp n+1 n=1 =1 ( ) n C n, ln αs MS (m } ) N π The coefficients C n, are nown exactly to NNLL accuracy [Gardi (5)] For N f = 4 C n, are: ? ?? ??? ???? ????? ?????? ??????? At a given order in α s the coefficients of subleading logs (lower ) get large... Is the fixed logarithmic accuracy approximation at LL / NLL / NNLL good? Einan Gardi (University of Cambridge) KEK, October 5 11

12 Conventional Sudaov resummation with NNLL accuracy { ( ) n 1 αs MS (m } ) Sud(N, m) = exp g n (λ) ; λ αms s (m ) β lnn π π n= g (λ) = C F [(1 λ) ln(1 λ) 1 ] β (1 λ) ln(1 λ) Sud(N, m) Corresponding spectra 1. 1 LL NLL NNLL N Einan Gardi (University of Cambridge) KEK, October 5 1

13 n Coefficients in the Sudaov exponent in the large β limit { Sud(N, m) = exp n+1 n=1 =1 ( ) n C n, ln αs MS (m } ) N π The part in C n, that is proportional to (β ) n 1 is nown to all orders: C n, increase for lower powers of lnn, building up n+1 =1 C n, ln N n!f n (N) Truncation at fixed logarithmic accuracy is not a good approximation. Renormalon divergence sets in already at low orders requires a prescription! Einan Gardi (University of Cambridge) KEK, October 5 13

14 m r expansion in α s (Q ) x 1 (large N) 1 3 p Dressed Gluon Exponentiation 4 large order n Dressed Gluon Renormalons 1/Q Single n!c F β n 1 αns multiple emission Sudaov Double Logs C n F αn sl n α s L 1 (W Λ ) Dressed Gluon Exponentiation (DGE) (Λ /W is not negligible) Q Einan Gardi (University of Cambridge) KEK, October 5 14

15 Dressed Gluon Exponentiation the jet function Borel representation of the Sudaov exponent: ln J N (Q; µ F ) = 1 dx xn x µ F (1 x)q dµ µ A α s (µ ) + B α s ((1 x)q ) = C F β du u Q u Λ B J (u)γ( u) (N u 1) + µ F Q u B A (u) N ln, we defined B J (u) B A (u) ub B (u) and used the Borel representation of the anomalous dimensions, A α s (µ ) = C F Λ du β u B µ A (u); B α s (µ ) = C F Λ u du B β µ B (u), 1 dxx N 1 (1 x) 1 u = Γ( u)γ(n) Γ(N u) Γ( u) N u (1 + O(1/N)). Inthe large β limit B J (u) = e 5 3 u sin πu πu u u/ 1 + O(u/β ). Infrared sensitivity appears as renormalon ambiguity in the Sudaov exponent parametrically enhanced power corrections O(NΛ /Q ) in the exponent Einan Gardi (University of Cambridge) KEK, October 5 15

16 Dressed Gluon Exponentiation the soft function Borel representation of the soft Sudaov exponent: ln S N (Q; µ F ) = 1 dx xn x µ F dµ (1 x) Q µ A α s (µ ) D α s ((1 x) Q ) = C F β du u u B Q S (u)γ( u) Λ N u 1 + µ F Q u B A (u) N ln, where we defined B S (u) B A (u) ub D (u). What does one gain? Resummation of running coupling effects beyond the available logarithmic accuracy Upon choosing a prescription (e.g. PV) for the Borel integral, the divergent sum is defined. Cancellation of certain renormalon ambiguities can then tae place. Landau singularities are absent. The pattern of power corrections (observable dependent) can be studied: singularities in Γ( u) = power corrections (NΛ/Q) in the exponent, except for B S (u) =. However, QCD perturbation theory gives the power expansion: B S (u) = 1 + s 1 u + For DGE one needs to now B S (u) also away from the origin involves assumptions! Einan Gardi (University of Cambridge) KEK, October 5 16

17 , integer Soft anomalous dimensions in the large β limit B S (u) = e 5 3 u sin πu πu b S(u) 1 + O(u/β ) Observable b S (u) B S (u) = power corrections Drell-Yan () Γ (1 u) Γ(1 u) u = 1, 3,... ΛN Q, even Heavy Jet Mass (1) / Thrust () 1 Q ΛN c parameter () Γ (1 + u) Γ(1 + u) ΛN, integer Q Heavy Quar Fragmentation (1) Heavy Quar Distribution (1) (Q = m ) (1 u) πu sin πu u = 1 ΛN, m Einan Gardi (University of Cambridge) KEK, October 5 17

18 F PT (N; µ) large N The quar distribution function Ψ(y)γ b(p + b Φ ) y (, y) Ψ() b(p =H(m b) b µ, µ) S ip + b y N Nµ m b S Nµ m b = exp CF β du u u B µ S (u)γ( u) Λ Nµ m b u 1 + B A (u) ln Nµ m b Wilson line y with B S (u) = e 5 3 u (1 u) 1 + O(u/β ) = 1 + s 1 u + s u /! + b quar field: zp A + = gauge Renormalon in the exponent and their interpretation: p On shell b quar Leading renormalon u = 1, O(ΛN/m b), is related to the mass of b(p b ) : e i δm y = e δm N/m b Higher renormalons u 3, (ΛN/m b) with 3, correspond to the difference between the momentum distribution in the on-shell quar and the (unambiguous) distribution in the meson: F(N; µ) = Ψ(y)γ B(P + B ) Φ y (, y)ψ() µ B(P B) ip + B y N + O(1/N) Einan Gardi (University of Cambridge) KEK, October 5 18

19 Cancellation of the leading renormalon ambiguity Owing to inematic power corrections, the resummed E γ spectrum is not influenced by the u = 1 O(NΛ/m b) ambiguity of the perturbative Sudaov exponent: 1 dγ = Γ tot de γ m b M B c+i c i c+i c i dn πi dn πi ( Eγ m b ( Eγ M B ) N H(m b ) J(m b/n; µ)s PT (m b /N; µ) }{{} Sud(m b,n) ambiguous The cancellation is exact in all the moments, but it requires ) N H(m)J(m b/n; µ)s PT (m b /N; µ) e (N 1) Λ/m b } {{ } u= 1 prescription independent renormalon resummation in the Sudaov exponent renormalon resummation in Λ = M B m b using the same prescription. Einan Gardi (University of Cambridge) KEK, October 5 19

20 Sudaov resummation beyond logarithmic accuracy Sud(m, N) PV = exp 1 u What do we now about B S (u)? { CF PV dut(u) β ( Λ m ) u [ B S (u)γ( u) ( N u 1 ) B J (u)γ( u)(n u 1) u NNLO in the full theory: B S (u) = 1 + s 1 1! + s u! + Renormalon cancellation in Sud(m,N) e (N 1) Λ/M implies: B S (u = 1/) is equal in magnitude and opposite in sign to the residue of the u = 1/ renormalon in m/m MS, which can be determined from the nown NNLO expansion in MS within a few percent. All orders in the large β limit: B S (u) = e 5 3 u (1 u) + O(1/β ). The vanishing of B S (u) at u = 1 is assumed to hold in general. ]}. Einan Gardi (University of Cambridge) KEK, October 5

21 B X s γ spectrum: from moment space to E γ CF Sud(m, N) PV = PV exp du T(u) β Λ m u 1 u B S (u)γ( u) N u 1 B J (u)γ( u) (N u 1). dγ(e γ ) de γ = m PV c+i c i dn πi H(m) Sud(m, N) PV Modified support properties: Sud(N, m) PV with various approx. for B S (u) N Eγ m PV Corresponding spectra Einan Gardi (University of Cambridge) KEK, October 5 1

22 B S (u) away from the origin Ansatz for B S (u) that is consistent with the nown O(u ) result in QCD (and the large β limit): B S (u) = e 5 3 u (1 u) exp Here t 1, are fixed requiring: c u + 1 c 3 c + C A 5 β 18 π ζ 3 u W(u) W(u) e t 1 u+1 t u 1 t 1 u + 1 (t 1 t )u = 1 + O(u 3 ). B S (u = 1/) =.914 ± 3%(computed); B S (u = 3/) =.3366 C, ] -3 del [GeV - dp + 1/Γ dγ/dp NNLL-DGE: C=.1 C=1. C=1. - P =3.5 GeV, El=1.5 GeV P [GeV] Einan Gardi (University of Cambridge) KEK, October 5

23 comparison to data: B Xs γ branching fraction Theoretical uncertainty on the total BF 1% Experimental cuts on E γ do not significantly increase the overall uncertainty. The measured BF is consistent with the Standard Model. 3 BR[B X s γ, E γ >E ] NNLL-DGE: Λ=.355 GeV Λ=.45 GeV Λ=.455 GeV BaBar data M B / Possible determination of m b! E [GeV] Einan Gardi (University of Cambridge) KEK, October 5 3

24 comparison to data: cut moments in B X s γ E γ Eγ>E n E γ E γ Eγ>E Eγ>E 1 Γ(E γ > E ) 1 Γ(E γ > E ) E de γ dγ(e γ ) de γ E de γ dγ(e γ ) de γ E γ n E γ E Eγ>E γ. [GeV] <E γ > Eγ >E Belle data BaBar data NNLL-DGE (residue fixed) Λ=.355 GeV Λ=.45 GeV Λ=.455 GeV ] [GeV > Eγ >E <(<E γ >-E γ) Belle data BaBar data NNLL-DGE (residue fixed) α s =.198 α s =.16 α s = [GeV] E [GeV] E The comparison suggests that power corrections are indeed small. In future: possible measurement of power corrections. Einan Gardi (University of Cambridge) KEK, October 5 4

25 - Integrated B X u l ν spectrum Integrating the spectrum with given experimental cuts: Hadronic Mass Cut: P + P < (1.7 GeV), E l > 1 GeV Small Lightcone Component Cut: P + <.66 GeV, E l > 1 GeV The effect of cuts on the P spectrum Sensitivity of the Event Fraction to C ] -1 [GeV 1/Γ dγ/dp NNLL-DGE: E =GeV, M x =M B E =1GeV, M x =M B E =1GeV, M =1.7GeV x E =1GeV, M =1.7GeV, fully diff. x + E max =.66GeV =1GeV, P + E =1GeV, P =.66GeV, fully diff. max Event Fraction NNLL-DGE: C=1 C=.1 C=1 E =1GeV - µ=p P [GeV] [GeV] M X Einan Gardi (University of Cambridge) KEK, October 5 5

26 Extraction of V ub from Belle data B( B X u l ν restricted phase space) = τ B Γ tot B X u l ν R cut. From Belle data B(P + P < (1.7 GeV), E l > 1 GeV) = (±13.4%) B(P + <.66 GeV, E l > 1 GeV) = (±17.%) and the computed event fraction R cut (P + P < (1.7 GeV), E l > 1 GeV) =.615 (±9.6%) R cut (P + <.66 GeV, E l > 1 GeV) =.535 (±15.%), we obtain V ub = 4.35 ±.8 [exp] ±.14 [th total (m MS )] b ±. [th cuts] 1 3 V ub = 4.39 ±.36 [exp] ±.14 [th total (m MS )] b ±.38 [th cuts] 1 3 Einan Gardi (University of Cambridge) KEK, October 5 6

27 Conclusions Resummed perturbation theory can be directly used as an approximation to inclusive B meson decay spectra, without a leading power non-perturbative function. The leading renormalon cancels out with inematic power corrections involving the pole mass. Requires renormalon resummation with the same prescription in both the Sudaov exponent and the pole mass. DGE yields definite predictions for decay spectra in the on-shell approximation. Beyond the logarithmic accuracy at hand (NNLL), the Borel sum of the exponent is constrained by information on renormalon residues. For the quar distribution in an on-shell heavy quar B S (u = 1/) was computed(!) and B S (u = 1) vanishes(?) Contrary to Sudaov resummation with fixed logarithmic accuracy, the DGE prediction is free of Landau singularities and stable. The DGE spectrum smoothly extends beyond the perturbative endpoint Its support is close to the physical one, provided that B S (u) is not too large at intermediate u. Application to B X s γ: Predictions for moments in the experimentally accessible range E γ > E agree well with data. Potential measurement of m b. Application to charmless semileptonic decay : The event fraction for an invariant mass cut P + P < (1.7 GeV) has ±1% accuracy. Consistent values for V ub are obtained from two different cuts. The program can be found at: andersen/bdk/bu Einan Gardi (University of Cambridge) KEK, October 5 7