Inclusive B decay Spectra by Dressed Gluon Exponentiation Plan of the talk Einan Gardi (Cambridge) Inclusive Decay Spectra Why do we need to compute decay spectra? Kinematics, the endpoint region and the conventional Shape Function approach Factorization and Sudakov resummation with NNLL accuracy Divergence of perturbation theory Dressed Gluon Exponentiation The Sudakov exponent as a Borel sum Higher renormalons ambiguities and renormalon inspired Shape Function Shape Functions for hadronization effects in event-shapes and fragmentation Application to inclusive B decay Using the quark distribution function in an on-shell heavy quark as an approximation to the one in the meson Cancellation of the leading renormalon ambiguity Modified support properties Numerical results, comparison to data Einan Gardi (Cambridge) Workshop on first principles non-perturbative QCD of hadron jets, Paris, January 26 1
Inclusive B decay Spectra by Dressed Gluon Exponentiation References Inclusive spectra in charmless semileptonic B decays by DGE, J.R. Andersen, E. Gardi, to appear in JHEP [hep-ph/5936]. Taming the B X s γ spectrum by Dressed Gluon Exponentiation, J.R. Andersen, E. Gardi, JHEP 56 3 (25) [hep-ph/52159]. On the quark distribution in an on-shell heavy quark and its all-order relations with the perturbative fragmentation function, E. Gardi, JHEP 52, 53 (25) [hep-ph/51257]. Radiative and semi-leptonic B-meson decay spectra: Sudakov resummation beyond logarithmic accuracy and the pole mass, E. Gardi, JHEP 44, 49 (24) [hep-ph/43249]. Einan Gardi (Cambridge) Workshop on first principles non-perturbative QCD of hadron jets, Paris, January 26 2
Weights / 1 MeV 4 radiative decay: B Xs γ Data Spectator Model Inclusive B decay Spectra Events / 12 MeV/c 2 15 125 1 75 5 25 semi-leptonic decay: B Xu l ν l signal region (a) -25 1.5 2.5 3.5 4.5 E (GeV) CLEO 1 2 3 4 5 M x (GeV/c 2 ) The spectra peak close to the endpoint (E γ M B /2; small M X ) where the hadronic system is jet-like. Measurements are restricted to this region. BELLE Example: extracting V ub from the semi-leptonic decay Precise measurements are restricted to the small M X region because of charm background. Determination of V ub relies on calculation of the spectrum. Einan Gardi (Cambridge) Workshop on first principles non-perturbative QCD of hadron jets, Paris, January 26 3
Kinematics in B X s γ In the B meson the b quark is close to its mass shell. = The inclusive spectrum can be computed in perturbation theory, where the initial state is an on-shell quark (up to power corrections...). photon q s quark jet x 2E γ m b ; At LO m 2 X =, 1 Γ tot dγ m 2 X = ( P B q) 2 dx = δ(1 x) LO Perturbative endpoint: x = 1 Physical endpoint: x = M B /m b > 1 b quark p k In the endpoint region the distribution is smeared by radiation and by the primordial motion of the quark = conventional approach: leading power NP shape function. P B Neubert; Bigi, Shifman, Uraltsev & Vainshtein (93) We will show that: It is possible and useful to compute the on-shell heavy quark decay spectrum. Distinguish: Additional energy available in the meson Λ = M B m b Dynamical structure of the meson Einan Gardi (Cambridge) Workshop on first principles non-perturbative QCD of hadron jets, Paris, January 26 4
Large x factorization in inclusive B decays Hard Jet Hard The spectrum can be computed in PT: infrared and collinear safe Dominated by Sudakov logs, ln(1 x) = ln(p + j /p j ) Quark distribution scales: Hard: p j = O(m b) P B meson (in PT: on shell b quark) Jet: m 2 X = (p b q) 2 = p j p+ j = m2 b (1 x) = m2 /N Soft = Quark Distribution: p + j = m b(1 x) = m/n Korchemsky & Sterman (94) P Spectral moments: Γ PT N 1 dxx N 1 1 dγ PT Γ PT tot dx = H(m)J(m 2 /N; µ)s PT (m/n; µ) + O(1/N) H(m) Sud(N, m) + O(1/N) Einan Gardi (Cambridge) Workshop on first principles non-perturbative QCD of hadron jets, Paris, January 26 5
n Coefficients in the Sudakov exponent { Sud(N, m) = exp n+1 n=1 k=1 ( ) n C n,k ln k αs MS (m 2 } ) N π The coefficients C n,k are known exactly to NNLL accuracy [Gardi (25)] For N f = 4 C n,k are: k 1.564.667 3.837.78 1.389? 2.579 6.339 3.376?? 116.464 33.24 9.42??? 597.221 138.6 25.955???? 2859.284 548.17 78.492????? 13141.289 2129.58 247.233?????? 58941.217 8238.359??????? 26391.559 At a given order in α s the coefficients of subleading logs (lower k) get large... Is the fixed logarithmic accuracy approximation at LL / NLL / NNLL good? Einan Gardi (Cambridge) Workshop on first principles non-perturbative QCD of hadron jets, Paris, January 26 6
Conventional Sudakov resummation with NNLL accuracy { ( ) n 1 αs MS (m 2 } ) Sud(N, m) = exp g n (λ) ; λ αms s (m 2 ) β lnn π π n= g (λ) = C F 2 [(1 λ) ln(1 λ) 12 ] β (1 2λ) ln(1 2λ) Sud(N, m) Corresponding spectra 1.2 1 LL NLL NNLL.8.6.4.2 5 1 15 2 25 3 35 N Einan Gardi (Cambridge) Workshop on first principles non-perturbative QCD of hadron jets, Paris, January 26 7
n Coefficients in the Sudakov exponent in the large β limit { Sud(N, m) = exp n+1 n=1 k=1 ( ) n C n,k ln k αs MS (m 2 } ) N π The part in C n,k that is proportional to (β ) n 1 is known to all orders: k 1.56.67 1.24.9 1.39 61.17 28.32 8.28 3.38 196.6 515.2 166.25 34.89 9.4 2399.23 178.43 3231.4 793.25 131.33 25.95 444615.21 221481.3 73268.94 17791.58 3514.66 482.12 78.49 11342675.74 5665794.49 1883129.5 46818.33 91361.3 158.79 1768.5 33432127.3 1669657.5 556962.17 1386774.58 276946.21 449959.1 63745.75 C n,k increase for lower powers of lnn, building up n+1 k=1 C n,k ln k 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 (Cambridge) Workshop on first principles non-perturbative QCD of hadron jets, Paris, January 26 8
m r expansion in α s (Q 2 ) x 1 (large N) k 1 k 2 k 3 p k Different directions in resummation 4 large order n Dressed Gluon Renormalons 1/Q Single 2 n!c F β n 1 αns multiple emission Sudakov Double Logs C n F αn sl 2n α s L 1 (W 2 Λ 2 ) Q Dressed Gluon Exponentiation (DGE) evolution kernel treated as an asymptotic series Q 2 W 2 but Λ 2 /W 2 is not negligible k Einan Gardi (Cambridge) Workshop on first principles non-perturbative QCD of hadron jets, Paris, January 26 9
Dressed Gluon Exponentiation the jet function Evolution equation for the jet (also for the DIS structure function F 2 ): d ln F N 2 (Q2 ) d ln Q 2 = d ln J N(Q; µ F ) d ln Q 2 = 1 dx xn 1 1 1 x J α s (1 x)q + O(1/N) 2 Usually the Sudakov anomalous dimension J(α s ) is computed order by order and the equation is solved to a fixed logarithmic accuracy. Infrared sensitivity from the x 1 limit appears through the coupling and is not regularized. Imagine that the anomalous dimension J(α s ) is known to all orders, Borel representation: J α s (µ 2 ) = C F Λ 2 u du B β µ 2 J (u). so the Borel representation of the evolution kernel: d ln J N (Q; µ F ) d ln Q 2 = C F β Λ 2 u du B Q 2 J (u)γ( u) Γ(N) Γ(N u) + 1 u + O(1/N) Infrared sensitivity appears as renormalon ambiguity in the evolution kernel parametrically enhanced power corrections O(NΛ 2 /Q 2 ) in the Sudakov exponent Einan Gardi (Cambridge) Workshop on first principles non-perturbative QCD of hadron jets, Paris, January 26 1
Dressed Gluon Exponentiation the soft function Evolution equation for the soft function and the corresponding Borel representation: d ln S N (Q; µ F ) d ln Q 2 = 1 = C F β dx xn 1 1 1 x S α s (1 x) 2 Q + O(1/N) 2 Λ 2 u du B Q 2 S (u)γ( 2u) Γ(N) Γ(N 2u) + 1 2u + O(1/N) What does one gain? (assuming B S (u) is known) All order resummation of running coupling effects renormalization scale invariance Upon choosing a prescription (e.g. PV) for the Borel integral, the divergent sum is defined. Cancellation of certain renormalon ambiguities can then take place. Landau singularities are absent. The pattern of power corrections observable dependent shape function can be studied: singularities in Γ( 2u) = power corrections (NΛ/Q) k 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 know B S (u) also away from the origin. Einan Gardi (Cambridge) Workshop on first principles non-perturbative QCD of hadron jets, Paris, January 26 11
DGE: Shape Functions and Sudakov anomalous dimensions d ln S N (Q; µ F ) d ln Q 2 = C F β Λ 2 u du B Q 2 S (u)γ( 2u) Γ(N) Γ(N 2u) + 1 2u Conjecture: B S (u) has no renormalon poles of its own S N (Q; µ F ) S N (Q; µ F ) exp k=1 ǫ k k! Q Λ k Π k j=1 (N j) µ ßÞ Ð NP Shape Function on the scale Q/N B S (u = k/2) = no ambiguous power term (NΛ/Q) k IR finiteness of the observable at the k power level Analytic results for B S (u) large β show that it indeed vanishes at some u = k/2 (k integers) Assumption: B S (u = k/2) large β = B S (u = k/2)= Write an ansatz for B S (u), e.g. B S (u) = B S (u) large β exp w 1 u w 2 u 2 + w i are O(C A /β ); determined from QCD PT expansion B S (u) = 1 + s 1 u + w 1 = C A π 2 β 12 1 3 is universal (cusp); w 2,3,... are observable dependent Einan Gardi (Cambridge) Workshop on first principles non-perturbative QCD of hadron jets, Paris, January 26 12
Soft anomalous dimensions in the large β limit: examples 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 e.g. in p p Z/γ l l 2 Γ2 (1 u) Γ(1 2u) u = 1 2, 3 2, 5 2,... ΛN Q k, k even Event Shapes in e + e jets Heavy Jet Mass Thrust c parameter 1 2 2 Γ2 (1 + u) Γ(1 + 2u) u = 1, 2, 3,... ΛN k, k odd Q Heavy Quark Fragmentation Heavy Quark Distribution (Q 2 = m 2 ) (1 u) πu sin πu u = 1 ΛN k,k 2 m Einan Gardi (Cambridge) Workshop on first principles non-perturbative QCD of hadron jets, Paris, January 26 13
Heavy Quark Fragmentation: Principal Value Borel sum Energy distribution in inclusive e + e B(E) + X DGE (NLL+large β ) matched to NLO; Exponent regularized by Principal Value Borel sum Leading renormalon ambiguity exp { (N 1)Λ/m b } Shift Moment space Energy Fraction x E = 2E/Q Shift is a good approximation; SF fit has small effect on the width Cacciari & Gardi (23) Einan Gardi (Cambridge) Workshop on first principles non-perturbative QCD of hadron jets, Paris, January 26 14
Shift and shape function fits in event shape variables at LEP Starting with the DGE result (NLL+large β ) matched to NLO Shift/SF parameters are determined based on the measured Thrust distribution These parameters are used to compute the NP Heavy Jet Mass distribution, assuming no correlation between hemispheres. Thrust (t = 1 T) Heavy Jet Mass ρ H Good agreement over a wide range of the variables t, ρ H and energies Shift is a good approximation to the right of the peak; SF fit has small effect on the width Gardi & Rathsman (22) Einan Gardi (Cambridge) Workshop on first principles non-perturbative QCD of hadron jets, Paris, January 26 15
The quark distribution function in an on-shell quark F PT (N; µ) large N Ψ(y)γ b(p + Nµ b Φ ) y (, y) Ψ() b(p = H(m b) b, µ) S PT µ ip + b y N m b Nµ S PT m b =exp CF β du u u B µ 2 S (u)γ( 2u) Λ 2 Nµ m b 2u Nµ 1 + B A (u) ln m b Computing the exponentiation kernel in S PT with a single dressed gluon: B S (u) = e 5 3 u (1 u) 1 + O(u/β ) In general it is known only as an expansion B S (u) = 1 + s 1 u + s 2 u 2 /2! + b quark field: zp Wilson line k y A + = gauge p On shell b quark S PT has O((ΛN/m b ) k ) ambiguities (k 2) in the exponent! The leading one exp (ΛN/m b ) corresponds to the overall shift of the E γ distribution by O(Λ)... Einan Gardi (Cambridge) Workshop on first principles non-perturbative QCD of hadron jets, Paris, January 26 16
For an on-shell quark we found: The quark distribution function in the meson F PT (N; µ) large N Ψ(y)γ b(p + b ) Φ y (, y)ψ() Nµ = H(m b, µ) S PT m b µ b(p b) ip + b y N where S PT Nµ Interpretation: m b has O((ΛN/m b ) k ) renormalon ambiguities in the Sudakov exponent! Leading renormalon u = 1 2, O(ΛN/m b), is related to the mass of b(p b ) : even in the absence of interaction a mass difference generates e i δm y = e δm N/m b Higher renormalons u 3 2, (ΛN/m b) k with k 3, correspond to the difference between the momentum distribution in the on-shell quark and the (unambiguous) distribution in the meson: F(N; µ) large N Ψ(y)γ B(P + B ) Φ y (, y)ψ() Nµ = H(m b, µ) S PT m b µ B(P B) ip + B y N e N Λ/m b NΛ S NP ; Λ = MB m b m b Einan Gardi (Cambridge) Workshop on first principles non-perturbative QCD of hadron jets, Paris, January 26 17
Cancellation of the leading renormalon ambiguity Owing to kinematic power corrections, the resummed E γ spectrum is not influenced by the u = 1 2 O(NΛ/m b) ambiguity of the perturbative Sudakov exponent: 1 dγ = 2 Γ tot de γ m b 2 M B c+i c i c+i c i dn 2πi dn 2πi ( 2Eγ m b ( 2Eγ M B ) N H(m b ) J(m 2 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 2 b/n; µ)s PT (m b /N; µ) e (N 1) Λ/m b } {{ } u= 1 2 prescription independent renormalon resummation in the Sudakov exponent renormalon resummation in Λ = M B m b using the same prescription. Einan Gardi (Cambridge) Workshop on first principles non-perturbative QCD of hadron jets, Paris, January 26 18
B X s γ spectrum: from moment space to E γ CF Sud(m, N) PV = PV exp du T(u) β Λ 2 m 2 u 1 u B S (u)γ( 2u) N 2u 1 B J (u)γ( u) (N u 1). dγ(e γ ) de γ = m PV 2 c+i c i dn 2πi H(m) Sud(m, N) PV Modified support properties: Sud(N, m) PV with various approx. for B S (u) N 2Eγ m PV Corresponding spectra Einan Gardi (Cambridge) Workshop on first principles non-perturbative QCD of hadron jets, Paris, January 26 19
DGE in inclusive B decays: summary The leading renormalon cancels out with kinematic power corrections involving the pole mass. Requires renormalon resummation with the same prescription in both the Sudakov 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 quark distribution in an on-shell heavy quark B S (u = 1/2) was computed(!) and B S (u = 1) vanishes. CF Sud(m, N) PV = PV exp du T(u) β Λ 2 m 2 u 1 u B S (u)γ( 2u) N 2u 1 B J (u)γ( u) (N u 1). Contrary to Sudakov 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. The renormalon analysis indicates that infrared sensitivity to the quark distribution in the meson appears only through the third power of NΛ/m b, i.e. it is very small up to N m b /Λ. Einan Gardi (Cambridge) Workshop on first principles non-perturbative QCD of hadron jets, Paris, January 26 2
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 ] 1.45.4.35.3.25.2.15.1.5 NNLL-DGE: Λ=.355 GeV Λ=.45 GeV Λ=.455 GeV BaBar data M B /2 Possible determination of m b! 1.6 1.8 2 2.2 2.4 2.6 2.8 E [GeV] Einan Gardi (Cambridge) Workshop on first principles non-perturbative QCD of hadron jets, Paris, January 26 21
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 2.45 2.4 2.35 Belle data BaBar data NNLL-DGE (residue fixed) Λ=.355 GeV Λ=.45 GeV Λ=.455 GeV ] 2 [GeV > Eγ >E 2 <(<E γ >-E γ).6.5.4.3 Belle data BaBar data NNLL-DGE (residue fixed) α s =.198 α s =.216 α s =.222 2.3.2 2.25.1 2.2 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 [GeV] E 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 [GeV] E The comparison suggests that power corrections are indeed small. In future: possible measurement of power corrections. Einan Gardi (Cambridge) Workshop on first principles non-perturbative QCD of hadron jets, Paris, January 26 22
- Integrated B X u l ν spectrum Integrating the triple differential B X u l ν spectrum with given experimental cuts: Hadronic Mass Cut: P + P < (1.7 GeV) 2, 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 B S (u = 3 2 ) ] -1 [GeV 1/Γ dγ/dp.35.3.25.2.15 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 1.9.8.7.6.5.4 NNLL-DGE: C=1 C=.1 C=1 E =1GeV - µ=p.1.3.5.2.1 1 2 3 4 5 - P [GeV] 1 1.5 2 2.5 3 3.5 [GeV] M X Einan Gardi (Cambridge) Workshop on first principles non-perturbative QCD of hadron jets, Paris, January 26 23
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) 2, E l > 1 GeV) = 1.24 1 3 (±13.4%) B(P + <.66 GeV, E l > 1 GeV) = 1.1 1 3 (±17.2%) and the computed event fraction R cut (P + P < (1.7 GeV) 2, E l > 1 GeV) =.615 (±9.6%) R cut (P + <.66 GeV, E l > 1 GeV) =.535 (±15.2%), we obtain V ub = 4.35 ±.28 [exp] ±.14 [th total (m MS )] b ±.22 [th cuts] 1 3 V ub = 4.39 ±.36 [exp] ±.14 [th total (m MS )] b ±.38 [th cuts] 1 3 Einan Gardi (Cambridge) Workshop on first principles non-perturbative QCD of hadron jets, Paris, January 26 24
Conclusions Resummed perturbation theory can be directly used as an approximation to inclusive B meson decay spectra, without a leading power non-perturbative function! 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) 2 has ±1% accuracy. Consistent values for V ub are obtained from two different cuts. The program can be found at: www.hep.phy.cam.ac.uk/ andersen/bdk/b2u Einan Gardi (Cambridge) Workshop on first principles non-perturbative QCD of hadron jets, Paris, January 26 25