Higher order corrections to inclusive semileptonic B decays

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Nuclear Physics B Proceedings Supplement 00 (2014) 1–4

Nuclear Physics B Proceedings Supplement

Higher order corrections to inclusive semileptonic B decays

Andrea Alberti

Universit`a di Torino, Dip. di Fisica&INFN Torino, I−10125, Italy

Abstract

We describe the computation of theO(α_S) corrections to the Wilson coefficients of the kinetic and chromomagnetic operators in inclusive semileptonicBdecays. We can thus evaluate the complete correction at orderα_SΛ²_QCD/m²_b to the semileptonic width and the first two leptonic moments.

Keywords:

OPE, HQET, B meson, semileptonic decay

1. Introduction

The study of semileptonic B decays (first initiated in [1, 2]) has its foundation on an Operator Product Expan- sion together with an heavy quark expansion. The OPE expresses the differential decay rate (and consequently the total width, leptonic and hadronic moments) as a double series inα_S andΛQCD/m_b. The leading term reproduces the free quark decay, while subsequent corrections take into account interactions inside the meson and further reduce the theoretical uncertainty.

Currently several corrections are known: up toO(α²_S) in the free decay [3, 4, 5], while the nonpertur- bative terms already calculated include O(Λ²_QCD/m²_b) [2], O(Λ³_QCD/m³_b) [6] and a first investigation of O((ΛQCD/m_b)^4,5) [7]. The terms in the heavy mass expansion are expressed as B-meson matrix elements of local operators, growing in dimension: they are custom- arily parametrized asµ²_π andµ²_Gat order O(Λ²_QCD/m²_b) andρ_D,ρ_LS atO(Λ³_QCD/m³_b).

Regarding the perturbative corrections to these para- menters, there has been a first investigation, aboutµ²_π alone, in reference [8] and then the complete calculation for bothµ²_πandµ²_Ghas been carried out in the two papers relevant to this proceeding, [9] and [10].

Quite recently the coefficient of µ²_Gfor the total width has been calculated again in the limitmc→0 [11], yield-

ing results compatible with our own.

The method we used to perform the calculation can be traced back to [12], where it was employed for the sim- pler process B → X_sγ, then it has been further devel- oped in [9] for the coefficient ofµ²_πand finally extended to the full calculation ofµ_G² in [10].

2. The inclusive semileptonic decay

The triple differential decay distribution for an inclusive semileptonic decay is

dΓ dq²dEνdEe

=1 4

X

X_c

X

spin

|<Xce¯ν|Hw|B¯>|

2m_B

2

δ⁴(pt) (1) wherept=pB−pe−pν−pXand labelse,νandXrepresent the final lepton, neutrino and hadronic stateXC;qis the transferred momentum to the leptonic couple and Hw

the weak Hamiltonian H_w= 4GF

√ 2

V_cb J^µ_L eγ¯ _µP_Lν_e; J^µ_L=cγ¯ ^µP_Lb (2) The first step to simplify eq. (1) is to express it as the product of an hadronic tensor W^αβand a leptonic tensor L^αβ, which is possible since leptons do not interact strongly (and we work at leading order in the ew interactions):

dΓ dq²dEνdEe

=2G²_F|Vcb|²WαβL^αβ (3)

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/Nuclear Physics B Proceedings Supplement 00 (2014) 1–4 2

the leptonic tensor is fairly simple L^αβ=2

p^α_ep^β_ν+p^α_νp^β_e−pe·pνg^αβ−i^ρβσαpeρpν σ (4) while the hadronic tensor still contains unknown matrix elements

W^αβ= (2π)³ 2mB

X

Xc

δ⁴(pt)<B¯|J^†α_L |Xc><Xc|J^β_L|B¯> (5) withJ_L^α the usual left-handed weak current (described in eq. (2)). To further manipulateW_αβwe can relate it to the discontinuity of a time-ordered product across a cut. So by defining

T^αβ=−i Z

d⁴xe^−iq·x 1

2m_B<B¯|Th

J_L^†α(x)J_L^β(0)i

|B>¯ (6) we obtain the relation

Wαβ=−1

πImTαβ (7)

Up to this point we are still handling tensors with two indices, in order to simplify things a bit in this regard we can express the whole tensorW^αβin terms of structure functionsWi:

W^αβ=−g^αβW1+v^αv^βW2−i^αβµνvµqνW3 (8) wherevis the four-velocity of the ¯Bmeson: p_B=m_Bv.

There would be two additional functionsW4andW5fac- toring structures containingq, but since we are dealing with massless leptons, the leptonic tensor yields zero when contracted withq:

q^αLαβ=q^βLαβ=0 (9) and so onlyW1,W2andW3contribute in the end.

Eq. (7) thus becomes W_i=−1

πImT_i; i=1,2,3 (10)

3. The operator product expansion

Having relatedWi to a product of currents (eq. (6) and eq. (10)), we can now employ an operator product expansion and express the latter as the sum of local operators:

Th

J^†α_L (x)J^β_L(0)i

=X

i

C_i(x)O_i(0) (11) in order to do so we consider an initialbquark which is slightly off-shell (being inside a ¯Bmeson),p_b=m_bv+k withkof the order ofΛQCD, and expand inkup toO(k²).

If we then use QCD perturbation theory to calculate the

coefficients of theOioperators, we obtain a double series inα_S andΛQCD/m_b:

Tⁱ=X

n≥3

X

j≥0

ΛQCD

m_b

!n−3

αS

π j

c^{i j}_(n)<B|O¯ ⁱ⁽ⁿ⁾_j |B¯> (12) wherenis the dimension of the operator.

It has to be noted that we could have followed the same steps with the addition of an extrernal soft gluon and we would have expected a result similar to eq. (12), but possibly with different coefficients and operators. This consideration will play an important role in the determi- nation of the coefficient ofµ_G².

4. Heavy quark effective theory

Even after having found all operators in eq. (12) with the respective coefficients, we must still calculate their matrix elements between ¯Bmeson states. This could be done in QCD, like the rest of the calculation so far, but it is convenient to switch to HQET, an effective theory specifically tailored around the idea of heavy quarks being very close to their mass shell.

In practice what we do is substituting the old quark field in terms of the new one

b(x)=e^−im^b^v·x 1+ iD/ 2mb

!

bv(v) (13) and then take advantage of all the properties that hold true inside HQET in order to simplify the result and minimize the number of operators we have to deal with.

The equation of motion is of particular importance iD·v bv=−(iD)²bv (14) it implies that no operator appears at O(ΛQCD/mb) which cannot be written in terms of higher dimension operators. So in the end we have one operator at dimension three, describing the decay of a free quark, which is better to express in QCD:

O^µ_b =bγ¯ ^µb (15)

and then two operators at dimension five, the kinetic operator and the chromomagnetic operator:

O^µν₂ = 1 2

b¯v{iD^µ,iD^ν}bv (16) O^µν₃ = 1

2

b¯_vg_SG^µ_ασ^ανb_v (17) As becomes apparent from eq. (17), the chromomagnetic operator contains a gluon field, so its cofficient can be calculated only from the current product emitting an

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Figure 1: One-loop diagrams contributing to the current correla- tor. The background gluon can be attached wherever a cross is marked.

external gluon.

The operators are parametrized as follows:

1 MB

<B|O¯ ^µ_b|B¯ >=2v^µ (18) 1

MB

<B|O¯ ^µν₂ |B¯ >=− 2µ²_π

d−1(g^µν−v^µv^ν) (19) 1

MB

<B|O¯ ^µν₃ |B¯ >= 2µ²_G

d−1(g^µν−v^µv^ν) (20) having neglected higher order power corrections and in- troducing dimensiond=4−2.

When expressing the final amplitude what really mat- ters, rather than the operators themselves, are their matrix elements evaluated between ¯Bstates, so we can now rewrite eq. (12) as

Wi=W_i⁽⁰⁾+ µ²_π

2m²_bW_i^(π,0)+ µ_G² 2m²_bW_i^(G,0) +α_S

π





W_i⁽¹⁾+ µ²_π

2m²_bW_i^(π,1)+ µ²_G 2m²_bW_i^(G,1)





 (21)

5. One loop calculation

At tree level the use of HQET is sufficient to solve most of the problems, but carring out the calculation at one loop (contributing diagrams shown in figure 1) presents many more challenges: namely lenghtier ex- pressions, loop integrals, and divergencies to be can- celled through renormalization. The latter deserve some deeper discussion, in order to get rid of all the poles we need to be precise when matching HQET with QCD.

We are equating Tⁱ=

c^i,j,0_{α} +αS

4πc^i,j,1_{α} +O(α²_S)

<O^{α}_j > (22)

where<O>=<B¯|O|B¯ >and{α}is a set of indices; the lhs contains the poles coming from the diagrams in figure 1 and the counterterms from mass and wave function renormalization, while on the rhs we haveO(α) contri- butions from the one-loop matrix elements (zero in the case of O2 andO3) as well as the renormalization of each operator.

After having taken into account all the counterterms, both the lhs and the rhs become ultraviolet free, but still retain infrared divergencies: since QCD and HQET re- produce the same infrared behaviour, these will cancel out and so the one-loop Wilson coefficientc^i,j,1_{α} (what we wanted to calculate in the first place) will be finite.

The relevant counterterms are:

δZ^µναβ_kin =−CF

3−ξ

(g^µν−2v^µv^ν)v^αv^βαS

4π +... (23) δZ^µναβ_chromo=CA

(g^µα−v^µv^α)g^νβαS

4π +... (24) whereξis the Feynman gauge and bare quantities are defined as follows:

hc_XµνO^µν_Xibare

=Z^OS_b

v Z_X^µναβc_XµνO_Xαβ (25)

6. Results

The calculation we have briefly sketched has been carried out in ref. [9] and [10], respectively for the one- loop coefficient ofµ²_πandµ²_G. The full analytic result for the structure functionsW_ican be found in the Appendix of each paper.

Numerical result are not as lenghty and can be dis- cussed here. Considering on-shell quark masses of mb = 4.6GeV and mc = 1.15GeV we obtain a total width of

ΓB→Xceν= Γ0







1−1.78α_S π





1− µ²_π 2m²_b







−

1.94+2.42αS

π

µ²_G(m_b) m²_b





 (26)

withΓ0 = G²_Fm⁵_b(1−8ρ+8ρ³−ρ⁴−12ρ²lnρ)/192π³ andρ = m²_b/m²_c. The parameterµ²_G is renormalized at the scaleµ =m_b. Assumingα_S = 0.25, the one-loop correction increases theµ²_Gcoefficient by 7%.

Then there are the first and second central leptonic moments

<E_`>=1.41GeV







1−0.02α_S π





1+ µ²_π 2m²_b







−

1.19+4.20αS

π

µ²_G(m_b) m²_b





 (27)

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`₂ =0.183GeV²





1−0.16α_S

π +4.98µ²_π m²_b

−0.37αS

π µ²_π m²_b−

2.89+8.44αS

π

µ_G²(mb) m²_b





 (28)

where`2 =<E_`² >−<E` >²; here the relative NLO correction to theµ²_G coefficient are bigger than in the width, amounting to+28% and+23%.

All the above can also be calculated inside the kinetic scheme (see [13, 14]) with cutoffµ_kin=1GeV and they yield:

ΓB→X_ceν= Γ0





1−1.11αS

π − µ²_π 2m²_b +0.99αS

π µ²_π m²_b −

1.94+3.46αS

π

µ_G²(m_b) m²_b





 (29)

<E`>=1.41GeV





1−0.01αS

π + µ²_π 2m²_b

−0.44α_S π

µ²_π m²_b −

1.19+3.21α_S π

µ_G²(mb) m²_b





 (30)

`2 =0.183GeV²





1−0.24αS

π +4.98µ²_π m²_b

−3.89α_S π

µ²_π m²_b −

2.89+7.01α_S π

µ_G²(mb) m²_b





 (31) hereµ_G² coefficients always increase with the addition of the NLO corrections, respectively by+15%,+20% and +20%.

7. Conclusions

We have calculated theO(αS) corrections to the Wil- son coefficients of the kinetic and chromomagnetic operators in inclusive semileptonic B decays. The com- pleteO(α_SΛ²_QCD/m²_b) contribution to the width is just a few per mill, but the corrections to the first two leptonic moments are comparable to the experimental errors. In order to estimate their effect on|V_cb|, these corrections have to be included in the global fit to the moments, which will be the subject of a future publication.

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