1

Next-to-leading Order Calculations of the Radiative and Semileptonic

Rare B Decays in the Standard Model and Comparison with Data

A. Alia∗

aDeutsches Elektronen-Synchrotron DESY,Notkestraße 85, D-22603 Hamburg, Federal Republic of Germany

We review some selected rare B decays calculated in next-to-leading order accuracy in the Standard Model(SM). These include the radiative decays B → (Xs, K

∗, ρ)γ and the semileptonic decays B → (Xs, K, K∗)�+�−,for which new data from the BABAR and BELLE collaborations have been presented at this conference. SM isin agreement with the current measurements within the theoretical and experimental errors. The impact of rareB-decays on the CKM phenomenology is quantitatively discussed.

1. INTRODUCTION AND OVERVIEW

First measurements of rare B decays B → K∗γ[1] and B → Xsγ [2] were reported by CLEOin 1993 and 1995, respectively.. Since then, awealth of data on these decays has become avail-able through the subsequent work of the CLEOcollaboration [3,4], the ALEPH collaboration atLEP [5], and more recently from the experimentsat the B factories, BABAR [6] and BELLE [7],which now dominate this field. We quantify theimpact of B → Xsγ measurement on the CKMphenomenology.

Concerning the exclusive decays B → K∗γ, theNLO calculations of the decay widths were com-pleted last year [8–10]. Radiative decays providea cleaner test of the underlying theoretical frame-work of perturbative QCD factorization whichhas been invoked initially to calculate the two-body exclusive non-leptonic B decays, such asB → ππ [11]. Quantitative rapport of theory andexperiment in these decays is on hold at presentdue to the imprecise knowledge of the form factor,as discussed later in this talk.

A big leap forward has been reported in the ex-perimental study of the Cabibbo-suppressed de-cays B → ργ and B → ωγ at this conference

∗Present address: CERN (-TH-), CH-1211 Geneva 23.

[6,7]. These decays which enact b → dγ transi-tion at the quark level yield complementary in-formation on the CKM parameters and hence arepotentially very important. We work out the con-straints following from the current upper limitson the B → ργ branching ratios [6,7]. Apartfrom these, the Cabibbo-suppressed radiative de-cays B → ργ allow us to study isospin- and CP-asymmetries, which in the SM have a sensitivedependence on the angle α of the unitarity tri-angle [12,8,9]. We present the SM expectationsfor the decay rates and asymmetries, which arealso potentially of interest in searching for physicsbeyond-the-SM [12,13].

The semileptonic decays B → (Xs, K, K∗)�+�−

(�± = e±, µ±) are the new additions to the flavourchanging neutral current FCNC decays, whereB factories have made first experimental inroads[7,14]. Of particular interest is the inclusive de-cay B → Xs�

+�−, whose measurement has beenreported at this conference by the BELLE collab-oration [7]. These decays have been calculated inthe SM to the same degree of theoretical precisionas their radiative counterpart B → Xsγ, and takeinto account the explicit O(αS) corrections to thedilepton invariant mass distribution [15,16], theforward-backward asymmetry of the charged lep-ton [17,18] and the leading power corrections in

ELSEVIERProceedings of ICHEP 2002, pp. 624–630

S. Bentvelsen, P. de Jong, J. Koch and E. Laenen (Editors) 31st INTERNATIONALCONFERENCEONHIGH ENERGY PHYSICS AMSTERDAM

c© 2003 Elsevier Science B.V. All rights reserved.doi:10.1016/S0920-5632(02)02158-8

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1/mb [19,20] and 1/mc [21]. Precise measure-ments of the dilepton invariant mass spectrumand the forward-backward asymmetry would al-low to extract the (effective) Wilson coefficients,providing precision tests of the SM and enablingsearches for physics beyond-the-SM [22].

What concerns the exclusive decays B →(K, K∗)�+�−, important theoretical progress hasbeen made recently [23,10], using ideas basedon the large energy expansion [24]. In particu-lar, both the dilepton invariant mass spectrumand the forward-backward asymmetry in B →K∗�+�− have been calculated in the lower regionof the dilepton invariant mass (s < m2

J/ψ) to thesame accuracy as is the case for B → Xs�

+�−, al-beit using perturbative QCD factorization. Thisframework has also been used to carry out a he-licity analysis of the decays B → K∗�+�− andB → ρ�ν� [25]. Data on B → (Xs, K

∗, K)�+�−

are compared with the SM estimates [26,27] andthe two are found to be compatible with eachother.

2. Inclusive Decay B → Xsγ

The theoretical framework for analyzing thetransition B → Xsγ is provided by the effectiveHamiltonian

Heff = −4GF√2

V ∗tsVtb

8∑i=1

Ci(µ)Oi, (1)

where GF is the Fermi coupling constant andVij are the CKM matrix elements. The effectiveHamiltonian is obtained from the SM by integrat-ing out all the particles that are much heavierthan the b-quark. The Wilson coefficients Ci(µ)play the role of effective coupling constants forthe interaction terms Oi. The generic structureof the operators Oi can be seen, for example, inRef. [27].

Perturbative calculations are used to determinethe Wilson coefficients in a given renormalizationscheme, such as the MS scheme, at a renormal-ization scale, typically µb ∼ O(mb),

Ci(µb) = C(0)i (µb) +

αs(µb)4π

C(1)i (µb) + . . . . (2)

Here, C(n)i (µb) depend on αs only via the ratio

η ≡ αs(µ0)/αs(µb), where µ0 ∼ mW . In theleading order (LO) calculations, everything butC

(0)i (µb) is neglected in Eq. (2). In NLO, one

takes in addition C(1)i (µb) into account.

It has become customary to quote the theoret-ical branching ratios for B → Xsγ decay with acut on the photon energy Eγ . In the MS scheme,the results in the SM (in units of 10−4) are [28,29]:

B(B → Xsγ ; Eγ > 1.6 GeV) = (3.57 ± 0.30),B(B → Xsγ ; Eγ > 1

20mb) = (3.70 ± 0.31).(3)

The present world averages (also in units of 10−4)

B(B → Xsγ ; Eγ > 1.6 GeV) =(3.28 +0.41

−0.36

),

B(B → Xsγ ; Eγ > 120mb) =

(3.40 +0.42

−0.37

)(4)

result from the following four measurements:BABAR [6,30], CLEO [3], BELLE [31], andALEPH [5]. They are in good agreement withthe NLO results in the MS scheme. However,there is a residual theoretical uncertainty on theSM estimates related to the scheme-dependenceof the quark masses. This can be judged from thetheoretical branching ratios in the MS scheme forwhich one gets B(B → Xsγ) = (3.73 ± 0.31) ×10−4 [28], and in the pole quark mass schemeyielding B(B → Xsγ) = (3.35±0.30)×10−4 [32].Reducing this theoretical uncertainty, which cur-rently represents an error of O(10%), requires cal-culation of the next-to-next-to-leading order con-tribution to the decay width (i.e., explicit O(αs)2

improvements), which is a formidable but doabletask in a concerted theoretical effort [33]. Antic-ipated experimental precision at the B factorieswill make it mandatory to undertake this heroiceffort.

The transition b → sγ is not expected to yielduseful information on the CKM-Wolfenstein pa-rameters ρ and η, which define the apex of theunitarity triangle of current interest. The test ofCKM-unitarity for the b → s transitions in rareB-decays lies in checking the relation λt � −λc,which holds up to corrections of order λ2. Thisis implicit in the theoretical decay rates quoted

NLO calculations of radiative and semileptonic rare B decays in the Standard Model and comparison with data 625

3

above. Alternatively, one can drop the explicituse of the CKM unitarity and calculate the var-ious contributions in the b → sγ amplitude fromthe current knowledge of λc � |Vcb| = (41.0 ±2.1) × 10−3 and λu from PDG [34], which thenyields λt � Vts = −(47 ± 8) × 10−3 [35]. This isconsistent with λt = −λc, but less precise. This isdue to the strong mixing of the operator O2 withO7 under QCD renormalization, which enhancesthe coefficient of the λc term in the b → sγ am-plitude in comparison with the coefficient of theλt term, thereby reducing the precision on λt.

3. EXCLUSIVE DECAYS B → (K∗, ρ)γ

To compute the branching ratios for B → V γ(V = K∗, ρ) reliably, one needs to calculate theexplicit O(αs) improvements to the lowest orderdecay widths. This requires the calculation ofthe renormalization group effects in the appro-priate Wilson coefficients in the effective Hamil-tonian [32], an explicit O(αs) calculation of thematrix elements involving the hard vertex correc-tions [36–38], annihilation contributions [39–41],which are more important in the decays B → ργbut have also been worked out for the B → K∗γdecays [42], and the so-called hard-spectator con-tributions involving (virtual) hard gluon radia-tive corrections off the spectator quarks in the B-,K∗-, and ρ-mesons [8–10]. We discuss the decaysB → K∗γ and B → ργ in turn.

3.1. B → K∗γ DecaysThe branching ratio for the decays B → K∗γ

can be written as [8–10]:

B(B → K∗γ) = τBG2

F α|VtbV∗ts|2

32π4m2

b,pole M3B

×[ξ(K∗)⊥ (0)

]2(

1 − m2K∗

M2

)3 ∣∣∣C(0)eff7 + A(1)(µ)

∣∣∣2 .

Here, MB is the B-meson mass and ξ(K∗)⊥ (0) is

the form factor in the large energy effective the-ory. It differs from the corresponding form fac-tor in QCD, T K∗

1 (0), by O(αs) terms calculatedin Ref. [23]. Numerically, this relation can beexpressed as T K∗

1 (0) � 1.08ξ(K∗)⊥ . The effect of

the O(αs) corrections is encoded in the function

A(1)(µ) and its numerical value can be expressedin terms of a K-factor [8–10]:

K =

∣∣∣C(0)eff7 + A(1)(µ)

∣∣∣2∣∣∣C(0)eff7

∣∣∣2 , with 1.5 ≤ K ≤ 1.7.

This yields a charge-conjugate averaged branch-ing ratio

〈B(B → K∗γ)〉 � (7.2 ± 1.1) × 10−5

×( mb,pole

4.65 GeV

)2(

ξ(K∗)⊥ (0)0.35

)2

,

where the default values of the most sensitiveparameters are indicated. The value given forξ(K∗)⊥ (0) is based on using the light-cone QCD

sum rules. Varying the parameters within rea-sonable ranges (see, for example, [8]) and addingthe errors in quadrature, one gets

〈B(B → K∗γ)〉 � (7.2 ± 2.7) × 10−5 , (5)

to be compared with the corresponding experi-mental measurement [4,6,7]:

〈B(B → K∗γ)〉 � (4.22 ± 0.28) × 10−5 . (6)

Given the large theoretical errors, there is agree-ment between theory and experiment. Since thereexists good agreement between the SM and exper-iment in the inclusive decay B → Xsγ, one coulduse the measured branching ratios for B → K∗γto determine the form factor T K∗

1 (0), which inthe stated theoretical context yields T K∗

1 (0) =0.27±0.04. This is to be compared with the light-cone QCD sum rule result T K∗

1 (0) = 0.38 ± 0.05[43,26], and with T K∗

1 (q2 = 0) = 0.32+0.04−0.02 [44],

obtained by using the Lattice UK-QCD simula-tions combined with a LC-QCD-sum-rule inspiredinput for extrapolation from q2 0, where theLattice simulations are actually done, to the phys-ical value q2 = 0. A preliminary result on T K∗

1 (0)using Lattice-QCD has also been reported at thisconference [45]. Due to the spread in the theoreti-cal estimates it is too early to draw a quantitativeconclusion on the precise value of the form factor,and hence on the perturbative QCD factorizationmethod underlying the theoretical estimates inB → K∗γ.

626 A. Ali

4

3.2. B → ργ DecaysThe effective Hamiltonian for the radiative de-

cays B → (ρ, ω)γ can be seen for the SM in [8].These transitions involve the CKM matrix ele-ments in the first and third column, with the uni-tarity constraints taking the form

∑u,c,t ξi = 0,

with ξi = VibV∗id. All three matrix elements are

of order λ3, with ξu � Aλ3(ρ − iη), ξc � −Aλ3,and ξt � Aλ3(1 − ρ − iη). This equation leads tothe same unitarity triangle as studied through theconstraints Vub/Vcb, ∆MBd

(or ∆MBd/∆MBs).

We shall concentrate here on B → ργ decays.As the absolute values of the form factors in thisdecay and in B → K∗γ decays discussed earlierare quite uncertain, it is advisable to calculateinstead the ratios of the branching ratios

R±(ργ/K∗γ) ≡ B(B± → ρ±γ)B(B± → K∗±γ)

, (7)

R0(ργ/K∗γ) ≡ B(B0 → ρ0γ)B(B0 → K∗0γ)

. (8)

They have been calculated in the NLO accuracyand the result can be expressed as [8]:

R±(ργ/K∗γ) =∣∣∣∣Vtd

Vts

∣∣∣∣2 [PS] ζ2(1 + ∆R±) ,

R0(ργ/K∗γ) =12

∣∣∣∣Vtd

Vts

∣∣∣∣2 [PS] ζ2(1 + ∆R0) ,

where [PS] = (M2B − M2

ρ )3/(M2B − M2

K∗)3 repre-sents a kinematic factor, which is 1 to a goodapproximation, and ζ = ξρ

⊥(0)/ξK∗⊥ (0), with

ξρ⊥(0)(ξK∗

⊥ (0)) being the form factors (at q2 = 0)in the effective heavy quark theory for the de-cays B → ρ(K∗)γ. Noting that in the SU(3)limit one has ζ = 1, we take ζ = 0.76 ± 0.10 -a range which straddles various theoretical esti-mates of this quantity [39,43,46,47]. The func-tions ∆R± and ∆R0, appearing on the r.h.s. ofthe above equations and whose parametric de-pendence is suppressed here for ease of writing,encode both the O(αs) and annihilation contri-butions. They have been evaluated as [8,13]:∆R± = 0.055 ± 0.130 and ∆R0 = 0.015 ± 0.110,where the errors also include the effect of varyingthe angle α in the currently allowed region.

There are two more observables of interest,namely the isospin-violating ratio ∆(ργ) definedas

∆(ργ) ≡ Γ±(B → ργ)2 Γ0(B → ργ)

− 1 , (9)

and the CP asymmetry in the decay rates, whichfor B± → ρ±γ decays is defined as

A±CP (ργ) ≡ B(B− → ρ−γ) − B(B+ → ρ+γ)

B(B− → ρ−γ) + B(B+ → ρ+γ).(10)

The expected values of the observables in theB → ργ system defined in Eqs. (7) - (10)and A0

CP (ργ), the CP asymmetry in the neutralmodes, have been recently updated in Ref. [13].Including the errors on the current determinationof the CKM parameters, the results are:

R±(ργ/K∗γ) = 0.023 ± 0.012 ,

R0(ργ/K∗γ) = 0.011 ± 0.006 ,

∆(ργ) = 0.04+0.14−0.07 ,

A±CP (ργ) = 0.10+0.03

−0.02 ,

A0CP (ργ) = 0.06 ± 0.02 . (11)

At this conference, the BABAR collaborationhas reported a significant improvement on theupper limits of the branching ratios for the de-cays B0(B0) → ρ0γ and B± → ρ±γ. Averagedover the charge conjugated modes, the current90% C.L. upper limits are [6]:

B(B0 → ρ0γ) < 1.4 × 10−6 , (12)B(B± → ρ±γ) < 2.3 × 10−6 , (13)

B(B0 → ωγ) < 1.2 × 10−6 . (14)

They have been combined, using isospin weightsfor B → ργ decays and assuming B(B0 →ωγ) = B(B0 → ρ0γ), to yield the improved upperlimit [6]

B(B → ργ) < 1.9 × 10−6 . (15)

Together with the current measurements of thebranching ratios for B → K∗γ decays byBABAR, this yields a 90% C.L. upper limit [6]

R(ργ/K∗γ) ≡ B(B → ργ)B(B → K∗γ)

< 0.047 . (16)

NLO calculations of radiative and semileptonic rare B decays in the Standard Model and comparison with data 627

5

Thus, we see that the current experimental upperlimit on the ratio R(ργ/K∗γ) is typically a factor2 away from the central values of the SM. Thiscan be seen graphically in Fig. 1, where we showthe fit of the unitarity triangle resulting from thevarious direct and indirect measurements, includ-ing the CP asymmetry aψKs , reported by theBABAR and BELLE collaboration at this con-ference, with the current world average beingaψKs = 0.734 ± 0.054 [48]. The (almost concen-tric) constraints in the (ρ, η) plane from the mea-sured value ∆MBd

= 0.503±0.006 ps−1 [49], andthe two competing bounds ∆MBs > 14.4 ps−1

[49], and R(ργ/K∗γ) < 0.047 [6] are also shown.The two set of curves for the constraints followingfrom ∆MBd

and ∆MBs , shown with and withoutthe chiral logs, reflect the current uncertainty in-herent in Lattice calculations due to chiral extrap-olations from the region ms to md [50]. We seethat the quantity R(ργ/K∗γ) is not yet compet-itive with the other two constraints from either∆MBd

or ∆MBs . However, this sure will changeas the B factory experiments will soon reach theSM-sensitivity on the decay mode B → ργ, andhence on R(ργ/K∗γ).

−1 −0.5 0 0.5 10

0.2

0.4

0.6

0.8

1

Μ

εk

∆MB d

VubVcb

Β∆

s

R(

/ K

)∗ργ

γ

ksψa

ksψa

without chiral logs

Figure 1. Unitary triangle fit in the SM and theresulting 95% C.L. contour in the ρ - η plane.The impact of the R(ργ/K∗γ) < 0.047 constraintis also shown. (From Ref. [13].)

4. THE DECAYS B → (Xs, K, K∗)�+�−

To discuss the FCNC semileptonic decays inthe SM, one has to extend the operator basisgiven in Eq. (1) by adding two semileptonic op-erators:

O9 =e2

g2s

(sLγµbL)∑

�

(�γµ�︸︷︷︸V

) ,

O10 =e2

g2s

(sLγµbL)∑

�

(�γµγ5�︸ ︷︷ ︸A

), (17)

where e (gs) is the QED (QCD) coupling con-stant and the subscript L refers to the left-handedcomponents of the fermion fields. There are ad-ditional non-local contributions, which can beadded to C9, defining an effective Wilson coeffi-cient, which is a misnomer as it is actually a func-tion of the dilepton mass squared (s = s/M2

B):Ceff

9 (s) = C9η(s)+Y (s), where η(s) = (1+O(αs))includes the real gluon bremsstrahlung and vir-tual contributions to the matrix element of theoperator O9 [51], and Y (s) contains perturba-tive charm loops and Charmonium resonances(J/ψ, ψ′, ...).

Concentrating on the perturbative contributiononly, which is expected to be the dominant con-tribution away from the resonances, and includ-ing the leading power corrections in 1/mb and1/mc, the dilepton invariant mass distribution forB → Xs�

+�− can be written as [27]:

dΓ(b → s�+�−)ds

=(αem

4π

)2 G2F m5

b,pole |λts|248π3

×(1 − s)2[(1 + 2s)

(∣∣∣Ceff9

∣∣∣2 +∣∣∣Ceff

10

∣∣∣2) G1(s)

+4 (1 + 2/s)∣∣∣Ceff

7

∣∣∣2 G2(s) + 12Re(Ceff

7 Ceff∗9

)×G3(s) + Gc(s)] , (18)

where the functions Gi(s) (i = 1, 2, 3) encode the1/mb corrections [19,20] and the function Gc(s)the corresponding 1/mc corrections [21]. Theirexplicit forms as well as of the other effective co-efficients in the above equation can be seen inRef. [27]. As noted in Ref. [16], inclusion of theexplicit O(αs) corrections reduces the branchingratio for B → Xs�

+�−, compared to the partial

628 A. Ali

6

O(αs)-corrected result [19] and its scale depen-dence is considerably reduced.

Data from the BELLE collaboration on thedilepton invariant mass spectrum has been an-alyzed using Ref. [27], which incorporates all theexplicit O(αs) and power corrections mentionedabove. The hadron invariant mass spectrum inB → Xs�

+�−, calculated in the heavy quark ef-fective theory (HQET) and in a phenomenologicalFermi motion model [52], has been used to esti-mate the effect of the experimental cuts on MXS .The results are given in Table 1, and are to beunderstood with a cut on the dilepton invariantmass M�+�− > 0.2 GeV. Given the current exper-imental errors, the agreement with the SM-basedestimates is reasonably good.

The corresponding results for the exclusive de-cays B → K�+�−, averaged over � = e, µ, andfor B → K∗�+�−, with � = e, µ, presented at thisconference [7,14] are also given in Table 1 andcompared with the SM-based estimates [26,27].One notes the comparatively larger theoreticaluncertainty in the exclusive decay rates due tothe form factors. However, within the theoret-ical and experimental errors, SM accounts wellthe current data on all the inclusive and exclu-sive rare B decays. This can be used to constrainmodels of physics beyond-the-SM [27].

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NLO calculations of radiative and semileptonic rare B decays in the Standard Model and comparison with data 629

7

Table 1SM predictions (O(αs) and leading power corrected) and comparison with data (in units of 10−6)

Decay Mode Theory (SM) [26,27] BELLE [7] BABAR [14]

B → K�+�− 0.35 ± 0.12 0.58+0.17−0.15 ± 0.06 0.78+0.24+0.11

−0.20−0.18

B → K∗e+e− 1.58 ± 0.52 < 5.1 1.68+0.68−0.58 ± 0.28

B → K∗µ+µ− 1.2 ± 0.4 < 3.0 < 3.0; weighted e+e−, µ+µ−

B → Xsµ+µ− 4.15 ± 0.70 7.9 ± 2.1+2.0

−1.5 –

B → Xse+e− 4.2 ± 0.7 5.0 ± 2.3+1.2

−1.1 –

B → Xs�+�− 4.18 ± 0.70 6.1 ± 1.4+1.3

−1.1 –

The inclusive measurements and the corresponding SM rates are given with a cut on the dilepton mass,M�+�− > 0.2 GeV.

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