Gauge invariance and form factors for the decay B --> gamma l^+l^-

10 Pages·0.19 MB·English

by  Frank Krüger

Gauge invariance and form factors for the decay B → γl+l−
Frank Kru¨ger1,∗ and Dmitri Melikhov2,†
1Physik Department, Technische Universit¨at Mu¨nchen, D-85748 Garching, Germany
2Institut fu¨r Theoretische Physik, Universit¨at Heidelberg,
Philosophenweg 16, D-69120, Heidelberg, Germany
WeanalysetheformfactorsfortheB→γl+l− weaktransition. Weshowthatmakinguseofthe
gauge invariance of the B → γl+l− amplitude, the structure of the form factors in the resonance
region,andtheirrelationsatlargevaluesofthephotonenergyresultsinefficientconstraintsonthe
behavior of the form factors. Based on these constraints, we propose a simple parametrization of
the form factors and apply it to the lepton forward-backward (FB) asymmetry in the B → γl+l−
decay. We find that the behavior of the FB asymmetry as a function of the photon energy, as well
as the location of its zero, depend only weakly on the B → γ form factors, and thus constitutes a
powerful tool for testing thestandard model.
3
0
PACSnumbers: 13.20.He,12.39.Ki,13.40.Hq
0
2

n
a
J I. INTRODUCTION

0
2
Recently, the decay B γl+l− has been the subject of a number of investigations [1–5], where it has been
→
pointed out that this process may serve as an important probe of the standard model (SM) and possible extensions.
2
v However,knowledge of the long-distanceQCD effects, which are inherently non-perturbative,is important to extract
6 quantitative information on the underlying short-distance interactions.
5 From the analyses of the decay B K∗l+l− it is known [6, 7] that the uncertainties due to the hadronic form
2 factors are considerably reduced if one→considers asymmetries such as the forward-backward(FB) asymmetry of the
8
lepton. This empirical observation later received an explanation within the large energy effective theory (LEET) [8].
0
According to LEET,all the heavy-to-lightmesontransitionformfactorsare givenatleading orderin 1/M and1/E
2 B
(E istheenergyofthelightmeson)intermsofafewuniversalformfactors,andthankstothattheformfactoreffects
0
/ largely drop out from the asymmetries.
h As for the radiative dilepton decay B γl+l−, one expects the same to be true. However, a surprisingly strong
p →
dependenceoftheFBasymmetryonthespecificformfactormodelcanbefoundintheexistingliterature[1–3],which
-
p needs better understanding.
e In this paper, we analyse the form factors for the B γ transition induced by vector, axial-vector, tensor, and
h →
pseudotensor currents.
:
v We show that important relations between form factors of different currents arise as a consequence of the gauge
i invariance of the B γ amplitude. We derive an exact relationbetween the form factors oftensor and pseudotensor
X currents at q2 = 0, →where q2 is the dilepton invariant mass in the decay B γl+l−. We note that the form factors
r →
a from a recent sum-rule calculation of Ref. [4] are inconsistent with this exact relation.
We investigate the behavior of the various form factors at large q2 and find interesting relations corresponding to
the resonance contributions to the form factors. We argue that these contributions signal substantial corrections to
the Isgur-Wise relations valid to 1/m accuracy at large q2 [9].
b
Combined with the relations among the form factors from LEET, the results obtained provide strong restrictions
on the B γ form factors. We propose a simple model for the form factors which is valid over the full range of the
→
photon energy, and which satisfies all known constraints. An important remark is in order here: It has been shown
recently that a proper account of collinear and soft gluons leads to a different effective theory – the so-called soft-
collinear effective theory (SCET) [10] which supersedes the LEET. Important for our discussion is that interactions
with collinear gluons preserve the relations for the soft part of the heavy-to-light form factors from LEET [11] [the
differences appear in the O(α ) part]. Our analysis is therefore fully compatible with SCET.
s
As an application of our form factor model, we examine the FB asymmetry in the decay B γµ+µ−. We find
s
→
thatthisasymmetry,andparticularlyitszeroarisingintheSM,canbepredictedwithsmalltheoreticaluncertainties.
∗Electronicaddress: [email protected]
†Electronicaddress: [email protected]
2
II. FORM FACTORS FOR THE B →γ TRANSITION
We are concerned with the amplitudes of the B γ transition resulting from various quark currents. In our
→
analysis, we adopt the following conventions:
i
γ5 =iγ0γ1γ2γ3, σ = [γ ,γ ], ǫ0123 = 1, (1)
µν µ ν
2 −
and accordingly
i
σ γ5 = ǫ σαβ. (2)
µν µναβ
−2
A. Transition to a virtual photon
We start with the amplitude describing the transition of the B (q = s,d) meson with momentum p to a virtual
q
photon with momentum k. In this case, the form factors depend on two variables: that is, the photon virtuality k2
andthesquareofthemomentumtransfer(p k)2. Asweshallsee,gaugeinvarianceandtheabsenceofsingularitiesin
theamplitudeleadtoseveralrelationsamon−gtheformfactorsatk2 =0,therebyreducingthenumberofindependent
form factors for the transition to a real photon.
(i) For the B γ∗ transition induced by the axial-vector current, the gauge-invariantamplitude (with respect to
q
the photon) conta→ins three form factors and can be written in the form1
k k k p k p
hγ∗(k)|q¯γµγ5b|B¯q(p)i=ieε∗α(k) gµα− αk2µ f +pµ pα− k·2 kα a1+kµ pα− k·2 kα a2 , (3)
(cid:26)(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:27)
where we have explicitly written the gauge-invariant Lorentz structures. In the above, εα denotes the polarization
vector of the photon, e = √4πα, and the form factors are defined according to Ref. [12]. Since the amplitude is a
regularfunctionatk2 =0,the requirementofgaugeinvarianceresultsinthefollowingconstraintsonthe formfactors
at k2 =0:
f +(k p) a =0, a =0. (4)
2 1
·
(ii) For the transitioninduced by the vector current,the amplitude is parametrizedin terms of a single formfactor
g; namely,
γ∗(k)q¯γ bB¯ (p) = 2egε∗α(k)ǫ pρkσ. (5)
µ q µαρσ
h | | i
(iii) For the B γ∗ transition induced by the pseudotensor current, the amplitude can be written in terms of
q
→
three form factors:
k k k k
γ∗(k)q¯σ γ bB¯ (p) = eε∗α(k) g α ν p g α µ p g
h | µν 5 | q i αν − k2 µ− αµ− k2 ν 1
(cid:26)(cid:20)(cid:18) (cid:19) (cid:18) (cid:19) (cid:21)
k p
+(gανkµ−gαµkν)g2+ pα− k·2 kα (kµpν −pνkµ)g0 . (6)
(cid:18) (cid:19) (cid:27)
At k2 =0, gauge invariance leads to the condition
g (k p) g =0. (7)
1 0
− ·
(iv) The amplitude for the transition induced by the tensor current can be obtained from Eq. (6) by applying the
identity in Eq. (2), and is given by
k k p
hγ∗(k)|q¯σµνb|B¯q(p)i = ieε∗α(k) ǫµναρpρ− kα2ǫµνσρkσpρ g1+ǫµνασkσg2+ pα− k·2 kα ǫµνρσpρkσg0 . (8)
(cid:26)(cid:18) (cid:19) (cid:18) (cid:19) (cid:27)
1 Noticethat fortheB-meson transitionto avirtualphoton the amplitudeofthe axial-vector current contains acontact term, whichis
proportional to the charge of the B meson. The contact term is thus present for the charged B-meson transition, but isabsent inthe
caseofaneutral B meson. Foradetaileddiscussionofformfactorsandcontact termsinthisamplitude,seeRef.[12].
3
B. Transition to a real photon
Forthe transitiontoarealphoton,thematrixelementofthevectorcurrentisgivenbyEq.(5)whilethe amplitude
for the axial-vector current, employing the relation in Eq. (4), reads
γ(k)q¯γ γ bB¯ (p) = ieε∗α(k)[g (p k) p k ]a (k2 =0). (9)
µ 5 q µα α µ 2
h | | i − · −
As for tensor and pseudotensor currents, their matrix elements have the form
γ(k)q¯σ γ bB¯ (p) = eε∗α(k) g k g k g + [g (p k) p k ]p [g (p k) p k ]p g ,
µν 5 q αν µ αµ ν 2 αν α ν µ αµ α µ ν 0
h | | i − · − − · −
(cid:26)(cid:18) (cid:19) (cid:18) (cid:19) (cid:27)
γ(k)q¯σ bB¯ (p) = ieε∗α(k) ǫ kσg + p ǫ pρkσ (p k)ǫ pρ g . (10)
µν q µνασ 2 α µνρσ µνρα 0
h | | i − ·
(cid:26) (cid:20) (cid:21) (cid:27)
It should be noted that in the case of a real photon the gauge-invariant amplitudes of the pseudotensor and tensor
currents contain two Lorentz structures and not only one as stated in Ref. [13]. Multiplying Eq. (10) by (p k)ν, we
−
arrive at the following set of amplitudes that describe the B γ transition:
→
F
γ(k)q¯γ γ bB¯ (p) = ieε∗α(k)[g (p k) p k ] A ,
µ 5 q µα α µ
h | | i · − M
Bq
F
γ(k)q¯γ bB¯ (p) = eε∗α(k)ǫ pρkσ V ,
µ q µαρσ
h | | i M
Bq
γ(k)q¯σ γ bB¯ (p) (p k)ν = eε∗α(k)[g (p k) p k ]F ,
µν 5 q µα α µ TA
h | | i − · −
γ(k)q¯σ bB¯ (p) (p k)ν = ieε∗α(k)ǫ pρkσF , (11)
µν q µαρσ TV
h | | i −
where the dimensionless form factors F are given in terms of the form factors g, a , g , g at k2 =0:
i 2 0 2
F =2M g, F = M a , (12)
V Bq A − Bq 2
F = g [p2 (p k)]g , F = g (p k)g . (13)
TV 2 0 TA 2 0
− − − · − − ·
Forarealphotoninthe finalstate,theformfactorsdependonthe squareofthemomentumtransfer,q2 (p k)2.
≡ −
Equivalently, one may consider the form factors as functions of the photon energy in the B-meson rest frame,
M q2
B
E = 1 . (14)
2 − M2
(cid:18) B(cid:19)
For massless leptons, the kinematically accessible range is
0 q2 M2, 0 E E =M /2. (15)
≤ ≤ B ≤ ≤ max B
Then, rewriting Eq. (13) in the form
F =F M (M 2E)g , (16)
TV TA B B 0
− −
we derive the following exact relation:
F (E ) = F (E ). (17)
TA max TV max
We stress that the equality of the form factors F and F is valid only at E =E .
TA TV max
C. LEET and form factors at large E
Interesting relations between the form factors emerge in the limit where the initial hadron is heavy and the final
photon has a large energy [8]. In this case, the form factors may be expanded in inverse powers of Λ /M and
QCD B
Λ /E. As a result, to leading-order accuracy, one finds
QCD
F F F F ζγ(E,M ), (18)
V ≃ A ≃ TA ≃ TV ≃ ⊥ B
4
whereζγ(E,M )istheuniversalformfactorfortheB γ transition(cf.Ref.[8]).2 Weemphasizethatthisrelation
⊥ B q →
is violated by terms of order O(Λ /M ) and O(Λ /E), as well as by radiative corrections of O(α ) [14].
QCD B QCD s
ForlargevaluesofE,the energydependence ofthe universalB γ formfactorcanbeobtainedfromperturbative
→
QCD, the result being [13]
ζγ(E,M ) f M /E. (19)
⊥ B ∝ B B
D. Heavy quark symmetry and form factors at large q2
As found by Isgur and Wise [9], in the region of large q2 M2 the form factors satisfy the relations
≃ B
1
g 2M g, g (2g+a ), (20)
2 B 0 2
≃− ≃ 2M
B
which are valid at leading order in Λ /m .3 Combining these expressions with Eqs. (12) and (13), we derive the
QCD b
following result for the tensor-type form factors in the region of small E:
M E
B
F F − (F F ),
TV V V A
≃ − 2M −
B
E
F F (F F ). (21)
TA V V A
≃ − 2M −
B
It is obvious that these relations are compatible with those in Eq. (18), which emerge in the region where E is large,
and hence are valid to leading order in Λ /m in the full range of E. Yet the leading-order Λ /m relations in
QCD b QCD b
Eq. (21) may not be sufficient to understand the behavior of the form factors F and F in the region where
V,A TA,TV
q2 M2.
≃ B
To explain this point, it is useful to express the form factors F in terms of the Wirbel-Stech-Bauer (WSB) form
i
factors[15]. The advantageofthe WSB formfactorsis thateachone hasdefinite spin andparity,andhence contains
contributions of resonances with the corresponding quantum numbers. Explicitly, we have
M M
F =2VB→γ, F = BAB→γ, F =2TB→γ, F = BTB→γ. (22)
V A E 1 TV 1 TA E 2
Notice that the relation F = F at q2 = 0 (or E = M /2) is just the well-known relation T (0) = T (0). Then,
TV TA B 1 2
makinguse ofthe constraintsin Eqs.(4)and(7), whicharedue to electromagneticgaugeinvariance,we obtainexact
relations between the WSB form factors that are relevant for the transition into a real photon; namely,
2E 2E
TB→γ = TB→γ, AB→γ = AB→γ. (23)
2 M 3 1 M 2
B B
By virtue of these relations, we may rewrite Eq. (22) in the form
F =2AB→γ, F =2TB→γ, (24)
A 2 TA 3
which exhibits the absence of a singular behavior of F and F .
A TA
We now examine the analytic structure of the form factors near q2 = M2, starting with V and T , which have
B 1
a pole at q2 = M2 . This pole is located very close to the upper boundary of the physical region, q2 = M2, since
B∗ B
MB∗ −MB =45MeV ∼O(Λ2QCD/mb). Moreover, as shown in Refs. [12, 16], the residues of the form factors V and
T in the pole at q2 = M2 are equal in the heavy quark limit m . As a consequence, F and F should be
ap1proximately equal andBri∗se steeply as q2 M2. b → ∞ V TV
As for the form factors F and F , th→ey arBe expected to have qualitatively different behavior near q2 = M2.
A TA B
Indeed, the masses of the resonances that correspond to A and T [Eq. (24)], denoted by B∗∗, are expected to be
2 3
2 Foramassiveparticleinthe finalstate onehas twouniversal formfactors ζ⊥(E,MB)andζ||(E,MB). Thelatter does notcontribute
inthecaseofamasslessfinalvectorparticle. Wealsonotetherelationζ⊥Bu→γ(E,MB)=Qu/Qdζ⊥Bd→γ(E,MB),whereQu/Qd=−2.
3 Itshouldbenotedthatatlargeq2thereareingeneralthreeindependentrelationsbetweentheB→V formfactors. Thethirdrelation,
however,isautomaticallysatisfiedinthecaseofaphotoninthefinalstate,duetothegauge-invarianceconstraintsinEqs.(4)and(7).
5
severalhundred MeV higher than MB, since MB∗∗ MB O(ΛQCD). Thus, singularities of the form factors FA and
− ∼
F are much farther from the physical region, compared to the form factors F and F . Consequently, F and
TA V TV A
F are relatively flat as q2 M2.
TA → B
It is now clear why the leading-order Λ /m relations (21) are not useful for understanding the behavior of the
QCD b
form factors near q2 = M2. As a matter of fact, at leading order in Λ /m all the bq¯resonances with different
B QCD b
spins have the same masses, so that all the form factors F have poles at q2 = M2 (i.e., at E = 0). This picture is
i B
fully consistent with Eq. (21), but it is far from reality,since O(Λ /m ) corrections to the form factor relations in
QCD b
Eq.(21)becomecrucialinthe regionnearq2 =M2. We expect,then,the followingrelationbetweenthe formfactors
B
near q2 =M2:
B
F F F F , (25)
A TA V TV
≃ ≪ ≃
in agreement with the resonance location.
E. The general picture of the B →γ form factors
Combining the above information on the form factors, the following picture of the B γ form factors emerges.
→
At E =E , the form factors F and F are equal, F (E )=F (E ).
max TA TV TA max TV max
•
In the region where E Λ , the form factors obey the LEET relation, which is valid to O(Λ /m ),
QCD QCD b
• ≫
O(Λ /E), and O(α ) accuracy:
QCD s
F F F F f M /E. (26)
V A TA TV B B
≃ ≃ ≃ ∝
At large q2 M2 (i.e., at small E), the following relation for the form factors should hold:
• ≃ B
F F F F . (27)
A TA V TV
≃ ≪ ≃
We expect these relations to work with 10–15% accuracy.
Given these features, we now turn to the analysis of existing predictions for the B γ form factors.
→
1. Form factors FA and FV
The form factors F and F for the B γ transition have been calculated in Ref. [12] within the dispersion
A V d
approach of Ref. [17].4 For large and interm→ediate values of E, these form factors can be well parametrized by a
particularly simple formula:
f M
B B
F(E)=β , (28)
∆+E
with f = 0.2 GeV, M = 5.28 GeV, and β β = O(1/m ) in accord with LEET. In order to use this formula
B B V A b
−
for the form factors in the full range of E, we should take into account that the form factors F and F have,
V A
rmeaspsseecstiovfeltyh,eppoolessitaivteq-p2a=ritMy sB2t∗ataensdwqit2h=higMhBe2r∗∗s,psinostBha∗t∗∆arVe n=otMkBn∗ow−nM, wBeacnadn∆usAe d=atMaBo∗n∗D−mMeBs.onAslttohoeustgihmtahtee
the mass difference ∆ 0.3–0.4 GeV; here we take ∆ =0.3 GeV. The numerical parameters are listed in Table I.
A A
≃
The maximal difference between the form factors at E = 0 is at the level of 50%. The difference between F and
A
F is around 10% for E 0.7 GeV, indicating Λ /m corrections to the LEET relation between the form factors
V QCD b
≥
at the level of 5–10%.
4 TheformfactorsFA,V fortheBu→γ transitionhavebeencalculatedinRefs.[18,19]usinglight-conesumrules. Theseresultsagree
withthe results fromthe dispersionapproach [12]. It shouldbe noted that the formfactors FA,V for the Bu →γ transition have the
oppositesignandareapproximatelytwiceasbigasthecorrespondingformfactorsFA,V fortheBd→γ transition; seethediscussion
inRef. [12].
6
2. Form factors FTA and FTV
There are several calculations of the B γ form factors F and F available in the literature [4, 5]. Let us
TA TV
→
check whether the results for the form factors satisfy the constraints derived above.
(i) The light-cone sum rule calculation of Ref. [4] predicts the form factors F F for large values of E,
TV TA
≫
including E , which points to a very strong violation of the LEET relations. More importantly, the exact relation
max
in Eq. (17) between the form factors at E is also drastically violated. Thus, we conclude that the calculation of
max
form factors performed in Ref. [4] cannot be trusted.
(ii)The quarkmodelcalculationofRef.[5]satisfiesthe exactconstraintinEq.(17),withvaluesofthe formfactors
F = F = 0.115 at q2 = 0 (or, equivalently, E = M /2) [20]. Taking into account the LEET relation (18),
TA TV max B
this value is in agreement with our results for the form factors F = 0.09 and F = 0.105 at q2 = 0. On the other
A V
hand, there are several features of the predicted form factors that do not seem realistic. First, as can be seen from
Fig. 1 of Ref. [5], the form factors F and F differ considerably, with F F for the values of E 0.5–1
TA TV TA TV
≫ ≃
GeV. This signals a very strong violation of the LEET condition, with corrections of the order of several hundred
percent. Let us recall that the form factors may indeed be very different in the region q2 M2, since F contains
a pole at q2 = M2 while F does not. But then one would expect the relation F ≃F B, oppositeTVto the one
obtained in Ref. [B5∗]. SecondT,Athe form factors F and F of Ref. [5] vanish at qT2V=≫M2T.A Taking into account
TA TV B
that the form factor F contains a pole at q2 = M2 , it seems very unlikely that the form factor F vanishes at
TV B∗ TV
q2 =M2. Therefore, the predictions of Ref. [5] for values of q2 10 GeV2 cannot be considered very reliable.
B ≥
To sum up: There are no fully convincing results for the form factors F and F ; thus, for the analysis of the
TA TV
FB asymmetry, we prefer to rely upon a simple model for the B γ transition formfactors which satisfies explicitly
→
all the constraints discussed above.
F. Model for the B →γ form factors
We assumethe ansatzinEq.(28)tobe validforallB γ formfactorswiththeir ownconstants. Togetherwith
d,s
→
the condition in Eq. (16), this leads to the following relation between the parameters of F and F :
TA TV
β β
TV TA
= , (29)
∆ +M /2 ∆ +M /2
TV B TA B
such that β β =O(1/m ). Furthermore, according to our arguments mentioned above, we set
TV TA b
−
∆ =∆ , ∆ =∆ . (30)
TV V TA A
The remaining parameter to be fixed is the constant β , for which we write
TV
β =(1+δ)β , (31)
TV V
and choose δ = 0.1 according to the result of Ref. [5]. This completes our simple model for the form factors which
are consistent with the exact relations at E , LEET at large E, and heavy quark symmetry at small E. Table I
max
contains the various numerical parameters for the B γ form factors, and Fig. 1 shows the form factors in our
d
→
model as a function of the scaled photon energy, x 2E/M .
B
≡
Parameter FV FTV FA FTA
β (GeV−1) 0.28 0.30 0.26 0.33
∆ (GeV) 0.04 0.04 0.30 0.30
TABLE I:Parameters of theB →γ form factors, as definedin Eq. (28).
d
For the B γ transition, we do not know the precise values of the form factors, but we shall assume that the
s
→
LEET-violating effects in the B γ form factors have the same structure as those in the B γ transition. With
s d
→ →
this assumption, the form factors as given in Table I are sufficient for the analysis of the FB asymmetry in the
B γl+l− decay presented in the next section.
s
→
7
0.6
0.5
0.4
Fi 0.3
0.2
0.1
PSfragrepla
ements 00.2 0.4 0.6 0.8 1
x
FIG.1: ThepredictedxdependenceoftheBd →γ formfactorsFV (solid curve),FA (dashedcurve),FTA (dottedcurve),and
FTV (dash-dottedcurve) according to ourmodel, as described in thetext (x≡2E/MB).
III. FORWARD-BACKWARD ASYMMETRY IN B →γl+l−
We now assess the implications of our form factor model for the FB asymmetry of µ− in the decay B¯ γµ+µ−.
s
→
Referring to Refs. [1, 4, 5, 19], the radiative dilepton decay receives various contributions. The main contribution in
the case of light leptons comes from the so-called structure-dependent (SD) part, where the photon is emitted from
the external quark line. Contributions coming from photons attached to charged internal lines are suppressed by
m2/M2 [19]. The bremsstrahlungcontributiondue toemissionofthe photonfromthe externalleptons issuppressed
b W
by the mass of the light leptons l=e,µ and affects the photon energy spectrum only in the low E region [1, 5].
Neglecting the bremsstrahlung contributions, the decay is then governed by the effective Hamiltonian describing
theb sl+l− decay,togetherwiththeformfactorsparametrizingtheB γ transition,asdiscussedinthepreceding
sectio→n. Using the effective Hamiltonian for b sl+l− in the SM [21], th→e matrix element is (m =0)
s
→
G α
= F V V∗ ceff¯lγµl+c ¯lγµγ l γ(k)s¯γ P bB¯ (p)
MSD √2π tb ts" 9 10 5 h | µ L | s i
(cid:0) (cid:1)
2ceffm
7 b γ(k)s¯iσ qνP bB¯ (p) ¯lγµl , (32)
− q2 h | µν R | s i #
where q = p k and P = (1 γ )/2. Within the SM, the Wilson coefficients, including next-to-leading-order
L,R 5
− ∓
corrections [21], have numerical values (m =166 GeV)
t
ceff = 0.330, ceff =c +Y(q2), c =4.182, c = 4.234, (33)
7 − 9 9 9 10 −
where the function Y denotes contributions from the one-loop matrix elements of four-quark operators (see Ref. [21]
for details), and has absorptive parts for q2 > 4m2. In addition to the short-distance contributions, there are also
c
cc¯resonant intermediate states such as J/ψ,ψ′, etc., which we will take into account by utilizing e+e− annihilation
data, as described in Ref. [22].
Defining the angle θ between the three-momentum vectors of µ− and the photon in the dilepton centre-of-mass
system, and recalling the scaled energy variable x 2E/M in the B rest frame, we obtain the differential decay
≡ Bs s
rate (mˆ m /M )
i ≡ i Bs
1/2
dΓ(B¯ γµ+µ−) G2α3M5 4mˆ2
dsx→dcosθ = F211π4Bs|VtbVt∗s|2x3 1− 1 µx! [B0(x)+B1(x)cosθ+B2(x)cos2θ]. (34)
−
Here, we have summed over the spins of the particles in the final state, and have introduced the auxiliary functions
B = (1 x+4mˆ2)(F +F ) 8mˆ2 c 2(F2 +F2),
0 − µ 1 2 − µ| 10| V A
8
0.6
0.4
0.2
FB 0
A
−0.2
−0.4
−0.6
PSfragrepla
ements
0.2 0.4 0.6 0.8
x
FIG. 2: SM prediction for the FB asymmetry of µ− in the decay B¯s →γµ+µ− as a function of x≡2E/MBs, using the form
factorsgiveninEq.(28)(solid curve)andincludingtheeffectsofcc¯resonances. Forcomparison,wealsoshowthedistribution
obtained byutilizing theleading-order LEET form factor relation in Eq. (18) (dashed curve).
1/2
4mˆ2
B = 8 1 µ Re c [ceff∗(1 x)F F +ceffmˆ (F F +F F )] ,
1 − 1 x! { 10 9 − V A 7 b V TA A TV }
−
B = (1 x 4mˆ2)(F +F ), (35)
2 − − µ 1 2
with the form factors defined in Eq. (11), and
4ceff 2mˆ2 4Re(ceffceff∗)mˆ
F = (ceff 2+ c 2)F2 + | 7 | bF2 + 7 9 bF F ,
1 | 9 | | 10| V (1 x)2 TV 1 x V TV
− −
4ceff 2mˆ2 4Re(ceffceff∗)mˆ
F2 = (|ce9ff|2+|c10|2)FA2 + (|17 |x)2bFT2A+ 17 9x bFAFTA. (36)
− −
Recall that the Wilson coefficient ceff [Eq. (33)] depends on x via q2 =M2 (1 x).
9 Bs −
The term odd in cosθ in Eq. (34) produces a FB asymmetry, defined as5
1 dΓ 0 dΓ
dcosθ dcosθ
dxdcosθ − dxdcosθ
A (x)= Z0 Z−1 , (37)
FB 1 dΓ 0 dΓ
dcosθ + dcosθ
dxdcosθ dxdcosθ
Z0 Z−1
which is given by
1/2
4mˆ2 Re c [ceff∗(1 x)F F +ceffmˆ (F F +F F )]
AFB(x)=3 1− 1−µx! {[(F101+9F2)(−1−x+V 2Amˆ2µ)−7 6mˆb2µ|cV10|T2(AFV2 +AFA2T)V] }. (38)
Note that there are also non-factorizable radiative corrections to the asymmetry which are not contained in the
transition form factors.6
5 Note that the FB asymmetry is equivalent to the asymmetry in the l+ and l− energy spectra discussed in Ref. [1]. Our results in
Eqs.(34)and(38)agreewiththosegiveninthatpaper.
6 InthecaseoftheB→K∗l+l− decaythesecorrectionswereanalysedin[14].
9
We plot in Fig. 2 the FB asymmetry as a function of the scaled photon energy x, by using our model for the form
factors,Eq.(28),andtheuniversalformfactors,Eq.(18). Inthelattercase,omittingthenon-factorizablecorrections
the dependence on the form factors drops out completely, and so the asymmetry is fully determined by the Wilson
coefficients.
From Fig. 2 one infers an interesting feature of A (x) in the SM: namely, for a given photon energy x=x , and
FB 0
farfromthecc¯resonances,theFBasymmetryvanishes. AscanbeseenfromFig.2,the1/M and1/E correctionsto
B
theformfactors,whicharetakenintoaccountbyourformfactormodel,Eq.(28),shiftthelocationofthezerobyonly
a few percent, and do not change the qualitative picture of the asymmetry for x & 0.4. Using the numerical values
of the standard model Wilson coefficients given in Eq. (33), together with m = 4.4 GeV, we obtain x 0.85–0.88
b 0
≃
depending on the form factors used. Notice again that the location of zero is further affected by non-factorizable
radiative corrections [14].
We would like to emphasize that the absence of the zero in the SM forward-backward asymmetry in the region
x 0.7 reportedin [3] is due to using the formfactorsofRef. [4], inconsistentwiththe rigorousconstraintsdiscussed
≥
in the present paper. In view of this, the various distributions and asymmetries calculated in Refs. [2, 3] with the
form factors of Ref. [4] should be revised.
IV. CONCLUSIONS
We have analysed the form factors that describe the B γ transition, and investigated their implications for the
FB asymmetry of the muon in the decay B¯ γµ+µ−, wi→thin the SM. Our results are as follows.
s
→
We have derivedan exactrelationfor the formfactorsF and F ofthe B γ transitioninduced by tensor
TA TV
• →
and pseudotensor currents at maximum photon energy:
F (E )=F (E ). (39)
TA max TV max
We have investigated the resonance structure of the form factors at q2 M2 and found that singularities of
• F and F are located much closer to the edge of the physical region (≃i.e., qB2 = M2) than those of the form
T TV B
factors F and F . Hence we expect F and F to rise rapidly as q2 M2 but F and F to remain
relativelyTflAat, so thAat at q2 M2 we havTe the relaTtVion → B A TA
≃ B
F F F F . (40)
A TA V TV
≃ ≪ ≃
This behavior indicates a strong violation of the Isgur-Wise relations between the form factors at large q2.
We have found a serious discrepancy between the just-mentioned constraints and existing calculations of the
•
form factors F and F from QCD sum rules [4] and quark models [5]:
TA TV
(i) the form factors of Ref. [4] violate both the exact constraint (39) and the relations expected from the large
energy effective theory (18);
(ii) the form factors of Ref. [5] signal a very strong violation of the LEET relation [Eq. (18)]. Moreover, the
vanishing of the form factors F and F at q2 =M2 in [5] contradicts the resonancestructure of these form
TA TV B
factors.
We would like to stress that there is an important relation between the universal form factors describing the
•
B γ and B γ transitions:
u d
→ →
ζBu→γ(E,M )=Q /Q ζBd→γ(E,M ), (41)
⊥ B u d ⊥ B
whereQ representthechargeofuanddquarks. Asaconsequenceofthisrelation,theformfactorsF
u,d A,V,TA,TV
oftheB γtransitionhaveoppositesign,andtheirmoduliareapproximatelytwiceasbigasthecorresponding
u
→
form factors of the B γ transition. Furthermore, it is worth emphasizing that this relation has not been
d
→
properly taken into account in Ref. [4] when using the B γ form factors of Ref. [19] for the description of
u
the B γl+l− decay. →
d,s
→
By using the exact relation between the form factors F and F at E , the resonance structure of the
TA TV max
• form factors in the region q2 M2, and the LEET relations F F F F valid for E Λ , we
≃ B A ≃ V ≃ TA ≃ TV ≫ QCD
proposed a simple parametrization for the form factors:
M f
B B
F (E)=β , i=A,V,TA,TV. (42)
i i
∆ +E
i
10
The numerical parameters (Table I) have been fixed by utilizing reliable data on the form factors at large and
intermediate values of the photon energy.
We have applied our form factor model to the FB asymmetry of the muon in B¯ γµ+µ− decay. Comparing
s
• →
the distribution of A based on these form factors with the one obtained by using the leading-order LEET
FB
form factors shows that the behavior of the FB asymmetry remains essentially unchanged in the region x =
2E/M &0.4.
Bs
Our analysis confirms the result of Ref. [1] that the shape of the FB asymmetry as well as the location of its
zeroaretypicalfortheSM.Wepointoutthattheasymmetriesanddistributionsreportedinanumberofrecent
publications[2,3]shouldberevisedastheyarebasedontheformfactorsofRef.[4]whichareinconsistentwith
the rigorous constraints on the form factors.
According to the above results we conclude that the FB asymmetry in the B γl+l− decay, particularly its zero
arising in the SM, can be predicted with small theoretical uncertainties. This is→similar to the decay B K∗l+l−,
→
where a full next-to-leading-order calculation (second reference in [14]) shows that a measurement of the zero of the
FB asymmetry would allow a determination of ceff/Re(ceff) at the 10% level. (This order of magnitude should also
7 9
hold in the case of the B γl+l− decay.)
To sum up, the study of→the decay B µ+µ−γ atfuture hadroncollider experiments willprovidecomplementary
s
information on the structure of the unde→rlying effective Hamiltonian describing b sl+l− transitions.
→
Acknowledgments
We are grateful to Berthold Stech for valuable discussions. F.K. has been supported by the Deutsche Forschungs-
gemeinschaft (DFG) under contractBu.706/1-1. D.M. would like to thank the Alexander von Humboldt-Stiftung for
financial support.
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