Chiral symmetry and exclusive B decays in the SCET

We describe a chiral formalism for processes involving both energetic hadrons and soft Goldstone bosons, which extends the application of soft-collinear effective theory to multibody B decays. The nonfactorizable helicity amplitudes for heavy meson decays into multibody final states satisfy symmetry relations analogous to the large energy form factor relations, which are broken at leading order in Lambda/mb by calculable factorizable terms. We use the chiral effective theory to compute the leading corrections to these symmetry relations in B ->M_n pi ell\bar\nu and B ->M_n pi e+e- decays.

We construct a chiral formalism for processes involving both energetic hadrons and soft Goldstone bosons, which extends the application of soft-collinear effective theory to multibody B decays. The nonfactorizable helicity amplitudes for heavy meson decays into multibody final states satisfy symmetry relations analogous to the large energy form factor relations, which are broken at leading order in Λ/m b by calculable factorizable terms. We use the chiral effective theory to compute the leading corrections to these symmetry relations in B → Mnπℓν and B → Mnπℓ + ℓ − decays. Introduction. The study of processes involving energetic quarks and gluons is simplified greatly by going over to an effective theory which separates the relevant energy scales. The soft-collinear effective theory (SCET) [1] simplifies the proof of factorization theorems and allows a systematic treatment of power corrections. SCET has been applied to both inclusive and exclusive hard processes with energetic final state particles.
In this paper we present a combined application of the SCET with chiral perturbation theory which can be used to study exclusive processes involving both energetic light hadrons and soft pseudo Goldstone bosons and photons. The main observation is that once the dynamics of the collinear degrees of freedom has been factorized from that of the soft modes, usual chiral perturbation theory methods can be applied to the latter, unhampered by the presence of the energetic collinear particles which might have upset the momentum power counting in p/Λ χ . The chiral formalism has been applied previously to compute matrix elements of operators appearing in hard scattering processes, such as DIS and DVCS [2,3,4]. Our paper extends these results to processes with both soft and collinear hadrons.
We focus here on exclusive B decays, which are described by three well-separated scales: hard Q ∼ m b , hard-collinear √ ΛQ and the QCD scale Λ ∼ 500 MeV. This requires the introduction of a sequence of effective theories QCD → SCET I → SCET II , containing degrees of freedom of successively lower virtuality [5]. The intermediate theory SCET I contains hard-collinear quarks ξ n and gluons A µ n with virtuality p 2 hc ∼ ΛQ and ultrasoft quarks and gluons q, A µ with virtuality Λ 2 . Finally, one matches onto SCET II which includes only soft q, A µ and collinear ξ n , A µ n modes with virtuality p 2 ∼ Λ 2 . The expansion parameter in both effective theories can be chosen as λ 2 ∼ Λ/m b .
In the low energy theory SCET II the soft and collinear modes decouple at leading order and the effective Lagrangian is simply a sum of the kinetic terms for each mode The matching of an arbitrary operator O onto SCET II can be written symbolically as [5] O where the ellipses denote power suppressed contributions. The first term is a 'factorizable' contribution, with O S , O C soft and collinear operators convolved with a Wilson coefficient T depending on the arguments of operators. Their precise form depends on the IR regulator adopted for SCET II ; for example, in dimensional regularization they might take the form of T products of operators involving messenger modes [6]. This formalism has been used to study exclusive B decays into energetic light hadrons (e.g. B → πℓν and B → K * γ) [5,7,8,9], and nonleptonic decays into 2 energetic light hadrons such as B → ππ [10,11]. This paper presents an extension of this formalism to describe multibody B decays to one energetic hadron plus multiple soft pions and photons. Such decays received increased attention recently [12,13] due to their ability to extend the reach of existing methods for determining weak parameters.
In Sec. 2 we introduce the SCET formalism and review the derivation of the large energy symmetry relations for the B → M form factors [5,14,16]. We show that similar relations exist for B decays into multibody final states containing one collinear hadron M n plus soft hadrons X S , B → M n X S . Sec. 3 develops a chiral formalism for computing the matrix elements of the soft operators in (2) X S |O S |B with X S containing only soft Goldstone bosons. As an application we discuss in Sec. 4 the semileptonic and rare radiative decays B → M n π S ℓν and B → M n π S ℓ + ℓ − .
2. Symmetry relations. The most general SCET I operator appearing in the matching of SM currentsqΓb for b → uℓν or b → sγ decays has the form (we neglect here light quark masses, which can be included as in [15]) These are the most general operators allowed by power counting and which contain a left-handed collinear quark. We neglect O(λ) operators of the formq n P † ⊥ Γb v which do not contribute below. The relevant modes are soft quarks and gluons with momenta k s ∼ Λ and collinear quarks and gluons moving along n. n µ ,n µ are unit light-cone vectors satisfying n 2 =n 2 = 0, n ·n = 2.
The O(λ) operators are defined as with {Γ The action of the collinear derivative i∂ µ on collinear fields is given by the momentum label operator P µ = 1 2 n µn · P + P ⊥ µ . The collinear gluon field tensor is The Wilson coefficients c i , b i depend on the Dirac structure of the QCD current Γ and are presently known to next-to-leading order in matching [1,17,18].
After matching onto SCET II , the effective current (4) contains the factorizable operators where J ⊥, are jet functions defined as in [11]. We de- This has the factorized form of Eq. (2), with the Wilson coefficient T given by b i ⊗ J ,⊥ . The nonfactorizable operator O nf in Eq. (2) arises from matching the LO SCET I operators onto SCET II [5]. The precise form of the latter operators is not essential for our argument, which depends only on the Dirac structure of the SCET I operators. Before proceeding to write down the SCET predictions for these matrix elements, we define more precisely the kinematics of the process. The transition B → M n X S induced by the current J µ =qΓ µ b can be parameterized in QCD in terms of 4 helicity amplitudes defined as with ε µ ±,0,t a set of four orthogonal unit vectors defined in the rest frame of v as ε µ . These definitions correspond to the choice n = (1, 0, 0, 1),n = (1, 0, 0, −1).
In the language of helicity amplitudes, the most general matrix elements of the nonfactorizable operators are given in terms of the 2 parameters ζ ⊥,0 (E M , X S ) are complex quantities depending on the momenta, spins and flavor of the particles in the final state.
The relations Eq. (7) imply several types of SCET predictions for the nonfactorizable contributions to the helicity amplitudes. The most important one is the vanishing of the right-handed (nonfactorizable) helicity amplitudes at leading order in 1/m b , for any current Γ coupling only to left chiral collinear quarks For decays to one-body states, this constraint leads to the well-known large energy form factor relations [14,16,18]. The argument above extends this result to hadrons of arbitrary spin and multibody states M n X S . Another prediction is a relation between the time-like and longitudinal nonfactorizable contributions to the helicity amplitudes for an arbitrary current Γ Finally, SCET predicts also the ratio of helicity amplitudes mediated by different currents, into any state M n X S containing one energetic collinear particle, e.g.
These relations are in general broken by the factorizable contributions from Eq. (5). For example, the helicity zeros (8) could disappear if the b 1R term gives a nonvanishing contribution (note that the b 1R(L) term in Eq. (5) contributes only to the H +(−) helicity amplitude). For a 1-body state, this is forbidden by angular momentum conservation since the collinear part of the operator can only produce a longitudinally polarized meson. However, this constraint does not apply for multibody final states M n X S (except in channels of well defined J P quantum numbers). In particular, this means that the helicity zero Eq. (8) receives corrections at leading order in 1/m b . These corrections are computed in Sec. 4.
The factorizable corrections to these relations are parameterized in terms of the soft functions Parity invariance of the strong interactions gives one relation among these functions in channels with X JΠ S of well-defined spin J and intrinsic parity (−) Π whereP is the parity operator andR π the rotation operator by 180 • around the y axis. Compared with the decays into one-body hadronic states, for which only the soft function S (0) is required, this represents an increase in the number of independent parameters. However, the total number is still less than in QCD (see Table 1), such that predictive power is retained. In the next section we construct a chiral formalism which can be used to compute these matrix elements for any state X S containing only soft pions.
3. Chiral formalism. We construct here the representation of the soft operator O S giving the soft functions in (12) in the low energy chiral theory. Since we are interested in B decays, the appropriate tool is the heavy hadron chiral perturbation theory developed in Refs. [20]. The main result is that the matrix elements of O S depend only on the B meson light cone wave function.
The effective Lagrangian that describes the low momentum interactions of the B mesons with the pseudo-Goldstone bosons π, K and η is invariant under chiral SU (3) L × SU (3) R symmetry and under heavy quark spin symmetry. This requires the introduction of the heavy quark doublet (B, B * ) as the relevant matter field. The chiral Lagrangian for matter fields such as the B ( * ) must be written in terms of velocity dependent fields, to preserve the validity of the chiral expansion.
The chiral effective Lagrangian describing the ground state mesons containing a heavy quark Q is [20] where the ellipsis denote light quark mass terms, O(1/m b ) operators associated with the breaking of heavy quark spin symmetry, and terms of higher order in the derivative expansion. The pseudoscalar and vector heavy meson fields P (Q) a and P * (Q) aµ form the matrix For Q = b, (P and describesB andB * mesons with definite velocity v. For simplicity of notation we will omit the subscript v on H, P and P * µ . The pseudo-Goldstone bosons appear in the Lagrangian 17) and the pion decay constant f ≃ 135 MeV. These fields transform as The Lagrangian Eq. (14) is the most general Lagrangian invariant under both the heavy quark and chiral symmetries to leading order in m q and 1/m Q .
The symmetries of the theory constrain also the form of operators such as currents. For example, the left handed current L ν a =q a γ ν P L Q in QCD can be written in the low energy chiral theory as [20] where the ellipsis denote higher dimension operators in the chiral and heavy quark expansions. The parameter α is obtained by taking the vacuum to B matrix element of the current, which gives α = f B √ m B (we use a nonrelativistic normalization for the |B ( * ) states as in [20]).
In the SCET we require also the matrix elements of nonlocal operators O S , which appear in Eq. (2). To leading order in 1/m b these operators are quark bilinears Under the chiral group they transform as (3 L , 1 R ) and (1 L , 3 R ). In analogy with the local current (19) we write for O a L,R (k + ) in the chiral theory where the most general form forα L,R (k + ) depends on eight unknown functions a i (k + ) The heavy quark symmetry constraint H (Q) v / = −H (Q) reduces the number of these functions to four. Taking the vacuum to B meson matrix element fixes the remaining functions aŝ where φ ± (k + ) are the usual light-cone wave functions of a B meson, defined by [19] dz − 4π We find thus the remarkable result that the B meson light-cone wave functions are sufficient to fix the pion matrix elements of the nonlocal operators O a L,R (k + ). The same result can be obtained also by considering only local operators. Let us consider the operator O a L (k + ) (the same results are obtained for O a R (k + )). Expanding in a power series of the distance along the light cone one is led to consider the matrix elements of the operator symmetric and traceless in its indices

−(traces)
Heavy quark and chiral symmetry constrain the chiral effective representation of this operator to be of the form where the sum over j includes the most general symmetric and traceless structures X formed from γ µ , v µ , g µν .
There are many such structures, but only 2 of them survive when contracted with n µ1 · · · n µN This gives the chiral representation of the projection of the operators (26) on the light-conē which makes it clear that the constants α N,0 , α N,1 are uniquely fixed in terms of the B → vacuum matrix elements of the operators (26). Assuming that the B lightcone wave functions are well behaved at large k + , these matrix elements are related to the moments of φ ± (k + ). Specifically, one finds In particular, for N = 1 this gives α 1,0 = − 8 3 f B √ m BΛ , which agrees with Ref. [21].
Beyond leading order in 1/m b many more operators can be written. For example, the matrix elements of O a L with one insertion of the chromomagnetic term in the HQET Lagrangian L m = gb v σ µν G µν b v gives structures of the form The proliferation of unknown constants (see also [23]) spoils the simple leading order result that knowledge of the B → vacuum matrix element is sufficient to fix all low energy constants. The operators in Eqs.

4.
Application:B → M n πℓν andB → M n πℓ + ℓ − . As an application we compute the factorizable corrections to the symmetry relations (8), (10) for the transverse helicity amplitudes inB → M n πℓν in the region of the phase space with one energetic meson M n = π, ρ, K * , etc. plus one soft pion.
The factorizable contribution to the transverse helicity amplitudes for B → M n π are given by the matrix elements of Eq. (5). Specifically, one has for M n a pseudoscalar meson and for M n a vector meson We used here the short notation b i J a φ a = dxdzdk + b i (x, z)J a (x, z, k + )φ + (k + )φ a (x). The isospin factor C depends on the collinear meson, e.g. C(ρ 0 ) = 1/ √ 2, C(ρ ± ) = 1. The corresponding results for the 1body factorizable decay amplitudes are obtained from these expressions by taking S R → 0 , S L → 1.
Inspection of the results (32)-(35) gives the following conclusions, valid to all orders in α s .
i) The null result in Eq. (33) means that the symmetry relation (10) forB → P n X S transitions is not broken by factorizable corrections and is thus exact to leading order in 1/m b . This leads, e.g., to a relation between B → (K n π S ) h=−1 e + e − andB → (π n π S ) h=−1 e −ν .
ii) The vanishing of the H + nonfactorizable helicity amplitudes inB decay Eq. (8) is violated by the factorizable terms Eqs. (32), (34). These terms are however calculable in chiral perturbation theory for X S containing only soft pions. For both M n = P, V , the pion carries m 3 = +1 angular momentum; the V n collinear meson is emitted longitudinally polarized.
The soft functions S R,L (p π ) in Eq. (32), (35) can be computed explicitly in terms of the chiral perturbation theory diagrams in Fig. 1. We find with ∆ = m B * −m B ≃ 50 MeV and Γ B * the width of the B * meson. While the soft matrix elements in Eq. (12) have a factorized form S (i) (k + , S X ) = φ + (k + )S i (S X ), the total factorizable amplitude is not simply the product B → B * π times B * → M n , due to the direct graph in Fig. 1a (nonvanishing only for S L ). At threshold, the relation Eq. (37) gives a soft pion theorem which fixes the soft function in B → M n π in terms of the factorizable contribution to the B → M n transition. Note that the B * width in the propagator is a source of strong phases at leading order in 1/m b . These results can be extended to final states containing multiple soft pions, without introducing any new unknown hadronic parameters.

5.
Conclusions. We presented in this paper the application of the soft-collinear effective theory to B decays into multibody final states, containing one energetic meson plus soft pseudo Goldstone bosons. The additional ingredient is the application of heavy hadron chiral perturbation theory [20] to compute the matrix elements with Goldstone bosons of the nonlocal soft operators obtained after factorization. (This assumes that the only SCET operators contributing to these decays are the same as those describing B → M n transitions [5].) Heavy quark and chiral symmetry are powerful constraints which fix all these couplings in terms of the usual B light-cone wave functions. This simplicity should be contrasted with the case of the twist-2 DIS and DVCS operators, whose matrix elements on nucleons plus soft pions require additional couplings not constrained by the nucleon structure functions [4].
Some of the symmetry predictions of SCET rely on angular momentum conservation arguments which are invalidated when the final hadronic state contains more than one hadron (see Eq. (8)), already at leading order in the 1/m b expansion. The chiral formalism presented here allows the systematic computation of these effects. We point out the existence of an exact relation Eq. (10) among left-handed helicity amplitudes inB → P n X S transitions induced by different b → q n currents.
These results extend the applicability of SCET to B decays into multibody states M n X S containing one energetic particle. It is interesting to note that the corrections to these predictions scale like max(Λ/E M , p S /Λ χpT ), rather than m X /E M . This suggests that the range of validity of factorization in these decays might be wider than previously thought, a fact noted empirically in Refs. [22] in the context of the B → DX decays. Many more problems can be studied using the formalism described here,