Polarization in B->VV Decays

Factorizable amplitudes in B decays to light vector meson pairs give a longitudinal polarization satisfying 1- f_L =O(1/m_b^2). This remains formally true when non-factorizable graphs are included in QCD factorization, and is numerically realized in B->Rho Rho. In \Delta S=1 decays a QCD penguin annihilation graph can effectively contribute at leading power to the transverse and longitudinal amplitudes. The observed longitudinal polarization, f_L (B->phi K^*) \approx 50%, can therefore be accounted for in the SM. The ratio of perpendicular to parallel transverse rates provides a sensitive test for new right-handed currents. The transverse b->sg dipole operator amplitudes are highly suppressed. CP violation measurements can therefore discriminate between new contributions to the dipole and four quark operators. SU(3)_F violation in QCD penguin amplitudes can easily be O}(1), in general, due to annihilation. Implications for B->Rho K^* polarization and New Physics searches are pointed out.


Introduction: 'helicity-flip' suppression
Polarization in B → V V decays should be sensitive to the V −A structure of the Standard Model due to the power suppression associated with the 'helicity-flip' of a collinear quark. For example, in the Standard Model the factorizable graphs forB → φK * are due to transition operators with chirality structures (sb) V −A (ss) V ∓A , see Figure. 1. There are three helicity amplitudes,Ā 0 ,Ā − , andĀ + , in which both vectors are longitudinally, negatively, and positively polarized, respectively. InĀ − a collinear s ors quark with positive helicity ends up in the negatively polarized φ, whereas inĀ + a second quark 'helicity-flip' is required in the form factor transition. In the case of new right-handed currents, e.g., (sb) V +A (ss) V ±A , the hierarchy is inverted, withĀ + andĀ − requiring one and two 'helicity-flips', respectively.
Helicity-flip suppression can be estimated by recalling that the probability for a positive helicity free fermion to have negative spin along some axis is given by sin 2 θ/2, where θ is the angle between the axis and the momentum vector. For a φ meson in a symmetric configuration the transverse momentum of the valence quarks is k ⊥ ∼ m φ /2, implying that the helicity suppression inĀ − is ∼ m φ /m B . The form factor helicity suppression inĀ + should be approximately p T /m b , where p T is the transverse momentum of the outgoing s quark. The latter can be estimated by identifying it with the transverse momentum of the b quark. In the 'Fermi momentum' model of [1] < p 2 T > 1/2 ≈ p F / √ 3. Using the equivalence of this model to a particular HQET based shape function ansatz [2] and for illustration takingΛ ≈ 500 MeV and −λ 1 ≈ 0.3 GeV 2 yields p F ≈ 400 MeV, or a helicity suppression of ∼ 0.05.
These simple estimates should be compared to naive factorization, supplemented by the large energy form factor relations [3] (also see [4]). ForB → φK * , The coefficientã = a 3 + a 4 + a 5 − 1 2 (a 7 + a 9 + a 10 ), where the a i are the usual naive factorization coefficients, see e.g. [5], and λ q p = V pb V * pq . The large energy relations imply (2) We use the sign convention V |qγ µ q|0 = −if V m V ǫ * µ . ζ and ζ ⊥ are the B → V form factors in the large energy limit [3]. Both scale as m −3/2 b in the heavy quark limit, implying that helicity suppression inĀ − is ≈ m φ /m B which is consistent with our estimate (the form factor transition contributes 2 ζ ⊥ in A − V 1 V 2 ). r ⊥ parametrizes the form factor helicity suppression. It is given by where A 1,2 and V are the axial-vector and vector current form factors, respectively. The large energy relations imply that it vanishes at leading power, because helicity suppression is O(1/m). Light-cone QCD sum rules [6], and lattice form factor determinations scaled to low q 2 using the sum rule approach [7], give r K * ⊥ ≈ 1 − 3 %; QCD sum rules give r K * ⊥ ≈ 5 % [8]; and the BSW model gives r K * ⊥ ≈ 10% [9]. These results are consistent with our simple estimate for form factor helicity suppression.
The large energy relations giving rise to (2) are strictly valid for the soft parts of the form factors, at leading power and at leading order in α s . However, the soft form factors are not significantly Sudakov suppressed in the Soft Collinear Effective Theory (SCET) [10]. The results of [4,11] thus imply that the O(α s ) form factor contributions, particularly the symmetry breaking corrections to the large energy relations, can be neglected. In fact, r ⊥ does not receive any perturbative corrections at leading power [12,4,11]; again, this is because form factor helicity suppression is O(1/m). Furthermore, power corrections to all of the form factor relations begin at O(1/m) (rather than 1/ √ m) in SCET [13]. Therefore, the above discussion of helicity suppression in naive factorization will not be significantly modified by perturbative and power corrections to the form factors.
In the transversity basis [14] the transvese amplitudes areĀ ⊥, = ( The polarization fractions satisfy in naive factorization, where the subscript L refers to longitudinal polarization, f i = Γ i /Γ total , and f L + f ⊥ + f = 1. The measured longitudinal fractions for B → ρρ are close to 1 [15,16,17]. This is clearly not the case for B → φK * 0 , for which full angular analyses yield f L = .43 ± .09 ± .04, f ⊥ = .41 ± .10 ± .04 [18] (5) Naively averaging the Belle and BaBar measurements (without taking large correlations into account) yields f ⊥ /f = 1.4 ± .7. In the charged mode, BaBar has measured f L (φK * + ) = 0.46 ± 0.12 ± 0.03 [16]. We must go beyond naive factorization in order to determine if the small values of f L (φK * ) could be due to the dominance of QCD penguin operators in ∆S = 1 decays.

QCD factorization for B → V V decays
In QCD factorization [20] exclusive two-body decay amplitudes are given in terms of convolutions of hard scattering kernels with meson light-cone distribution amplitudes.
At leading power this leads to factorization of short and long-distance physics. This factorization breaks down at sub-leading powers with the appearance of logarithmic infrared divergences. Nevertheless, the power-counting for all amplitudes can be obtained. The extent to which it holds numerically can be checked by introducing an infrared hadronic scale cutoff, and assigning large uncertainties. Non-perturbative quantities are thus roughly estimated via single gluon exchange. In general, large uncertainties should be expected for polarization predictions, given that the transverse amplitudes begin at O(1/m). However, we will find that this is not the case for certain polarization observables, particularly after experimental constraints, e.g., total rate or total transverse rate, are imposed. Our results differ substantially from previous studies of B → V V in QCD factorization [23,24]. Of particular note is the inclusion of annihilation topologies. The complete expressions for the helicity amplitudes are lengthy and will be given in [25].
Expressions for a few contributions are included below.
In QCD factorization, the Standard Model effective Hamiltonian matrix elements can be written as [21,22] where h labels the vector meson helicity, and D = s(d) for ∆S = 1(0). T B gives rise to annihilation topoplogy amplitudes, to be discussed shortly, and where p = u, c, and V ′ The coefficients a p,h i contain contributions from naive factorization, vertex corrections, penguin contractions, and hard spectator interactions. The transition operators j i , where q is summed over u, d, s. For i=1,2; 3-6; and 7-10 they originate from the current-current Q 1,2 ; QCD penguin Q 3,..,6 ; and electroweak penguin operators Q 7,..,10 , respectively. For i = 6, 8, The c coefficients contain factors of ±1, ±1/ √ 2, arising from the vector meson flavor structures. V ′ 2 (V ′ 1 ) is the 'emission' ('form factor') vector meson, see Figure 1. The i = 6, 8 matrix elements vanish at tree-level, i.e., at leading order in α s , as local scalar current vacuum-to-vector matrix elements vanish. Due to the underlying flavor structure, the effects of a 3 -a 10 are describable in terms of a reduced set of coefficients [22] α h 3(3EW) = a h 3(9) + a h 5(7) , α p, h 4(4EW) = a p, h 4(10) + a ′p, h 6(8) , where a ′p, h 6(8) = ia p, h 6(8) j are next-to-leading order matrix elements in α s , in which j 2 again forms the emission particle V 2 . The arguments (V 1 V 2 ) are understood throughout.
At next-to-leading order, the coefficients a (′) p,h i can be written as [22] a (′) p,h i where the upper (lower) signs apply when i is odd (even). The superscript 'p' appears for i = 4, 6, 8, 10. The N i are tree-level naive factorization coefficients ( Figure 1); at nextto-leading order the V h i account for one-loop vertex corrections, the P p,h i for penguin contractions (Figure 2), and the H h i for hard spectator interactions ( Figure 3). They are given in terms of convolutions of hard scattering kernels with vector meson and B meson light-cone distribution amplitudes. For each i, the corresponding graphs have the same quark helicity structure.
Two twist-2 light-cone distribution amplitudes φ (u) and φ ⊥ (u), and four two-particle twist-3 distributions (and their derivatives) enter the longitudinal and tranverse vector meson projections [26]. The argument u (ū ≡ 1 − u) is the quark (antiquark) light-cone momentum fraction. The two-particle twist-3 distributions can be expressed in terms of φ , ⊥ (u) via Wandura-Wilzcek type equations of motion [26], if higher Fock states are ignored. The twist-3 vector meson projections then depend on the three distributions, Φ a and Φ b project onto transversely polarized vectors in which the quark and antiquark flips helicity, respectively. Φ v (u), defined in [22], projects onto longitudinally polarized vectors in which either the quark or antiquark flips helicity. Light quark mass effects are ignored, and a discussion of twist-4 distribution amplitudes and higher Fock state effects is deferred [25]. The leading-twist distribution amplitudes are given in terms of an expansion in Gegenbauer polynomials [26,6], Our numerical results include the first two moments α 1, i , α 2, i . The asymptotic forms of the twist-3 distribution amplitudes are 3ū 2 , 3u 2 , and  twist-3. The quantities P i (V ) are the V P counterparts defined in [22], and f ⊥ V is the scale-dependent tensor-current decay constant. The transverse penguin contractions are P ±, p 6, 8 = 0 to twist-4, and at twist-3, where and G(s, x) is the well known penguin function, see e.g., [21]. The penguin contractions account for approximately 30% and 20% of the magnitudes of α c, 0 4 and α c − 4 (for default input parameters), respectively, before including the hard spectator interactions.
The dipole operators Q 8g , Q 7γ do not contribute to the transverse penguin amplitudes at O(α s ) due to angular momentum conservation: the dipole tensor current couples to a transverse gluon, but a 'helicity-flip' for q orq in Figure 2 would require a longitudinal gluon coupling. Formally, this result follows from the Wandura-Wilczek relations and the large energy relations between the tensor-current and vector-current form factors [25]. For example, the integrand of the convolution integral for P − 4 vanishes identically, Note that transverse amplitudes in which a vector meson contains a collinear higher Fock state gluon also vanish at O(α s ) . This can be seen from the vanishing of the corresponding partonic dipole operator graphs in the same momentum configurations. Transverse O(α 2 s ) spectator interaction contributions are highly suppressed and are studied in [25].
The hard spectator interaction quantities H h i contain logarithmically divergent integrals beyond twist-2, corresponding to the soft spectator limit in V 1 , see e.g., Figure  3. We integrate the quark light-cone momentum fraction in V 1 over the range [0, 1 − ε], and replace the divergent quantities ln ε with complex parameters X H . As in [21,22], these are modeled as arises via a twist-4 V 2 × twist-2 V 1 projection. The basic building blocks for annihilation are matrix elements of the operators , respectively, see e.g., Figure 3. The first quark bilinear corresponds to theB meson, the superscript i (f ) indicates a gluon attached to the initial (final) state quarks in the weak vertex, and by convention V 2 (V 1 ) contains a quark (antiquark) from the weak vertex. A f,0 3 and A f,− 3 dominate the ∆S = 1 QCD penguin annihilation amplitudes. The latter are expressed as The arguments (V 1 V 2 ) have been suppressed. The c coefficients are again determined by the vector meson flavor structures. For the electroweak penguin annihilation amplitude b h 3 EW , substitute C 3,4,5,6 → C 9,10,7,8 , respectively. A f,0 and change the sign of the second term. The integrals over u and v are logarithmically divergent, corresponding to the soft gluon limitū, v → 0. For simplicity, the asymptotic distribution amplitudes are used, as in [21,22]; non-asymptotic SU(3) F violating effects will be discussed shortly. The logarithmic divergences are again replaced with complex parameters, increase by more than an order of magnitude. A summary of power counting at next-to-leading order is given in [25,27]. As expected, each quark 'helicity-flip' costs 1/m in association with either one unit of twist, or form factor suppression. A ±1 change in vector meson helicity due to a collinear gluon in a higher Fock state also costs one unit of twist, or 1/m. In addition, annihilaton graphs receive an overall 1/m suppression. An apparent exception is provided by the (twist-3) 2 contributions to A i, − 1,2 ; they contain a linear infrared divergence which would break the power counting. (A i, − 1,2 would be promoted to O(1/m 2 ) but would remain numerically small, as can been by parametrizing the divergence as (m B /Λ h )κe iϕ with, e.g., κ < ∼ 3). However, the divergence should be canceled by twist-4×twist-2 effects, see below. Regardless, (4) remains formally true in QCD factorization. The first relation in (4) has also been confirmed recently in SCET [28]. We expect that the power counting obtained in QCD factorization will be reproduced for all corresponding graphs in SCET.
Amplitudes involving twist-4 vector meson projections remain to be explicitly evaluated [25]. Twist-4×twist-2 projections give rise to H + 6,8 . However, these effects should be similar in magnitude to (twist-3) 2 contributions to the positive helicity hard spectator amplitudes, which were found to be small. The twist-4×twist-2 contributions to A f ± 1,2 must cancel the non-vanishing (twist-3) 2 contributions, since A f ± 1,2 must vanish by equations of motion. This condition leads to new Wandura-Wilczek type relations between the products of twist-4×twist-2 and (twist-3) 2 light-cone distribution amplitudes. These relations should insure cancelation of the aforementioned linear divergence in A i− 1,2 by twist-4×twist-2 effects [25]. Finally, twist-4×twist-3 projections give rise to A i, + 3 and A f, + 3 ; however, these amplitudes should be both formally and numerically suppressed by O(1/m 2 ) compared to A i, − 3 and A f, − 3 , respectively. We have also not explicitly considered graphs in which higher twist two-body vector meson projections are replaced with higher Fock-state projections of same twist containing collinear gluons, e.g.,qqg. The latter are expected to receive additional suppression at each twist, e.g., 20% [29]. These corrections, especially the tree-level twist-3 contributions to the coefficients α (p), − i , should be included [25]. However, they will not alter our conclusions, given the large uncertainties that have already been assigned to the power corrections.
Expressions for a fewB → V V amplitudes are given below, where i have been suppressed, but it is understood that they are to be identified with the subscripts (V 1 V 2 ) of the prefactors A h V 1 V 2 . The new annihilation coefficients b h i , and amplitudes for other decays are given in [25]. analogous electroweak annihilation coefficients.
Averaging the Belle and BaBarB → φK * 0 (CP-averaged) measurements [18,19,16]  to the QCD annihilation amplitudes can be O(1) numerically even though they are formally O(1/m 2 ). This can be traced to the quadratic dependence on the divergences (X 2 A ) and the large coefficient N c C 6 in b f, h 3 . The quantities X 0 A and X − A , as well as the renormalization scales and form factors enteringĀ 0 andĀ − are, a-priori, unrelated. Figure 4 therefore implies that the measurements of Br L and Br T can easily be accounted for simultaneously. According to Figure 4, f L (φK * − ) ≈ 50% can also be accounted for given that the φK * − and φK * 0 amplitudes only differ by a small current-current operator annihilation graph.

A test for right-handed currents
In Figure 5 (left) the predicted ranges for f ⊥ /f and Br T are studied simultaneaously forB → φK * 0 in the Standard Model. The 'default' curve is again obtained by varying ̺ − A in the range [0, 1], keeping all other inputs at their default values, and the error bands are obtained by adding uncertainties in quadrature as in Figure 4. Evidently, the second relation in (4) holds at next-to-leading order, particularly at larger values of Br T where QCD annihilation dominates Br ⊥ and Br . We also plot the maximum values attained for f ⊥ /f under simultaneous variation of all inputs. The result is sensitive to r ⊥ , as it largely determines the relative signs and magnitudes of the 'form factor' terms inĀ − andĀ + , see (2). The thick black curve (corresponding to Br max T in Figure 4) and blue curve give maxima for r ⊥ ≥ 0, in accord with existing model determinations, and r ⊥ ≥ −.10, respectively. A ratio in excess of the Standard Model range, e.g., f ⊥ /f > 1.5 if r ⊥ > 0, would signal the presence of new right-handed currents. We mention that nonvanishing CP-violating triple products in pure penguin decays likeB → φK * would not be a signal for right-handed currents if significant strong phase differences ( = 0 mod π) existed betweenĀ 0, andĀ ⊥ [33,34]. There is some experimental indication for such phase differences [19], which is to be expected if annihilation amplitudes are important.
Right-handed currents are conventionally associated with effective operatorsQ i , obtained from the Standard Model operators Q i by interchanging V −A ↔ V + A. The final states in A 0, (A ⊥ ) are parity-even (parity-odd), so that the i'th pair of Wilson coefficients enters as [35] The different combinations allow for large modifications to (4). f L suffers from prohibitively large theoretical uncertainties. However, f ⊥ /f is a much cleaner observable.

Dipole operators versus four-quark operators
The suppression of dipole operator effects in the transverse modes has important implications for New Physics searches. For example, in pure penguin decays to CPconjugate final states f , e.g.,B → φ (K * 0 → K s π 0 ), if the transversity basis timedependent CP asymmetry parameters (S f ) ⊥ and (S f ) are consistent with (sin 2β) J/ψKs , and (S f ) 0 is not, then this would signal new CP violating contributions to the chromomagnetic dipole operators. However, deviations in (S f ) ⊥ or (S f ) would signal new CP violating four-quark operator contributions. If the triple-products A 0 T and A T [33,34] do not vanish and vanish, respectively, in pure penguin decays, then this would also signal new CP violating contributions to the chromomagnetic dipole operators. (This assumes that a significant strong phase difference is measured betweenĀ andĀ ⊥ .) However, non-vanishing A T , or non-vanishing transverse direct CP asymmetries would signal the intervention of four-quark operators. The above would help to discriminate between different explanations for an anomalous time-dependent CP asymmetry in B → φK s , i.e., S φKs , which fall broadly into two categories: radiatively generated dipole operators, e.g., supersymmetric loops; or tree-level four-quark operators, e.g., flavor changing (leptophobic) Z ′ exchange [36], R-parity violating couplings [37], or color-octet exchange [38]. Finally, a large f ⊥ /f would be a signal for right-handed four-quark operators.

SU(3) F violation and B → ρK *
We have seen that the large transverse φK * polarization can be accounted for in the Standard Model via the QCD penguin annihilation graphs. Would this necessarily imply large transverse ρK * polarizations? To answer this question we need to address SU(3) F flavor symmetry breaking in annihilation. For simplicity, we have thus far estimated the annihilation amplitudes using asymptotic light-cone distribution amplitudes [20,22]. However, for light mesons containing a single strange quark, non-asymptotic effects should shift the weighting of the distribution amplitudes towards larger strange quark momenta. SU(3) F violation in processes involving ss popping versus light quark popping, e.g., annihilation, can therefore be much larger than the canonical 20%, due to the appearance of inverse moments of the distribution amplitudes [39]. This can account for the order of magnitude hierarchy between theB → D 0 π 0 andB → D + s K − rates [39]. Similar considerations may also explain the O(1) flavor violation empirically observed in high energy e + e − fragmentation, e.g., in kaon versus pion multiplicities, or K * versus ρ multiplicities at the Z. In particular, the relative probability for ss popping versus uū or dd popping in JETSET fragmentation Monte Carlo's must be tuned to ≈ .3 [40].
The dominant B → V V QCD annihilation amplitudes A f, − 3 and A f, 0 3 involve products of inverse moments, see (17). The SU(3) F violation discussed above can be estimated by including the second and third terms in the Gegenbauer expansions for the distribution amplitudes. The first Gegenbauer moments α 1, ⊥ , α 1, determine the asymmetries of the corresponding leading-twist distribution amplitudes, i.e., the inverse moments of φ ⊥ are given by ū −1 ⊥ , u −1 ⊥ = 3(1 ± α 1,⊥ + α 2,⊥ ), and similarly for φ . Note that the first moments vanish for the symmetric φ and ρ mesons. For illustration, two sets of intervals for ϕ A = 0, for full (green), halved (blue) and default (solid line) ranges of Gegenbauer moments, see text.

Conclusion
We have presented an analysis of polarization in B decays to light vector meson pairs beyond naive factorization, using QCD factorization. Formally, the longitudinal polarization satisfies 1 − f L = O(1/m 2 ), as in naive factorization. However, we saw that the contributions of a particular QCD penguin annihilation graph which is formally O(1/m 2 ) can be O(1) numerically in longitudinal and negative helicity ∆S = 1B decays. Consequently, the observation of f L (φK * 0,− ) ≈ 50% can be accounted for in the Standard Model, with large theoretical errors. However, f L (ρ + ρ 0 ) and f L (ρ + ρ − ) are predicted to be close to 1 with small theoretical errors, in agreement with observation. We have shown that the ratio of transverse rates in the transversity basis satisfies Γ ⊥ /Γ = 1 + O(1/m), in agreement with naive power counting. A ratio in excess of the predicted Standard Model range would signal the presence of new right-handed currents in dimension-6 fourquark operators. The maximum ratio attainable in the Standard Model is sensitive to the B → V form factor combination (1 + m V /m B )A 1 − (1 − m V /m B )V or r ⊥ , see (3), which controls helicity suppression in form factor transitions. Existing model determinations give a positive sign for r ⊥ , which would imply Γ ⊥ (φK * )/Γ (φK * ) < 1.5 in the Standard Model. However, the maximum would increase for negative values. The magnitude and especially the sign of r ⊥ is an important issue which needs to be clarified with dedicated lattice studies.
The contributions of the b → sg dipole operators to the transverse modes were found to be highly suppressed, due to angular momentum conservation. Comparison of CP-violation involving the longitudinal modes with CP-violation only involving the transverse modes, in pure penguin ∆S = 1 decays, could therefore distinguish between new contributions to the dipole and four-quark operators. More broadly, this could distinguish between scenarios in which New Physics effects are loop induced and scenarios in which they are tree-level induced, as it is difficult to obtain O(1) CP-violating effects from dimension-6 operators beyond tree-level.
We have seen that the asymmetry of the K ( * ) meson light-cone distributions generically leads to O(1) SU(3) F flavor symmetry violation in annihilation amplitudes, as pointed out in [39]. In particular, ss popping can be substantially suppressed relative to light quark popping. This implies that the longitudinal polarizations should satisfy f L (ρ ± K * 0 ) < ∼ f L (φK * ) in the Standard Model. Consequently, f L (ρ ± K * 0 ) ≈ 1 would indicate that U-spin violating New Physics entering mainly in the b → sss channel is at least partially responsible for the small values of f L (φK * ). One possibility would be right-handed vector currents; they could interfere constructively (destructively) in the perpendicular (longitudinal and parallel) transversity amplitudes. Alternatively, a parity symmetric realization would only affect, and increase the perpendicular amplitude [35]. Either case would lead to Γ ⊥ > Γ , and could thus be ruled out. A more exotic possibility is tensor currents; they would contribute to the longitudinal and transverse amplitudes at subleading and leading power, respectively. If left-handed, i.e., of the form sσ µν (1 + γ 5 )bsσ µν (1 ± γ 5 )s, then Γ ⊥ ≈ Γ would be maintained. Finally, O(1) SU(3) F violation is possible in all QCD penguin amplitudes, given that the annihilation topology components can be comparable to, or greater than the penguin topology components. This is especially true of decays to V V and V P final states which, unlike decays to P P final states, do not receive large contributions from (S − P )(S + P ) chirality penguin topology matrix elements. Certain applications of SU(3) F symmetry in B decays should therefore be reexamined.