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Factorizable cc̄ contribution to the radiative B→K∗γ decay
6 September 2001
Physics Letters B 516 (2001) 61–64 www.elsevier.com/locate/npe
Factorizable cc¯ contribution to the radiative B → K ∗γ decay Dmitri Melikhov 1 Institut für Theoretische Physik, Universität Heidelberg, Philosophenweg 16, D-69120 Heidelberg, Germany Received 17 April 2001; received in revised form 2 July 2001; accepted 24 July 2001 Editor: P.V. Landshoff
Abstract The sum of the factorizable long- and short-distance cc¯ contributions to the B → K ∗ γ amplitude vanishes as a consequence of the gauge invariance. Moreover, the short- and long-distance contributions vanish, separately, if defined in a gauge-invariant way. In view of this, we dispute a popular estimate of the long-distance effects in the B → K ∗ γ decay in terms of the B → K ∗ semileptonic form factors and the decay constants of the cc¯ resonances. We show that a nonzero value obtained within this approach is an artifact of using a gauge-dependent prescription for the resonance contribution extended to q 2 = 0. 2001 Elsevier Science B.V. All rights reserved.
The understanding of the long-distance effects in rare B-decays is an important theoretical problem. In the exclusive radiative decay B → K ∗ γ there are two types of such effects: (i) contributions due to the penguin operators, and (ii) contributions due to the four-quark operators in the effective Hamiltonian (for details see [1], and references therein). We are going to discuss one of the effects induced by the 4-quark operators, namely, contributions of the intermediate cc¯ states to the B → K ∗ γ amplitude ¯ ν (1 − γ5 )c|B . A ∼ K ∗ γ |¯s γν (1 − γ5 )b · cγ
(1)
The cc¯ contribution can be represented as the sum of the nonperturbative (long-distance) contribution of the resonances (ψ, ψ , etc.) and the perturbativelycalculable (short-distance) contribution of the cc¯ continuum. E-mail address: [email protected] (D. Melikhov). 1 On leave from Nuclear Physics Institute, Moscow State University, 119899, Moscow, Russia.
The cc¯ resonances dominate the B → (K, K ∗ )l + l − amplitude in the resonance region of q 2 , q the lepton pair momentum. Various models [2–4] describe the cc¯ contribution as function of q 2 based on the factorization [5]. A challenging problem is to describe the longdistance effects in the B → K ∗ γ decay which occurs at q 2 = 0. A popular method to estimate the size of such effects is based on extending the factorizable resonance contribution to q 2 = 0. In this way, an estimate for the long-distance effects in the radiative B → K ∗ γ decays in terms of the semileptonic B → K ∗ form factors and the leptonic decay constants of the resonances has been obtained [1,2]. We show, however, that a nonzero value obtained within this approach is an artifact of using the gaugedependent prescription for the resonance contribution. In fact, gauge invariance requires the sum of the long- and short-distance factorizable cc¯ contributions to vanish at q 2 = 0. Each of these contributions vanish also separately, if defined in a gauge-invariant way.
0370-2693/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 ( 0 1 ) 0 0 9 3 1 - 5
62
D. Melikhov / Physics Letters B 516 (2001) 61–64
Let us recall the analysis of Refs. [1,2]. In the factorization approximation, the amplitude (1) takes the form ¯ ν (1 − γ5 )c|0 . (2) Afact ∼ K ∗ |¯s γν (1 − γ5 )b|B γ |cγ The B → K ∗ amplitude in this expression is given in terms of the known B → K ∗ weak transition form factors. The photon amplitude can be written as follows cc¯ (q), γ |cγ ¯ ν (1 − γ5 )c|0 = Qc µ (q)Πµν
(3)
where Qc is the electric charge of the charm quark and cc¯ Πµν (q) = i dx eiqx 0|T (cγ ¯ µ c(x), cγ ¯ ν c(0))|0 (4) is the charm contribution to the vacuum polarization. cc¯ contains poles at q 2 = M 2 , where The function Πµν n Mn is the mass of the cc¯ vector resonance (ψn = ψ, ψ , . . .). Neglecting the resonance widths we obtain in the region q 2 Mn2 qµ qν 1 cc¯ Πµν (q) = −Mn2 fn2 gµν − Mn2 Mn2 − q 2 + regular terms, (5) where fn is the leptonic decay constant of the vector resonance defined as follows 0|cγ ¯ µ c|ψn = µ(n) fn Mn .
(6)
Then one calculates the individual resonance contribution to the factorized B → K ∗ γ amplitude by extending Eq. (5) to q 2 = 0, takes the sum over all cc¯ resonances and obtains in this way Afact LD [1]: Afact B → K ∗ ψn → K ∗ γ Afact n LD = n
∼
2 n fn
∗µ
∗ν ∗ MB + MK ∗ (γ ) (K ) × iµναβ q α P β V (0) + gµν A1 (0)(MB + MK ∗ )2 + Pµ qν A2 (0) ,
(7)
where P = pB + pK ∗ , q = pB − pK ∗ , the polarization vectors. V (0), A1 (0), and A2 (0) are the B → K ∗ weak form factors at q 2 = 0 (see definition in [6]). Following [1,2] Afact LD is expected to describe the long-distance contribution to the amplitude of the radiative decay.
Clearly, this amplitude (as well as the contribution of an individual resonance) is not gauge-invariant if the form factors do not satisfy the relation A1 (0) = −
MB − MK ∗ A2 (0). MB + MK ∗
(8)
Ref. [1] gives arguments that this relation is satisified in the leading order of the large-energy limit. However, this does not help, since one needs the relation (8) to be exact. But the B → K ∗ form factors have no reason to satisfy this relation precisely, and therefore both Afact n 2 and Afact LD are not gauge-invariant. This means that the amplitude Afact LD of Eq. (7) is a gauge-dependent quantity and has no clear physical interpretation. In particular, it cannot be used as an estimate of the longdistance effects in the radiative decay. Let us understand better the origin of the difficulty. We need to take properly into account a resonance cc¯ . contribution to the gauge-invariant amplitude Πµν Clearly, this contribution is gauge-invariant near the pole where the resonance dominates the amplitude. Far from the pole, however, one can describe the individual resonance contribution in different ways: it is one of many regular contributions to the amplitude. As one of the possibilities, the contribution of a resonance far from the pole can be defined in a gaugedependent way — and that is what happened in the example above. Obviously, this is allowed: nothing prevents us from splitting the gauge-invariant quantity into many gauge-dependent parts. The only requirement is that the full quantity — in our example the sum of the resonance and continuum cc¯ states — is gauge invariant. Working with such gauge-dependent parts is however inconvenient and can lead to a confusion in the interpretation of the results. Much better way is to define the contribution of the individual resonance in an explicitly gauge-invariant way. The conservation of the charm vector current cc¯ such ¯ µ c) = 0 leads to the transversity of Πµν ∂µ (cγ that it takes the form cc¯ Πµν (q) = gµν − qµ qν /q 2 Π cc¯ q 2 .
(9)
2 This becomes even more obvious when nonzero q 2 are
considered: in this case transversity of the resonance amplitude requires exact relations between the form factors valid for all q 2 . Clearly, the form factors do not satisfy such relations.
D. Melikhov / Physics Letters B 516 (2001) 61–64
The invariant amplitude Π cc¯ (q 2 ) satisfies an exact relation Π cc¯ q 2 = 0 = 0, (10) which corresponds to the nonrenormalizability of the photon mass. To take this property into account, it is convenient to work with the spectral representation for the vacuum polarization which according to (10) requires a subtraction at q 2 = 0 q2 ds Π cc¯ q 2 = (11) Im Π cc¯ (s). π (s − q 2 )s The imaginary part contains contributions of the resonances and the continuum states cc¯ Im Π cc¯ (s) = π fn2 δ s − Mn2 + Im Πcont (s). (12) n
The expressions (11) and (12) for Π cc¯ (q 2 ) lead to an explicitly gauge-invariant contribution of the individual resonance to the B → K ∗ γ ∗ (q 2 ) amplitude 3 B → K ∗ ψn → K ∗ γ ∗ A¯ fact n ∼
(fn /Mn )2 ∗µ ∗ν ∗ MB + MK ∗ (γ ) (K )
× iµναβ P α q β V q 2 q 2 + gµν q 2 − qµ qν A1 q 2 (MB + MK ∗ )2 + Pµ q 2 − qµ P q qν A2 q 2 . (13)
Let us point out that the amplitude is gauge-invariant for any form factors A1 and A2 . Eq. (13) describes the factorizable resonance contribution for any q 2 . Most 2 interesting for us is that now A¯ fact n = 0 for q = 0. So, the explanation looks as follows: The gauge invariance requires Π cc¯ (0) = 0, and as a result of this relation we find for q 2 = 0 fact (14) Afact n + Acontinuum = 0.
63
and only their sum has the physical interpretation. If we define both contributions in a gauge-invariant way as given by (13) then each of them vanishes for the real photon emission. Summarizing: • The sum of the long- and short-distance factorizable cc-contributions ¯ to the B → K ∗ γ amplitude 2 vanishes at q = 0 as a direct consequence of gauge invariance. If defined in gauge-invariant way, each of these contributions vanishes, separately. • The estimate for the long-distance cc¯ contributions in terms of the B → K ∗ semileptonic form factors and the decay constants of the cc¯ resonances obtained in [1,2] gives a nonzero result because of using a gauge-dependent prescription for the resonance contribution. • The long-distance cc-contribution ¯ to the radiative decay is completely nonfactorizable [8,9]. Therefore, it cannot be expressed in terms of the B → K ∗ transition form factors, but requires other relevant quantities. For example, in [8] the long-distance cc¯ contribution was expressed in terms of the B → K ∗ matrix element of the quark–gluon–photon operator. ¯ do Obviously, at q 2 = 0 factorizable cc-contributions not vanish. A gauge-invariant modelling of a factorizable contribution of an individual resonance applicable at any q 2 is given by Eq. (13). We discussed the B → K ∗ γ decay, but the same arguments apply to other weak radiative B → V γ decays, V the vector meson.
Acknowledgements I am grateful to O. Nachtmann and B. Stech for discussions. The work was supported by the Alexander-von-Humboldt Stifting.
n
If we do not take care about the gauge invariance, then each of these contributions, separately, is ambiguos 3 Recall that there is in addition a nonfactorizable contribution. Following [7], one can multiply the (gauge-invariant) factorizable resonance contribution given by Eq. (13) by a phenomenological constant κ to describe correctly the branching ratio BR(B → ψX → l + l − X) = BR(B → ψX) BR(ψ → l + l − ).
References [1] B. Grinstein, D. Pirjol, Phys. Rev. D 62 (2000) 093002. [2] N.G. Deshpande, J. Trampetic, K. Panose, Phys. Lett. B 214 (1988) 467. [3] F. Krüger, L.M. Sehgal, Phys. Lett. B 380 (1996) 199. [4] M. Ahmady, Phys. Rev. D 53 (1996) 2843. [5] M. Neubert, B. Stech, in: A. Buras, M. Lindner (Eds.), Heavy Flavours II, World Scientific, Singapore, hep-ph/9705292.
64
D. Melikhov / Physics Letters B 516 (2001) 61–64
[6] D. Melikhov, N. Nikitin, S. Simula, Phys. Rev. D 57 (1998) 6814; D. Melikhov, B. Stech, Phys. Rev. D 62 (2000) 014006, and references therein. [7] A. Ali, T. Mannel, T. Morozumi, Phys. Lett. B 273 (1991) 505.
[8] A. Khodjamirian, R. Rückl, G. Stoll, D. Wyler, Phys. Lett. B 402 (1997) 167. [9] D. Melikhov, N. Nikitin, S. Simula, Phys. Lett. B 430 (1998) 332.
Physics Letters B 516 (2001) 61–64 www.elsevier.com/locate/npe
Factorizable cc¯ contribution to the radiative B → K ∗γ decay Dmitri Melikhov 1 Institut für Theoretische Physik, Universität Heidelberg, Philosophenweg 16, D-69120 Heidelberg, Germany Received 17 April 2001; received in revised form 2 July 2001; accepted 24 July 2001 Editor: P.V. Landshoff
Abstract The sum of the factorizable long- and short-distance cc¯ contributions to the B → K ∗ γ amplitude vanishes as a consequence of the gauge invariance. Moreover, the short- and long-distance contributions vanish, separately, if defined in a gauge-invariant way. In view of this, we dispute a popular estimate of the long-distance effects in the B → K ∗ γ decay in terms of the B → K ∗ semileptonic form factors and the decay constants of the cc¯ resonances. We show that a nonzero value obtained within this approach is an artifact of using a gauge-dependent prescription for the resonance contribution extended to q 2 = 0. 2001 Elsevier Science B.V. All rights reserved.
The understanding of the long-distance effects in rare B-decays is an important theoretical problem. In the exclusive radiative decay B → K ∗ γ there are two types of such effects: (i) contributions due to the penguin operators, and (ii) contributions due to the four-quark operators in the effective Hamiltonian (for details see [1], and references therein). We are going to discuss one of the effects induced by the 4-quark operators, namely, contributions of the intermediate cc¯ states to the B → K ∗ γ amplitude ¯ ν (1 − γ5 )c|B . A ∼ K ∗ γ |¯s γν (1 − γ5 )b · cγ
(1)
The cc¯ contribution can be represented as the sum of the nonperturbative (long-distance) contribution of the resonances (ψ, ψ , etc.) and the perturbativelycalculable (short-distance) contribution of the cc¯ continuum. E-mail address: [email protected] (D. Melikhov). 1 On leave from Nuclear Physics Institute, Moscow State University, 119899, Moscow, Russia.
The cc¯ resonances dominate the B → (K, K ∗ )l + l − amplitude in the resonance region of q 2 , q the lepton pair momentum. Various models [2–4] describe the cc¯ contribution as function of q 2 based on the factorization [5]. A challenging problem is to describe the longdistance effects in the B → K ∗ γ decay which occurs at q 2 = 0. A popular method to estimate the size of such effects is based on extending the factorizable resonance contribution to q 2 = 0. In this way, an estimate for the long-distance effects in the radiative B → K ∗ γ decays in terms of the semileptonic B → K ∗ form factors and the leptonic decay constants of the resonances has been obtained [1,2]. We show, however, that a nonzero value obtained within this approach is an artifact of using the gaugedependent prescription for the resonance contribution. In fact, gauge invariance requires the sum of the long- and short-distance factorizable cc¯ contributions to vanish at q 2 = 0. Each of these contributions vanish also separately, if defined in a gauge-invariant way.
0370-2693/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 ( 0 1 ) 0 0 9 3 1 - 5
62
D. Melikhov / Physics Letters B 516 (2001) 61–64
Let us recall the analysis of Refs. [1,2]. In the factorization approximation, the amplitude (1) takes the form ¯ ν (1 − γ5 )c|0 . (2) Afact ∼ K ∗ |¯s γν (1 − γ5 )b|B γ |cγ The B → K ∗ amplitude in this expression is given in terms of the known B → K ∗ weak transition form factors. The photon amplitude can be written as follows cc¯ (q), γ |cγ ¯ ν (1 − γ5 )c|0 = Qc µ (q)Πµν
(3)
where Qc is the electric charge of the charm quark and cc¯ Πµν (q) = i dx eiqx 0|T (cγ ¯ µ c(x), cγ ¯ ν c(0))|0 (4) is the charm contribution to the vacuum polarization. cc¯ contains poles at q 2 = M 2 , where The function Πµν n Mn is the mass of the cc¯ vector resonance (ψn = ψ, ψ , . . .). Neglecting the resonance widths we obtain in the region q 2 Mn2 qµ qν 1 cc¯ Πµν (q) = −Mn2 fn2 gµν − Mn2 Mn2 − q 2 + regular terms, (5) where fn is the leptonic decay constant of the vector resonance defined as follows 0|cγ ¯ µ c|ψn = µ(n) fn Mn .
(6)
Then one calculates the individual resonance contribution to the factorized B → K ∗ γ amplitude by extending Eq. (5) to q 2 = 0, takes the sum over all cc¯ resonances and obtains in this way Afact LD [1]: Afact B → K ∗ ψn → K ∗ γ Afact n LD = n
∼
2 n fn
∗µ
∗ν ∗ MB + MK ∗ (γ ) (K ) × iµναβ q α P β V (0) + gµν A1 (0)(MB + MK ∗ )2 + Pµ qν A2 (0) ,
(7)
where P = pB + pK ∗ , q = pB − pK ∗ , the polarization vectors. V (0), A1 (0), and A2 (0) are the B → K ∗ weak form factors at q 2 = 0 (see definition in [6]). Following [1,2] Afact LD is expected to describe the long-distance contribution to the amplitude of the radiative decay.
Clearly, this amplitude (as well as the contribution of an individual resonance) is not gauge-invariant if the form factors do not satisfy the relation A1 (0) = −
MB − MK ∗ A2 (0). MB + MK ∗
(8)
Ref. [1] gives arguments that this relation is satisified in the leading order of the large-energy limit. However, this does not help, since one needs the relation (8) to be exact. But the B → K ∗ form factors have no reason to satisfy this relation precisely, and therefore both Afact n 2 and Afact LD are not gauge-invariant. This means that the amplitude Afact LD of Eq. (7) is a gauge-dependent quantity and has no clear physical interpretation. In particular, it cannot be used as an estimate of the longdistance effects in the radiative decay. Let us understand better the origin of the difficulty. We need to take properly into account a resonance cc¯ . contribution to the gauge-invariant amplitude Πµν Clearly, this contribution is gauge-invariant near the pole where the resonance dominates the amplitude. Far from the pole, however, one can describe the individual resonance contribution in different ways: it is one of many regular contributions to the amplitude. As one of the possibilities, the contribution of a resonance far from the pole can be defined in a gaugedependent way — and that is what happened in the example above. Obviously, this is allowed: nothing prevents us from splitting the gauge-invariant quantity into many gauge-dependent parts. The only requirement is that the full quantity — in our example the sum of the resonance and continuum cc¯ states — is gauge invariant. Working with such gauge-dependent parts is however inconvenient and can lead to a confusion in the interpretation of the results. Much better way is to define the contribution of the individual resonance in an explicitly gauge-invariant way. The conservation of the charm vector current cc¯ such ¯ µ c) = 0 leads to the transversity of Πµν ∂µ (cγ that it takes the form cc¯ Πµν (q) = gµν − qµ qν /q 2 Π cc¯ q 2 .
(9)
2 This becomes even more obvious when nonzero q 2 are
considered: in this case transversity of the resonance amplitude requires exact relations between the form factors valid for all q 2 . Clearly, the form factors do not satisfy such relations.
D. Melikhov / Physics Letters B 516 (2001) 61–64
The invariant amplitude Π cc¯ (q 2 ) satisfies an exact relation Π cc¯ q 2 = 0 = 0, (10) which corresponds to the nonrenormalizability of the photon mass. To take this property into account, it is convenient to work with the spectral representation for the vacuum polarization which according to (10) requires a subtraction at q 2 = 0 q2 ds Π cc¯ q 2 = (11) Im Π cc¯ (s). π (s − q 2 )s The imaginary part contains contributions of the resonances and the continuum states cc¯ Im Π cc¯ (s) = π fn2 δ s − Mn2 + Im Πcont (s). (12) n
The expressions (11) and (12) for Π cc¯ (q 2 ) lead to an explicitly gauge-invariant contribution of the individual resonance to the B → K ∗ γ ∗ (q 2 ) amplitude 3 B → K ∗ ψn → K ∗ γ ∗ A¯ fact n ∼
(fn /Mn )2 ∗µ ∗ν ∗ MB + MK ∗ (γ ) (K )
× iµναβ P α q β V q 2 q 2 + gµν q 2 − qµ qν A1 q 2 (MB + MK ∗ )2 + Pµ q 2 − qµ P q qν A2 q 2 . (13)
Let us point out that the amplitude is gauge-invariant for any form factors A1 and A2 . Eq. (13) describes the factorizable resonance contribution for any q 2 . Most 2 interesting for us is that now A¯ fact n = 0 for q = 0. So, the explanation looks as follows: The gauge invariance requires Π cc¯ (0) = 0, and as a result of this relation we find for q 2 = 0 fact (14) Afact n + Acontinuum = 0.
63
and only their sum has the physical interpretation. If we define both contributions in a gauge-invariant way as given by (13) then each of them vanishes for the real photon emission. Summarizing: • The sum of the long- and short-distance factorizable cc-contributions ¯ to the B → K ∗ γ amplitude 2 vanishes at q = 0 as a direct consequence of gauge invariance. If defined in gauge-invariant way, each of these contributions vanishes, separately. • The estimate for the long-distance cc¯ contributions in terms of the B → K ∗ semileptonic form factors and the decay constants of the cc¯ resonances obtained in [1,2] gives a nonzero result because of using a gauge-dependent prescription for the resonance contribution. • The long-distance cc-contribution ¯ to the radiative decay is completely nonfactorizable [8,9]. Therefore, it cannot be expressed in terms of the B → K ∗ transition form factors, but requires other relevant quantities. For example, in [8] the long-distance cc¯ contribution was expressed in terms of the B → K ∗ matrix element of the quark–gluon–photon operator. ¯ do Obviously, at q 2 = 0 factorizable cc-contributions not vanish. A gauge-invariant modelling of a factorizable contribution of an individual resonance applicable at any q 2 is given by Eq. (13). We discussed the B → K ∗ γ decay, but the same arguments apply to other weak radiative B → V γ decays, V the vector meson.
Acknowledgements I am grateful to O. Nachtmann and B. Stech for discussions. The work was supported by the Alexander-von-Humboldt Stifting.
n
If we do not take care about the gauge invariance, then each of these contributions, separately, is ambiguos 3 Recall that there is in addition a nonfactorizable contribution. Following [7], one can multiply the (gauge-invariant) factorizable resonance contribution given by Eq. (13) by a phenomenological constant κ to describe correctly the branching ratio BR(B → ψX → l + l − X) = BR(B → ψX) BR(ψ → l + l − ).
References [1] B. Grinstein, D. Pirjol, Phys. Rev. D 62 (2000) 093002. [2] N.G. Deshpande, J. Trampetic, K. Panose, Phys. Lett. B 214 (1988) 467. [3] F. Krüger, L.M. Sehgal, Phys. Lett. B 380 (1996) 199. [4] M. Ahmady, Phys. Rev. D 53 (1996) 2843. [5] M. Neubert, B. Stech, in: A. Buras, M. Lindner (Eds.), Heavy Flavours II, World Scientific, Singapore, hep-ph/9705292.
64
D. Melikhov / Physics Letters B 516 (2001) 61–64
[6] D. Melikhov, N. Nikitin, S. Simula, Phys. Rev. D 57 (1998) 6814; D. Melikhov, B. Stech, Phys. Rev. D 62 (2000) 014006, and references therein. [7] A. Ali, T. Mannel, T. Morozumi, Phys. Lett. B 273 (1991) 505.
[8] A. Khodjamirian, R. Rückl, G. Stoll, D. Wyler, Phys. Lett. B 402 (1997) 167. [9] D. Melikhov, N. Nikitin, S. Simula, Phys. Lett. B 430 (1998) 332.