Very rare, exclusive radiative decays of W and Z bosons in QCD - PowerPoint PPT Presentation

The factorization formula Which of the Dirac structures Γ i contributes, depends on the type of meson and there is exactly one Dirac structure for a given meson. We denote the corresponding Wilson coefficient by C M ( t , µ ) and define the Fourier-transformed Wilson coefficient, called the hard function, as: � dt C M ( t , µ ) e ixt ¯ n · k H M ( x , µ ) = Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

The factorization formula Which of the Dirac structures Γ i contributes, depends on the type of meson and there is exactly one Dirac structure for a given meson. We denote the corresponding Wilson coefficient by C M ( t , µ ) and define the Fourier-transformed Wilson coefficient, called the hard function, as: � dt C M ( t , µ ) e ixt ¯ n · k H M ( x , µ ) = The factorization formula now reads: 1 power � A = − if M E dx H M ( x , µ ) φ M ( x , µ ) + corrections 0 Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

The factorization formula Define : Projectors M M , can be applied to partonic amplitudes directly. In a practical calculation each Feynman diagram gives an expression of the form: ¯ u ( k 1 ) A ( q , k 1 , k 2 ) v ( k 2 ) = Tr [ v ( k 2 )¯ u ( k 1 ) A ( q , k 1 , k 2 )] Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

The factorization formula Define : Projectors M M , can be applied to partonic amplitudes directly. In a practical calculation each Feynman diagram gives an expression of the form: ¯ u ( k 1 ) A ( q , k 1 , k 2 ) v ( k 2 ) = Tr [ v ( k 2 )¯ u ( k 1 ) A ( q , k 1 , k 2 )] The projection is then: 1 � ¯ u ( k 1 ) A ( q , k 1 , k 2 ) v ( k 2 ) → dx Tr [ M M ( k , x , µ ) A ( q , k 1 , k 2 )] 0 The projector M M depends on the type of meson (pseudoscalar, vector meson [longitudinal/tranverse polarization]). Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

The factorization formula For a pseudoscalar meson, the projector to twist-3-order is given by: M P ( k , x , µ ) = if P � � / k γ 5 φ P ( x , µ ) − µ P ( µ ) γ 5 φ p ( x , µ ) 4 k µ ¯ n ν φ ′ σ ( x , µ ) + i σ µν k µ φ σ ( x µ ) ∂ � � − i σ µν + 3-part. k · ¯ n 6 6 ∂ k ⊥ ν where φ p ( x , µ ) = 1 φ σ ( x , µ ) = 6 x (1 − x ) when three-particle LCDAs are neglected (Wandzura-Wilczek approximation). [Wandzura, Wilczek ( 1977 ), Phys. Lett. B 72, 195] Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

QCD factorization Light cone distributions for mesons Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

Gegenbauer expansion of the LCDAs The LCDA can be interpreted as the amplitude for finding a quark with longitudinal momentum fraction x Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

Gegenbauer expansion of the LCDAs The LCDA can be interpreted as the amplitude for finding a quark with longitudinal momentum fraction x Defined by local matrix element (here example for pseudo-scalar) 1 n ) / n ¯ � 2 γ 5 [ t ¯ dx e ixt ¯ n · k φ M ( x , µ ) � P ( k ) | ¯ q ( t ¯ n , 0] q (0) | 0 � = − if M E 0 Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

Gegenbauer expansion of the LCDAs The LCDA can be interpreted as the amplitude for finding a quark with longitudinal momentum fraction x Defined by local matrix element (here example for pseudo-scalar) 1 n ) / n ¯ � 2 γ 5 [ t ¯ dx e ixt ¯ n · k φ M ( x , µ ) � P ( k ) | ¯ q ( t ¯ n , 0] q (0) | 0 � = − if M E 0 For light mesons information about the LCDAs has to be extracted from lattice QCD or sum rules. For mesons containing a heavy quark (or for heavy quarkonia), this can be addressed with HQET (or NRQCD). Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

Gegenbauer expansion of the LCDAs We expand the LCDAs in the basis of Gegenbauer polynomials: ∞ � � � a M n ( µ ) C (3 / 2) φ M ( x , µ ) = 6 x (1 − x ) 1 + (2 x − 1) n n =1 where C ( α ) ( x ) are the Gegenbauer polynomials. The scale-dependence of n the LCDA is in the Gegenbauer moments a M n ( µ ) Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

Gegenbauer expansion of the LCDAs We expand the LCDAs in the basis of Gegenbauer polynomials: ∞ � � � a M n ( µ ) C (3 / 2) φ M ( x , µ ) = 6 x (1 − x ) 1 + (2 x − 1) n n =1 where C ( α ) ( x ) are the Gegenbauer polynomials. The scale-dependence of n the LCDA is in the Gegenbauer moments a M n ( µ ) We need φ at the scale µ ∼ M Z while the a M n ( µ ) are obtained at µ ∼ Λ QCD Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

Gegenbauer expansion of the LCDAs We expand the LCDAs in the basis of Gegenbauer polynomials: ∞ � � � a M n ( µ ) C (3 / 2) φ M ( x , µ ) = 6 x (1 − x ) 1 + (2 x − 1) n n =1 where C ( α ) ( x ) are the Gegenbauer polynomials. The scale-dependence of n the LCDA is in the Gegenbauer moments a M n ( µ ) We need φ at the scale µ ∼ M Z while the a M n ( µ ) are obtained at µ ∼ Λ QCD → RG evolution important AND works in our favor Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

RG evolution of the LCDAs The Gegenbauer expansion yields a diagonal scale-evolution of the coefficients: � α s ( µ ) � γ n / 2 β 0 a M a M n ( µ ) = n ( µ 0 ) α s ( µ 0 ) Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

RG evolution of the LCDAs The Gegenbauer expansion yields a diagonal scale-evolution of the coefficients: � α s ( µ ) � γ n / 2 β 0 a M a M n ( µ ) = n ( µ 0 ) α s ( µ 0 ) Every anomalous dimension γ n is strictly positive ⇒ a M n ( µ → ∞ ) → 0 ⇒ φ M ( x , µ → ∞ ) → 6 x (1 − x ) Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

RG evolution of the LCDAs a) K LCDA b) J /ψ LCDA c) B LCDA LCDAs for mesons at different scales, dashed lines: φ M ( x , µ = µ 0 ) , solid lines: φ M ( x , µ = m Z ) , grey dotted lines: φ M ( x , µ → ∞ ) Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

RG evolution of the LCDAs a) K LCDA b) J /ψ LCDA c) B LCDA LCDAs for mesons at different scales, dashed lines: φ M ( x , µ = µ 0 ) , solid lines: φ M ( x , µ = m Z ) , grey dotted lines: φ M ( x , µ → ∞ ) At high scales compared to Λ QCD (e.g. µ ∼ m Z ) the sensitivity to poorly-known a M is greatly reduced! n Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

Heavy mesons: quarkonia For heavy quarkonium states M ∼ ( Q ¯ Q ) the LCDA peaks at x = 1 / 2 . In the limit of m Q → ∞ , the width of the LCDA vanishes and φ M → δ ( x − 1 2 ) . Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

Heavy mesons: quarkonia For heavy quarkonium states M ∼ ( Q ¯ Q ) the LCDA peaks at x = 1 / 2 . In the limit of m Q → ∞ , the width of the LCDA vanishes and φ M → δ ( x − 1 2 ) . Using NRQCD, the LCDA can be related to a local matrix element [Caswell, Lepage ( 1986 ), Phys. Lett. B 167, 437] [Bodwin, Braaten, Lepage ( 1995 ), Phys. Rev. D 51, 1125] One finds: 1 dx (2 x − 1) 2 φ M ( x , µ 0 ) = � v 2 � M � + O ( v 4 ) 3 0 [Braguta, Likhoded, Luchinsky ( 2007 ), Phys. Lett. B 646, 80] Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

Heavy mesons: quarkonia For heavy quarkonium states M ∼ ( Q ¯ Q ) the LCDA peaks at x = 1 / 2 . In the limit of m Q → ∞ , the width of the LCDA vanishes and φ M → δ ( x − 1 2 ) . Using NRQCD, the LCDA can be related to a local matrix element [Caswell, Lepage ( 1986 ), Phys. Lett. B 167, 437] [Bodwin, Braaten, Lepage ( 1995 ), Phys. Rev. D 51, 1125] One finds: 1 dx (2 x − 1) 2 φ M ( x , µ 0 ) = � v 2 � M � + O ( v 4 ) 3 0 [Braguta, Likhoded, Luchinsky ( 2007 ), Phys. Lett. B 646, 80] Our model at the low scale: � − 6( x − 1 2 ) 2 � φ M ( x , µ 0 ) = x (1 − x ) exp × normalization � v 2 � Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

Heavy mesons: heavy-light states For heavy-light mesons M ∼ ( q ¯ Q ) , one defines: 1 dx φ M ( x , µ 0 ) m M � = λ M ( µ 0 ) + . . . x 0 [Beneke, Buchalla, Neubert, Sachrajda ( 1999 ), Phys. Rev. Lett. 83, 1914] where m M is the meson mass and the parameter λ M is a (poorly known) hadronic parameter and we have to use estimates. [Braun, Ivanov, Korchemsky ( 2004 ), Phy. Rev. D 69, 034014] [Ball, Jones, Zwicky ( 2007 ), Phys. Rev. D 75, 054004] Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

Heavy mesons: heavy-light states For heavy-light mesons M ∼ ( q ¯ Q ) , one defines: 1 dx φ M ( x , µ 0 ) m M � = λ M ( µ 0 ) + . . . x 0 [Beneke, Buchalla, Neubert, Sachrajda ( 1999 ), Phys. Rev. Lett. 83, 1914] where m M is the meson mass and the parameter λ M is a (poorly known) hadronic parameter and we have to use estimates. [Braun, Ivanov, Korchemsky ( 2004 ), Phy. Rev. D 69, 034014] [Ball, Jones, Zwicky ( 2007 ), Phys. Rev. D 75, 054004] As model LCDA we employ � − x m M � φ M ( x , µ 0 ) = x (1 − x ) exp × normalization λ M [Grozin, Neubert ( 1997 ), Phys. Rev. D 55, 272] Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

Heavy meson LCDAs and RGE Heavy meson LCDAs at the low scale µ 0 = 1 GeV : � � − x m M φ M ( x , µ 0 ) = x (1 − x ) exp × normalization λ M − 6( x − 1 2 ) 2 � � φ M ( x , µ 0 ) = x (1 − x ) exp × normalization � v 2 � Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

Heavy meson LCDAs and RGE Heavy meson LCDAs at the low scale µ 0 = 1 GeV : � � − x m M φ M ( x , µ 0 ) = x (1 − x ) exp × normalization λ M − 6( x − 1 2 ) 2 � � φ M ( x , µ 0 ) = x (1 − x ) exp × normalization � v 2 � The Gegenbauer expansion can be inverted to give: 1 2(2 n + 3) � dx C (3 / 2) a M n ( x , µ ) = (2 x − 1) φ M ( x , µ ) n 3( n + 1)( n + 2) 0 Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

Heavy meson LCDAs and RGE Heavy meson LCDAs at the low scale µ 0 = 1 GeV : � � − x m M φ M ( x , µ 0 ) = x (1 − x ) exp × normalization λ M − 6( x − 1 2 ) 2 � � φ M ( x , µ 0 ) = x (1 − x ) exp × normalization � v 2 � The Gegenbauer expansion can be inverted to give: 1 2(2 n + 3) � dx C (3 / 2) a M n ( x , µ ) = (2 x − 1) φ M ( x , µ ) n 3( n + 1)( n + 2) 0 For light mesons, only the first few moments are known (we use up to n = 2 ). For heavy mesons, we calculate the first 20 Gegenbauer moments to resolve the peak structure of the LCDAs. Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

Decays of electroweak gauge bosons Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

The Z → M + γ decay amplitude Diagrams at O ( α s ) : γ Z 0 Z 0 γ + analogous QCD corrections for second graph Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

The Z → M + γ decay amplitude Let us go through the steps of the calculation: Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

The Z → M + γ decay amplitude Let us go through the steps of the calculation: Compute the hard interactions at desired loop-order: xk γ + xk Z Z ¯ xk γ xk ¯ xk ) κ ( x ) � γ ν � v q − a q γ 5 � p γ µ � i A ∝ ¯ q ( xk ) q (¯ / x + κ (¯ x ) � p ′ γ ν � v q − a q γ 5 �� γ µ / ¯ q ( xk ) q (¯ xk ) ¯ x Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

The Z → M + γ decay amplitude Let us go through the steps of the calculation: Compute the hard interactions at desired loop-order: xk γ + xk Z Z xk ¯ γ xk ¯ contains O ( α s ) corrections xk ) κ ( x ) � γ ν � v q − a q γ 5 � p γ µ � i A ∝ ¯ q ( xk ) q (¯ / x + κ (¯ x ) � p ′ γ ν � v q − a q γ 5 �� γ µ / ¯ q ( xk ) q (¯ xk ) ¯ x Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

The Z → M + γ decay amplitude Dirac structure of the amplitude is of the form: p γ µ − a q γ ν / Γ = v q γ ν / p γ µ γ 5 Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

The Z → M + γ decay amplitude Dirac structure of the amplitude is of the form: p γ µ − a q γ ν / Γ = v q γ ν / p γ µ γ 5 The leading-twist two-particle projectors are: M P = i f P k γ 5 4 φ P ( x , µ ) / M V = − i f V 4 φ V ( x , µ ) / k V = i f ⊥ V ( µ ) ǫ V ∗ M ⊥ φ ⊥ V ( x , µ ) / k / ⊥ 4 Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

The Z → M + γ decay amplitude Dirac structure of the amplitude is of the form: p γ µ − a q γ ν / Γ = v q γ ν / p γ µ γ 5 The leading-twist two-particle projectors are: M P = i f P k γ 5 4 φ P ( x , µ ) / M V = − i f V 4 φ V ( x , µ ) / k V = i f ⊥ V ( µ ) ǫ V ∗ M ⊥ φ ⊥ V ( x , µ ) / k / ⊥ 4 At leading twist only P and V � allowed! (recall: projecting involves Tr[ M Γ] ) Subleading twist contributions strongly power-suppressed! Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

The Z → M + γ decay amplitude At the end of the day, we find: � � k µ q ν ε α Z ε ∗ β � γ − q · ε Z k · ε ∗ � egf M γ γ F M F M i A = ± i ǫ µναβ 1 − ε Z · ε ∗ 2 2 cos θ W k · q k · q with the form factors ∞ 1 = Q M � F M 6 [ I M + ( m Z ) + ¯ I M C (+) 2 n ( m Z , µ ) a M + ( m Z )] = Q M 2 n ( µ ) n =0 ∞ 2 = Q ′ � − ( m Z ) + ¯ − ( m Z )] = −Q ′ C ( − ) F M 6 [ I M M I M 2 n +1 ( m Z , µ ) a M 2 n +1 ( µ ) M n =0 Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

The Z → M + γ decay amplitude At the end of the day, we find: � � k µ q ν ε α Z ε ∗ β � γ − q · ε Z k · ε ∗ � egf M γ γ F M F M i A = ± i ǫ µναβ 1 − ε Z · ε ∗ 2 2 cos θ W k · q k · q + for pseudoscalar, - for vector with the form factors ∞ 1 = Q M � F M 6 [ I M + ( m Z ) + ¯ I M C (+) 2 n ( m Z , µ ) a M + ( m Z )] = Q M 2 n ( µ ) n =0 ∞ 2 = Q ′ � − ( m Z ) + ¯ − ( m Z )] = −Q ′ C ( − ) F M 6 [ I M M I M 2 n +1 ( m Z , µ ) a M 2 n +1 ( µ ) M n =0 Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

The Z → M + γ decay amplitude At the end of the day, we find: � � k µ q ν ε α Z ε ∗ β � γ − q · ε Z k · ε ∗ � egf M γ γ F M F M i A = ± i ǫ µναβ 1 − ε Z · ε ∗ 2 2 cos θ W k · q k · q with the form factors ∞ 1 = Q M � F M 6 [ I M + ( m Z ) + ¯ I M C (+) 2 n ( m Z , µ ) a M + ( m Z )] = Q M 2 n ( µ ) n =0 ∞ 2 = Q ′ � − ( m Z ) + ¯ − ( m Z )] = −Q ′ C ( − ) F M 6 [ I M M I M 2 n +1 ( m Z , µ ) a M 2 n +1 ( µ ) M n =0 quark couplings to photon and Z boson Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

The Z → M + γ decay amplitude At the end of the day, we find: � � k µ q ν ε α Z ε ∗ β � γ − q · ε Z k · ε ∗ � egf M γ γ F M F M i A = ± i ǫ µναβ 1 − ε Z · ε ∗ 2 2 cos θ W k · q k · q with the form factors ∞ 1 = Q M � F M 6 [ I M + ( m Z ) + ¯ I M C (+) 2 n ( m Z , µ ) a M + ( m Z )] = Q M 2 n ( µ ) n =0 ∞ 2 = Q ′ � − ( m Z ) + ¯ − ( m Z )] = −Q ′ C ( − ) F M 6 [ I M M I M 2 n +1 ( m Z , µ ) a M 2 n +1 ( µ ) M n =0 Convolution of LCDA with the hard function: 1 � I M ± ( m V ) = dx H ± ( x , m V , µ ) φ M ( x , µ ) 0 Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

The Z → M + γ decay amplitude At the end of the day, we find: � � k µ q ν ε α Z ε ∗ β � γ − q · ε Z k · ε ∗ � egf M γ γ F M F M i A = ± i ǫ µναβ 1 − ε Z · ε ∗ 2 2 cos θ W k · q k · q with the form factors ∞ 1 = Q M � F M 6 [ I M + ( m Z ) + ¯ I M C (+) 2 n ( m Z , µ ) a M + ( m Z )] = Q M 2 n ( µ ) n =0 ∞ 2 = Q ′ � − ( m Z ) + ¯ − ( m Z )] = −Q ′ C ( − ) F M 6 [ I M M I M 2 n +1 ( m Z , µ ) a M 2 n +1 ( µ ) M n =0 Sums over even and odd Gegenbauer moments and a coefficient function C ( ± ) ( m V , µ ) n Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

The Z → M + γ decay amplitude Coefficient functions: ( m V , µ ) = 1 + C F α s ( µ ) � m V � C ( ± ) c ( ± ) + O ( α 2 s ) n n 4 π µ with: � � log m 2 � � m V 2 � � c ( ± ) V = ( n + 1)( n + 2) − 4 H n +1 + 3 µ 2 − i π n µ n +1 − 4 ( H n +1 − 1) ± 1 2 + 4 H 2 ( n + 1)( n + 2) + ( n + 1) 2 ( n + 2) 2 − 9 Large logs are resummed to all orders by choosing µ ∼ m Z ! Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

The Z → M + γ decay amplitude The combination C ( ± ) ( m V , µ ) a M n ( µ ) is formally scale independent! n n = 1 n = 2 NLO NLO LO LO The form factors become: Re F M 0 . 94 + 1 . 05 a M 2 ( m Z ) + 1 . 15 a M 4 ( m Z ) + 1 . 22 a M � � = Q M 6 ( m Z ) + . . . 1 � 0 . 94 + 0 . 41 a M 2 ( µ 0 ) + 0 . 29 a M 4 ( µ 0 ) + 0 . 23 a M � = Q M 6 ( µ 0 ) + . . . F M = 0 2 Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

The Z → M + γ decay amplitude The combination C ( ± ) ( m V , µ ) a M n ( µ ) is formally scale independent! n n = 1 n = 2 NLO NLO LO LO The form factors become: moments at the high scale Re F M 0 . 94 + 1 . 05 a M 2 ( m Z ) + 1 . 15 a M 4 ( m Z ) + 1 . 22 a M � � = Q M 6 ( m Z ) + . . . 1 � 0 . 94 + 0 . 41 a M 2 ( µ 0 ) + 0 . 29 a M 4 ( µ 0 ) + 0 . 23 a M � = Q M 6 ( µ 0 ) + . . . F M = 0 2 Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

The Z → M + γ decay amplitude The combination C ( ± ) ( m V , µ ) a M n ( µ ) is formally scale independent! n n = 1 n = 2 NLO NLO LO LO The form factors become: Re F M 0 . 94 + 1 . 05 a M 2 ( m Z ) + 1 . 15 a M 4 ( m Z ) + 1 . 22 a M � � = Q M 6 ( m Z ) + . . . 1 � 0 . 94 + 0 . 41 a M 2 ( µ 0 ) + 0 . 29 a M 4 ( µ 0 ) + 0 . 23 a M � = Q M 6 ( µ 0 ) + . . . F M = 0 → sensitivity strongly reduced! 2 Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

Results for Z → M γ For the branching ratios BR( Z → M γ ) we find: Z → . . . Branching ratio asym. LO π 0 γ (9 . 80 + 0 . 09 − 0 . 14 µ ± 0 . 03 f ± 0 . 61 a 2 ± 0 . 82 a 4 ) · 10 − 12 7.71 14.67 ρ 0 γ (4 . 19 + 0 . 04 − 0 . 06 µ ± 0 . 16 f ± 0 . 24 a 2 ± 0 . 37 a 4 ) · 10 − 9 3.63 5.68 (2 . 89 + 0 . 03 − 0 . 05 µ ± 0 . 15 f ± 0 . 29 a 2 ± 0 . 25 a 4 ) · 10 − 8 ωγ 2.54 3.84 (8 . 63 + 0 . 08 − 0 . 13 µ ± 0 . 41 f ± 0 . 55 a 2 ± 0 . 74 a 4 ) · 10 − 9 φγ 7.12 12.31 (8 . 02 + 0 . 14 + 0 . 39 · 10 − 8 J /ψ γ − 0 . 15 µ ± 0 . 20 f − 0 . 36 σ ) 10.48 6.55 Υ(1 S ) γ (5 . 39 + 0 . 10 + 0 . 11 · 10 − 8 − 0 . 10 µ ± 0 . 08 f − 0 . 08 σ ) 7.55 4.11 Υ(4 S ) γ (1 . 22 + 0 . 02 + 0 . 02 · 10 − 8 − 0 . 02 µ ± 0 . 13 f − 0 . 02 σ ) 1.71 0.93 Υ( nS ) γ (9 . 96 + 0 . 18 + 0 . 20 · 10 − 8 − 0 . 19 µ ± 0 . 09 f − 0 . 15 σ ) 13.96 7.59 Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

Results for Z → M γ For the branching ratios BR( Z → M γ ) we find: Z → . . . Branching ratio asym. LO π 0 γ (9 . 80 + 0 . 09 − 0 . 14 µ ± 0 . 03 f ± 0 . 61 a 2 ± 0 . 82 a 4 ) · 10 − 12 7.71 14.67 ρ 0 γ (4 . 19 + 0 . 04 − 0 . 06 µ ± 0 . 16 f ± 0 . 24 a 2 ± 0 . 37 a 4 ) · 10 − 9 3.63 5.68 (2 . 89 + 0 . 03 − 0 . 05 µ ± 0 . 15 f ± 0 . 29 a 2 ± 0 . 25 a 4 ) · 10 − 8 ωγ 2.54 3.84 (8 . 63 + 0 . 08 − 0 . 13 µ ± 0 . 41 f ± 0 . 55 a 2 ± 0 . 74 a 4 ) · 10 − 9 φγ 7.12 12.31 (8 . 02 + 0 . 14 + 0 . 39 · 10 − 8 J /ψ γ − 0 . 15 µ ± 0 . 20 f − 0 . 36 σ ) 10.48 6.55 Υ(1 S ) γ (5 . 39 + 0 . 10 + 0 . 11 · 10 − 8 − 0 . 10 µ ± 0 . 08 f − 0 . 08 σ ) 7.55 4.11 Υ(4 S ) γ (1 . 22 + 0 . 02 + 0 . 02 · 10 − 8 − 0 . 02 µ ± 0 . 13 f − 0 . 02 σ ) 1.71 0.93 Υ( nS ) γ (9 . 96 + 0 . 18 + 0 . 20 · 10 − 8 − 0 . 19 µ ± 0 . 09 f − 0 . 15 σ ) 13.96 7.59 scale dependence Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

Results for Z → M γ For the branching ratios BR( Z → M γ ) we find: Z → . . . Branching ratio asym. LO π 0 γ (9 . 80 + 0 . 09 − 0 . 14 µ ± 0 . 03 f ± 0 . 61 a 2 ± 0 . 82 a 4 ) · 10 − 12 7.71 14.67 ρ 0 γ (4 . 19 + 0 . 04 − 0 . 06 µ ± 0 . 16 f ± 0 . 24 a 2 ± 0 . 37 a 4 ) · 10 − 9 3.63 5.68 (2 . 89 + 0 . 03 − 0 . 05 µ ± 0 . 15 f ± 0 . 29 a 2 ± 0 . 25 a 4 ) · 10 − 8 ωγ 2.54 3.84 (8 . 63 + 0 . 08 − 0 . 13 µ ± 0 . 41 f ± 0 . 55 a 2 ± 0 . 74 a 4 ) · 10 − 9 φγ 7.12 12.31 (8 . 02 + 0 . 14 + 0 . 39 · 10 − 8 J /ψ γ − 0 . 15 µ ± 0 . 20 f − 0 . 36 σ ) 10.48 6.55 Υ(1 S ) γ (5 . 39 + 0 . 10 + 0 . 11 · 10 − 8 − 0 . 10 µ ± 0 . 08 f − 0 . 08 σ ) 7.55 4.11 Υ(4 S ) γ (1 . 22 + 0 . 02 + 0 . 02 · 10 − 8 − 0 . 02 µ ± 0 . 13 f − 0 . 02 σ ) 1.71 0.93 Υ( nS ) γ (9 . 96 + 0 . 18 + 0 . 20 · 10 − 8 − 0 . 19 µ ± 0 . 09 f − 0 . 15 σ ) 13.96 7.59 scale dependence decay constant Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

Results for Z → M γ For the branching ratios BR( Z → M γ ) we find: Z → . . . Branching ratio asym. LO π 0 γ (9 . 80 + 0 . 09 − 0 . 14 µ ± 0 . 03 f ± 0 . 61 a 2 ± 0 . 82 a 4 ) · 10 − 12 7.71 14.67 ρ 0 γ (4 . 19 + 0 . 04 − 0 . 06 µ ± 0 . 16 f ± 0 . 24 a 2 ± 0 . 37 a 4 ) · 10 − 9 3.63 5.68 (2 . 89 + 0 . 03 − 0 . 05 µ ± 0 . 15 f ± 0 . 29 a 2 ± 0 . 25 a 4 ) · 10 − 8 ωγ 2.54 3.84 (8 . 63 + 0 . 08 − 0 . 13 µ ± 0 . 41 f ± 0 . 55 a 2 ± 0 . 74 a 4 ) · 10 − 9 φγ 7.12 12.31 (8 . 02 + 0 . 14 + 0 . 39 · 10 − 8 J /ψ γ − 0 . 15 µ ± 0 . 20 f − 0 . 36 σ ) 10.48 6.55 Υ(1 S ) γ (5 . 39 + 0 . 10 + 0 . 11 · 10 − 8 − 0 . 10 µ ± 0 . 08 f − 0 . 08 σ ) 7.55 4.11 Υ(4 S ) γ (1 . 22 + 0 . 02 + 0 . 02 · 10 − 8 − 0 . 02 µ ± 0 . 13 f − 0 . 02 σ ) 1.71 0.93 Υ( nS ) γ (9 . 96 + 0 . 18 + 0 . 20 · 10 − 8 − 0 . 19 µ ± 0 . 09 f − 0 . 15 σ ) 13.96 7.59 scale dependence LCDA shape decay constant Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

Results for Z → M γ For the branching ratios BR( Z → M γ ) we find: Z → . . . Branching ratio asym. LO π 0 γ (9 . 80 + 0 . 09 − 0 . 14 µ ± 0 . 03 f ± 0 . 61 a 2 ± 0 . 82 a 4 ) · 10 − 12 7.71 14.67 ρ 0 γ (4 . 19 + 0 . 04 − 0 . 06 µ ± 0 . 16 f ± 0 . 24 a 2 ± 0 . 37 a 4 ) · 10 − 9 3.63 5.68 (2 . 89 + 0 . 03 − 0 . 05 µ ± 0 . 15 f ± 0 . 29 a 2 ± 0 . 25 a 4 ) · 10 − 8 ωγ 2.54 3.84 (8 . 63 + 0 . 08 − 0 . 13 µ ± 0 . 41 f ± 0 . 55 a 2 ± 0 . 74 a 4 ) · 10 − 9 φγ 7.12 12.31 (8 . 02 + 0 . 14 + 0 . 39 · 10 − 8 J /ψ γ − 0 . 15 µ ± 0 . 20 f − 0 . 36 σ ) 10.48 6.55 Υ(1 S ) γ (5 . 39 + 0 . 10 + 0 . 11 · 10 − 8 − 0 . 10 µ ± 0 . 08 f − 0 . 08 σ ) 7.55 4.11 Υ(4 S ) γ (1 . 22 + 0 . 02 + 0 . 02 · 10 − 8 − 0 . 02 µ ± 0 . 13 f − 0 . 02 σ ) 1.71 0.93 Υ( nS ) γ (9 . 96 + 0 . 18 + 0 . 20 · 10 − 8 − 0 . 19 µ ± 0 . 09 f − 0 . 15 σ ) 13.96 7.59 obtained when using only asymptotic form of LCDA φ M ( x ) = 6x ( 1 − x ) Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

Results for Z → M γ For the branching ratios BR( Z → M γ ) we find: Z → . . . Branching ratio asym. LO π 0 γ (9 . 80 + 0 . 09 − 0 . 14 µ ± 0 . 03 f ± 0 . 61 a 2 ± 0 . 82 a 4 ) · 10 − 12 7.71 14.67 ρ 0 γ (4 . 19 + 0 . 04 − 0 . 06 µ ± 0 . 16 f ± 0 . 24 a 2 ± 0 . 37 a 4 ) · 10 − 9 3.63 5.68 (2 . 89 + 0 . 03 − 0 . 05 µ ± 0 . 15 f ± 0 . 29 a 2 ± 0 . 25 a 4 ) · 10 − 8 ωγ 2.54 3.84 (8 . 63 + 0 . 08 − 0 . 13 µ ± 0 . 41 f ± 0 . 55 a 2 ± 0 . 74 a 4 ) · 10 − 9 φγ 7.12 12.31 (8 . 02 + 0 . 14 + 0 . 39 · 10 − 8 J /ψ γ − 0 . 15 µ ± 0 . 20 f − 0 . 36 σ ) 10.48 6.55 Υ(1 S ) γ (5 . 39 + 0 . 10 + 0 . 11 · 10 − 8 − 0 . 10 µ ± 0 . 08 f − 0 . 08 σ ) 7.55 4.11 Υ(4 S ) γ (1 . 22 + 0 . 02 + 0 . 02 · 10 − 8 − 0 . 02 µ ± 0 . 13 f − 0 . 02 σ ) 1.71 0.93 Υ( nS ) γ (9 . 96 + 0 . 18 + 0 . 20 · 10 − 8 − 0 . 19 µ ± 0 . 09 f − 0 . 15 σ ) 13.96 7.59 obtained when using only LO hard functions Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

The W → M + γ decay amplitude Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

W → M + γ The decay W → M + γ is similar to the Z → M + γ decay, except for an additional local contribution: γ W + W + W + γ γ The form factor decomposition now looks as follows: k µ q ν ε α W ε ∗ β � � i A ( W + → M + γ ) = ± egf M γ F M 1 − ε ⊥ W · ε ⊥∗ γ F M √ i ǫ µναβ 2 V ij 2 k · q 4 Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

W → M + γ The decay W → M + γ is similar to the Z → M + γ decay, except for an additional local contribution: γ W + W + W + γ γ The form factor decomposition now looks as follows: k µ q ν ε α W ε ∗ β � � i A ( W + → M + γ ) = ± egf M γ F M 1 − ε ⊥ W · ε ⊥∗ γ F M √ i ǫ µναβ 2 V ij 2 k · q 4 Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

W → M + γ The decay W → M + γ is similar to the Z → M + γ decay, except for an additional local contribution: γ W + W + W + γ γ The form factor decomposition now looks as follows: k µ q ν ε α W ε ∗ β � � i A ( W + → M + γ ) = ± egf M γ F M 1 − ε ⊥ W · ε ⊥∗ γ F M √ i ǫ µναβ 2 V ij 2 k · q 4 + for pseudoscalar, - for vector Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

Results for W → M + γ For the branching ratios W ± → M ∓ γ , we find: mode Branching ratio asym. LO π ± γ (4 . 00 + 0 . 06 − 0 . 11 µ ± 0 . 01 f ± 0 . 49 a 2 ± 0 . 66 a 4 ) · 10 − 9 2.45 8.09 ρ ± γ (8 . 74 + 0 . 17 − 0 . 26 µ ± 0 . 33 f ± 1 . 02 a 2 ± 1 . 57 a 4 ) · 10 − 9 6.48 15.12 K ± γ (3 . 25 + 0 . 05 − 0 . 09 µ ± 0 . 03 f ± 0 . 24 a 1 ± 0 . 38 a 2 ± 0 . 51 a 4 ) · 10 − 10 1.88 6.38 K ∗± γ (4 . 78 + 0 . 09 − 0 . 14 µ ± 0 . 28 f ± 0 . 39 a 1 ± 0 . 66 a 2 ± 0 . 80 a 4 ) · 10 − 10 3.18 8.47 (3 . 66 + 0 . 02 + 1 . 47 − 0 . 82 σ ) · 10 − 8 D s γ − 0 . 07 µ ± 0 . 12 CKM ± 0 . 13 f 0.98 8.59 D ± γ (1 . 38 + 0 . 01 − 0 . 30 σ ) · 10 − 9 + 0 . 50 − 0 . 02 µ ± 0 . 10 CKM ± 0 . 07 f 0.32 3.42 B ± γ (1 . 55 + 0 . 00 + 0 . 68 − 0 . 45 σ ) · 10 − 12 − 0 . 03 µ ± 0 . 37 CKM ± 0 . 15 f 0.09 6.44 Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

Results for W → M + γ For the branching ratios W ± → M ∓ γ , we find: mode Branching ratio asym. LO π ± γ (4 . 00 + 0 . 06 − 0 . 11 µ ± 0 . 01 f ± 0 . 49 a 2 ± 0 . 66 a 4 ) · 10 − 9 2.45 8.09 ρ ± γ (8 . 74 + 0 . 17 − 0 . 26 µ ± 0 . 33 f ± 1 . 02 a 2 ± 1 . 57 a 4 ) · 10 − 9 6.48 15.12 K ± γ (3 . 25 + 0 . 05 − 0 . 09 µ ± 0 . 03 f ± 0 . 24 a 1 ± 0 . 38 a 2 ± 0 . 51 a 4 ) · 10 − 10 1.88 6.38 K ∗± γ (4 . 78 + 0 . 09 − 0 . 14 µ ± 0 . 28 f ± 0 . 39 a 1 ± 0 . 66 a 2 ± 0 . 80 a 4 ) · 10 − 10 3.18 8.47 (3 . 66 + 0 . 02 + 1 . 47 − 0 . 82 σ ) · 10 − 8 D s γ − 0 . 07 µ ± 0 . 12 CKM ± 0 . 13 f 0.98 8.59 D ± γ (1 . 38 + 0 . 01 + 0 . 50 − 0 . 30 σ ) · 10 − 9 − 0 . 02 µ ± 0 . 10 CKM ± 0 . 07 f 0.32 3.42 B ± γ (1 . 55 + 0 . 00 + 0 . 68 − 0 . 45 σ ) · 10 − 12 − 0 . 03 µ ± 0 . 37 CKM ± 0 . 15 f 0.09 6.44 flavour off-diagonal mesons allowed Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

Results for W → M + γ For the branching ratios W ± → M ∓ γ , we find: mode Branching ratio asym. LO π ± γ (4 . 00 + 0 . 06 − 0 . 11 µ ± 0 . 01 f ± 0 . 49 a 2 ± 0 . 66 a 4 ) · 10 − 9 2.45 8.09 ρ ± γ (8 . 74 + 0 . 17 − 0 . 26 µ ± 0 . 33 f ± 1 . 02 a 2 ± 1 . 57 a 4 ) · 10 − 9 6.48 15.12 K ± γ (3 . 25 + 0 . 05 − 0 . 09 µ ± 0 . 03 f ± 0 . 24 a 1 ± 0 . 38 a 2 ± 0 . 51 a 4 ) · 10 − 10 1.88 6.38 K ∗± γ (4 . 78 + 0 . 09 − 0 . 14 µ ± 0 . 28 f ± 0 . 39 a 1 ± 0 . 66 a 2 ± 0 . 80 a 4 ) · 10 − 10 3.18 8.47 (3 . 66 + 0 . 02 + 1 . 47 − 0 . 82 σ ) · 10 − 8 D s γ − 0 . 07 µ ± 0 . 12 CKM ± 0 . 13 f 0.98 8.59 D ± γ (1 . 38 + 0 . 01 + 0 . 50 − 0 . 30 σ ) · 10 − 9 − 0 . 02 µ ± 0 . 10 CKM ± 0 . 07 f 0.32 3.42 B ± γ (1 . 55 + 0 . 00 + 0 . 68 − 0 . 45 σ ) · 10 − 12 − 0 . 03 µ ± 0 . 37 CKM ± 0 . 15 f 0.09 6.44 introduces uncertainties from CKM elements Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

Decays of electroweak gauge bosons Z decays as BSM probes Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

Z → M + γ decays as BSM probes Our analysis can straight-forwardly be generalized to the case of non-SM Z boson couplings to quarks! Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

Z → M + γ decays as BSM probes Our analysis can straight-forwardly be generalized to the case of non-SM Z boson couplings to quarks! γ Z 0 Z 0 γ Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

Z → M + γ decays as BSM probes Our analysis can straight-forwardly be generalized to the case of non-SM Z boson couplings to quarks! γ Z 0 Z 0 γ At LEP, | a b | and | a c | have been measured to 1% , using our predictions, | a s | , | a d | and | a u | could be measured to ∼ 6% Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

Z → M + γ decays as BSM probes Our analysis can straight-forwardly be generalized to the case of non-SM Z boson couplings to quarks! γ Z 0 Z 0 FCNC FCNC γ At LEP, | a b | and | a c | have been measured to 1% , using our predictions, | a s | , | a d | and | a u | could be measured to ∼ 6% Introducing FCNC couplings allows the production of flavor off-diagonal mesons Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

Z → M + γ decays as FCNC probes γ Z 0 Z 0 γ Model independent predictions for flavor off-diagonal mesons: Decay mode Branching ratio SM background Z 0 → K 0 γ (7 . 70 ± 0 . 83) | v sd | 2 + (0 . 01 ± 0 . 01) | a sd | 2 � � · 10 − 8 λ α π ∼ 2 · 10 − 3 sin 2 θ W Z 0 → D 0 γ − 0 . 43 ) | v cu | 2 + (0 . 62 + 0 . 36 � (5 . 30 + 0 . 67 − 0 . 23 ) | a cu | 2 � · 10 − 7 λ π ∼ 2 · 10 − 3 α sin 2 θ W Z 0 → B 0 γ − 0 . 41 ) | v bd | 2 + (0 . 77 + 0 . 38 λ 3 � (2 . 08 + 0 . 59 − 0 . 26 ) | a bd | 2 � · 10 − 7 α π ∼ 8 · 10 − 5 sin 2 θ W Z 0 → B s γ − 0 . 52 ) | v bs | 2 + (0 . 87 + 0 . 51 λ 2 � (2 . 64 + 0 . 82 − 0 . 33 ) | a bs | 2 � · 10 − 7 α π ∼ 4 · 10 − 4 sin 2 θ W Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

Z → M + γ decays as FCNC probes γ W Z 0 W Z 0 γ Z 0 Z 0 W γ γ Model independent predictions for flavor off-diagonal mesons: Decay mode Branching ratio SM background Z 0 → K 0 γ (7 . 70 ± 0 . 83) | v sd | 2 + (0 . 01 ± 0 . 01) | a sd | 2 � � · 10 − 8 λ α π ∼ 2 · 10 − 3 sin 2 θ W Z 0 → D 0 γ − 0 . 43 ) | v cu | 2 + (0 . 62 + 0 . 36 � (5 . 30 + 0 . 67 − 0 . 23 ) | a cu | 2 � · 10 − 7 λ α π ∼ 2 · 10 − 3 sin 2 θ W Z 0 → B 0 γ − 0 . 41 ) | v bd | 2 + (0 . 77 + 0 . 38 λ 3 � (2 . 08 + 0 . 59 − 0 . 26 ) | a bd | 2 � · 10 − 7 α π ∼ 8 · 10 − 5 sin 2 θ W Z 0 → B s γ − 0 . 52 ) | v bs | 2 + (0 . 87 + 0 . 51 λ 2 � (2 . 64 + 0 . 82 − 0 . 33 ) | a bs | 2 � · 10 − 7 α π ∼ 4 · 10 − 4 sin 2 θ W Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

Z → M + γ decays as FCNC probes FCNCs would induce tree-level neutral-meson mixing, strongly constrained: � ( v sd ± a sd ) 2 �� � ( v sd ) 2 − ( a sd ) 2 �� � Re � � Re � < 2 . 9 · 10 − 8 < 3 . 0 · 10 − 10 � � � ( v sd ± a sd ) 2 �� � ( v sd ) 2 − ( a sd ) 2 �� � Im � < 1 . 0 · 10 − 10 � Im � < 4 . 3 · 10 − 13 � � � ( v cu ) 2 − ( a cu ) 2 � � � ( v cu ± a cu ) 2 � � < 2 . 2 · 10 − 8 < 1 . 5 · 10 − 8 � � � � ( v bd ± a bd ) 2 � � � ( v bd ) 2 − ( a bd ) 2 � < 4 . 3 · 10 − 8 < 8 . 2 · 10 − 9 � � � ( v bs ± a bs ) 2 � � � � ( v bs ) 2 − ( a bs ) 2 � < 5 . 5 · 10 − 7 < 1 . 4 · 10 − 7 � � [Bona et al. ( 2007 ), JHEP 0803, 049] [Bertone et al. ( 2012 ), JHEP 1303, 089] [Carrasco et al. ( 2013 ), JHEP 1403, 016] These bounds push our branching ratios down to 10 − 14 , rendering them unobservable. Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

Decays of electroweak gauge bosons Weak radiative Z decays to M + W Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

The Z → M + W decay The contributing diagrams in this case look similar to the W → M γ decays: W − Z 0 Z 0 Z 0 W − W − Form factor decomposition: g 2 f M � 1 − m 2 � i A ( Z → M + W − ) = ± W √ V ij m 2 4 2 c W Z Z ε ∗ β � k µ q ν ε α 2 + q · ε Z k · ε ∗ � F M 1 − ε Z · ε ∗ W F M F M W W × i ǫ µναβ 3 k · q k · q Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

The Z → M + W decay The contributing diagrams in this case look similar to the W → M γ decays: W − Z 0 Z 0 Z 0 W − W − Form factor decomposition: g 2 f M � 1 − m 2 � i A ( Z → M + W − ) = ± W √ V ij m 2 4 2 c W Z Z ε ∗ β � k µ q ν ε α 2 + q · ε Z k · ε ∗ � F M 1 − ε Z · ε ∗ W F M F M W W × i ǫ µναβ 3 k · q k · q now allowed because W can be longitudinally polarized Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

The Z → M + W decay The contributing diagrams in this case look similar to the W → M γ decays: W − Z 0 Z 0 Z 0 W − W − Form factor decomposition: g 2 f M � 1 − m 2 � i A ( Z → M + W − ) = ± W √ V ij m 2 4 2 c W Z Z ε ∗ β � k µ q ν ε α 2 + q · ε Z k · ε ∗ � F M 1 − ε Z · ε ∗ W F M F M W W × i ǫ µναβ 3 k · q k · q Allows the QCD factorization approach to be tested at lower scale ( m Z − m W ) ≈ 10 GeV ! Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

Results for Z → M + W For the decay rates, we find: Γ( Z → M + W − ) = πα 2 ( m Z ) f 2 | V ij | 2 s 2 � 3 2 + 3 W + 227 � 2 s 2 180 s 4 W + 0 . 003 a M M W 1 + . . . c 2 48 m Z W Our predictions for the branching ratios are: Decay mode Branching ratio Z 0 → π ± W ∓ (1 . 51 ± 0 . 005 f ) · 10 − 10 Z 0 → ρ ± W ∓ (4 . 00 ± 0 . 15 f ) · 10 − 10 Z 0 → K ± W ∓ (1 . 16 ± 0 . 01 f ) · 10 − 11 Z 0 → K ∗± W ∓ (1 . 96 ± 0 . 12 f ) · 10 − 11 Z 0 → D s W ∓ (6 . 04 ± 0 . 20 CKM ± 0 . 22 f ) · 10 − 10 Z 0 → D ± W ∓ (1 . 99 ± 0 . 14 CKM ± 0 . 10 f ) · 10 − 11 Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

Results for Z → M + W For the decay rates, we find: Γ( Z → M + W − ) = πα 2 ( m Z ) f 2 | V ij | 2 s 2 � 3 2 + 3 W + 227 � 2 s 2 180 s 4 W + 0 . 003 a M M W 1 + . . . c 2 48 m Z W very small sensitivity to LCDA Our predictions for the branching ratios are: Decay mode Branching ratio Z 0 → π ± W ∓ (1 . 51 ± 0 . 005 f ) · 10 − 10 Z 0 → ρ ± W ∓ (4 . 00 ± 0 . 15 f ) · 10 − 10 Z 0 → K ± W ∓ (1 . 16 ± 0 . 01 f ) · 10 − 11 Z 0 → K ∗± W ∓ (1 . 96 ± 0 . 12 f ) · 10 − 11 Z 0 → D s W ∓ (6 . 04 ± 0 . 20 CKM ± 0 . 22 f ) · 10 − 10 Z 0 → D ± W ∓ (1 . 99 ± 0 . 14 CKM ± 0 . 10 f ) · 10 − 11 Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

Results for Z → M + W For the decay rates, we find: Γ( Z → M + W − ) = πα 2 ( m Z ) f 2 | V ij | 2 s 2 � 3 2 + 3 W + 227 � 2 s 2 180 s 4 W + 0 . 003 a M M W 1 + . . . c 2 48 m Z W very small sensitivity to LCDA Our predictions for the branching ratios are: Decay mode Branching ratio Z 0 → π ± W ∓ (1 . 51 ± 0 . 005 f ) · 10 − 10 Z 0 → ρ ± W ∓ (4 . 00 ± 0 . 15 f ) · 10 − 10 Z 0 → K ± W ∓ (1 . 16 ± 0 . 01 f ) · 10 − 11 Z 0 → K ∗± W ∓ (1 . 96 ± 0 . 12 f ) · 10 − 11 Z 0 → D s W ∓ (6 . 04 ± 0 . 20 CKM ± 0 . 22 f ) · 10 − 10 Z 0 → D ± W ∓ (1 . 99 ± 0 . 14 CKM ± 0 . 10 f ) · 10 − 11 The O ( α s ) corrections to this are an interesting project left for future work, in particular the scale dependence of the result. Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

Conclusions, summary and outlook Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

Conclusions, summary and outlook To summarize: Decay mode Branching ratio Decay mode Branching ratio Z 0 → π 0 γ W ± → π ± γ (9 . 80 ± 1 . 03) · 10 − 12 (4 . 00 ± 0 . 83) · 10 − 9 Z 0 → ρ 0 γ W ± → ρ ± γ (4 . 19 ± 0 . 47) · 10 − 9 (8 . 74 ± 1 . 91) · 10 − 9 Z 0 → ωγ W ± → K ± γ (2 . 89 ± 0 . 41) · 10 − 8 (3 . 25 ± 0 . 69) · 10 − 10 Z 0 → φγ W ± → K ∗± γ (8 . 63 ± 1 . 01) · 10 − 9 (4 . 78 ± 1 . 15) · 10 − 10 Z 0 → J /ψ γ W ± → D s γ (8 . 02 ± 0 . 45) · 10 − 8 (3 . 66 + 1 . 49 − 0 . 85 ) · 10 − 8 Z 0 → Υ(1 S ) γ W ± → D ± γ (5 . 39 ± 0 . 16) · 10 − 8 (1 . 38 + 0 . 51 − 0 . 33 ) · 10 − 9 Z 0 → Υ(4 S ) γ W ± → B ± γ (1 . 22 ± 0 . 13) · 10 − 8 (1 . 55 + 0 . 79 − 0 . 60 ) · 10 − 12 For Z → V γ → µ + µ − γ , one can trigger on muons and the photon Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

Conclusions, summary and outlook To summarize: Decay mode Branching ratio Decay mode Branching ratio Z 0 → π 0 γ W ± → π ± γ (9 . 80 ± 1 . 03) · 10 − 12 (4 . 00 ± 0 . 83) · 10 − 9 Z 0 → ρ 0 γ W ± → ρ ± γ (4 . 19 ± 0 . 47) · 10 − 9 (8 . 74 ± 1 . 91) · 10 − 9 Z 0 → ωγ W ± → K ± γ (2 . 89 ± 0 . 41) · 10 − 8 (3 . 25 ± 0 . 69) · 10 − 10 Z 0 → φγ W ± → K ∗± γ (8 . 63 ± 1 . 01) · 10 − 9 (4 . 78 ± 1 . 15) · 10 − 10 Z 0 → J /ψ γ W ± → D s γ (8 . 02 ± 0 . 45) · 10 − 8 (3 . 66 + 1 . 49 − 0 . 85 ) · 10 − 8 Z 0 → Υ(1 S ) γ W ± → D ± γ (5 . 39 ± 0 . 16) · 10 − 8 (1 . 38 + 0 . 51 − 0 . 33 ) · 10 − 9 Z 0 → Υ(4 S ) γ W ± → B ± γ (1 . 22 ± 0 . 13) · 10 − 8 (1 . 55 + 0 . 79 − 0 . 60 ) · 10 − 12 For Z → V γ → µ + µ − γ , one can trigger on muons and the photon We expect O (100) J /ψ γ events at the LHC Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

Conclusions, summary and outlook To summarize: Decay mode Branching ratio Decay mode Branching ratio Z 0 → π 0 γ W ± → π ± γ (9 . 80 ± 1 . 03) · 10 − 12 (4 . 00 ± 0 . 83) · 10 − 9 Z 0 → ρ 0 γ W ± → ρ ± γ (4 . 19 ± 0 . 47) · 10 − 9 (8 . 74 ± 1 . 91) · 10 − 9 Z 0 → ωγ W ± → K ± γ (2 . 89 ± 0 . 41) · 10 − 8 (3 . 25 ± 0 . 69) · 10 − 10 Z 0 → φγ W ± → K ∗± γ (8 . 63 ± 1 . 01) · 10 − 9 (4 . 78 ± 1 . 15) · 10 − 10 Z 0 → J /ψ γ W ± → D s γ (8 . 02 ± 0 . 45) · 10 − 8 (3 . 66 + 1 . 49 − 0 . 85 ) · 10 − 8 Z 0 → Υ(1 S ) γ W ± → D ± γ (5 . 39 ± 0 . 16) · 10 − 8 (1 . 38 + 0 . 51 − 0 . 33 ) · 10 − 9 Z 0 → Υ(4 S ) γ W ± → B ± γ (1 . 22 ± 0 . 13) · 10 − 8 (1 . 55 + 0 . 79 − 0 . 60 ) · 10 − 12 For Z → V γ → µ + µ − γ , one can trigger on muons and the photon We expect O (100) J /ψ γ events at the LHC Ideas for reconstructing ( ρ , ω and φ ) + γ exist [Kagan et al. ( 2014 ), arXiv:1406.1722] Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

Conclusions, summary and outlook To summarize: Decay mode Branching ratio Decay mode Branching ratio Z 0 → π 0 γ W ± → π ± γ (9 . 80 ± 1 . 03) · 10 − 12 (4 . 00 ± 0 . 83) · 10 − 9 Z 0 → ρ 0 γ W ± → ρ ± γ (4 . 19 ± 0 . 47) · 10 − 9 (8 . 74 ± 1 . 91) · 10 − 9 Z 0 → ωγ W ± → K ± γ (2 . 89 ± 0 . 41) · 10 − 8 (3 . 25 ± 0 . 69) · 10 − 10 Z 0 → φγ W ± → K ∗± γ (8 . 63 ± 1 . 01) · 10 − 9 (4 . 78 ± 1 . 15) · 10 − 10 Z 0 → J /ψ γ W ± → D s γ (8 . 02 ± 0 . 45) · 10 − 8 (3 . 66 + 1 . 49 − 0 . 85 ) · 10 − 8 Z 0 → Υ(1 S ) γ W ± → D ± γ (5 . 39 ± 0 . 16) · 10 − 8 (1 . 38 + 0 . 51 − 0 . 33 ) · 10 − 9 Z 0 → Υ(4 S ) γ W ± → B ± γ (1 . 22 ± 0 . 13) · 10 − 8 (1 . 55 + 0 . 79 − 0 . 60 ) · 10 − 12 For Z → V γ → µ + µ − γ , one can trigger on muons and the photon We expect O (100) J /ψ γ events at the LHC Ideas for reconstructing ( ρ , ω and φ ) + γ exist [Kagan et al. ( 2014 ), arXiv:1406.1722] Reconstructing W decays at the LHC is more challenging [Mangano, Melia ( 2014 ), arXiv:1410.7475] Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

Conclusions and outlook A few things that I did not talk about today, but are featured in the paper: In some older papers, the authors speculated about a “possible huge enhancement” of the decays W , Z → P γ coming from an unsuppressed contribution from the axial anomaly. [Jacob, Wu ( 1989 ), Phys. Lett. B 232, 529] [Keum,Pham ( 1994 ), Mod. Phys. Lett. A 9, 1545] We find that such claims are false. Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

Conclusions and outlook A few things that I did not talk about today, but are featured in the paper: In some older papers, the authors speculated about a “possible huge enhancement” of the decays W , Z → P γ coming from an unsuppressed contribution from the axial anomaly. [Jacob, Wu ( 1989 ), Phys. Lett. B 232, 529] [Keum,Pham ( 1994 ), Mod. Phys. Lett. A 9, 1545] We find that such claims are false. We have derived decay constants for several mesons from updated experimental data, decreasing the uncertainty of our predictions. Very rare, exclusive radiative decays of W and Z bosons in QCD factorization

Conclusions and outlook We have derived predictions for the decay rates of exclusive radiative decays V → M + γ in the framework of QCD factorization. The branching ratios are small, between O (10 − 12 ) to O (10 − 9 ) . Very rare, exclusive radiative decays of W and Z bosons in QCD factorization