Possible explanation of BaBar anomaly with the use of Sudakov vertex - PowerPoint PPT Presentation

Possible explanation of BaBar anomaly with the use of Sudakov vertex E. A. Kuraev 1 , Yu. M. Bystritskiy 1 , V. V. Bytev 1 , A. N. Ilyichev 2 1 JINR, BLTP, Dubna, Russia 2 National Scientific and Educational Centre of Particle and High Energy Physics of the Belarusian State University, Minsk, Belarus International Workshop ”Bogoliubov readings” 2010 September 24, 2010 E. A. Kuraev, Yu. M. Bystritskiy, V. V. Bytev, A. N. Ilyichev ( JINR, BLTP, Dubna, Russia National Scientific and Educational Centre of Particle and High Energy Possible explanation of BaBar anomaly with the use of Sudakov vertex September 24, 2010 1 / 7

Process e + e − → π 0 e + e − The process e + e − → π 0 e + e − : e – (p – ) l – (q – ) e – (p – ) e – (p – ′ ) π (q π ) γ (q 1 ) π (q π ) γ * (q) γ (q 1 ) γ * (q) e + (p + ) e + (p + ′ ) e + (p _ ) l + (q + ) is described my two diagrams: M ann = (4 πα ) 2 M sc = 2 s (4 πα ) 2 1 q 2 J µ J ν ( l ) V ( s ) ǫ µναβ q α q β [ q × q 1 ] z V ( Q 2 ) N + N − , 1 , (1) q 2 q 2 q 2 1 where p µ � γ µ v ( p + ) � , J µ = [¯ − v � p ′ v ( p + ) γ µ u ( p − )] , � ¯ N + = + s p µ J ν ( l ) = [¯ + p ′ u l ( q − ) γ ν v l ( q + )] , � � � � N − = ¯ u γ µ u ( p − ) , (2) − s and V ( Q 2 ) is the vertex, which related with the neutral pion form factor F ( Q 2 ) as: M 2 � � Q 2 q V ( Q 2 ) = 2 π 2 F π Q 2 F . (3) M 2 q E. A. Kuraev, Yu. M. Bystritskiy, V. V. Bytev, A. N. Ilyichev ( JINR, BLTP, Dubna, Russia National Scientific and Educational Centre of Particle and High Energy Possible explanation of BaBar anomaly with the use of Sudakov vertex September 24, 2010 2 / 7

Pion Form Factor Pion form factor for vertex π 0 → 2 γ can be parameterized in different manners. In the approach based on QCD collinear factorization theorem (G. P. Lepage and S. J. Brodsky, Phys. Lett. B87 , 359 (1979)) γ (q 1 ) 1 V BL ( Q 2 ) = 2 F π � dx xQ 2 φ π ( x, s ) , (4) q(k) 3 0 π (q π ) q(k+q 1 ) and in the papers S. V. Mikhailov and N. G. Stefanis, Nucl. Phys. B821 , 291 (2009); M. V. Polyakov, JETP Lett. 90 , 228 (2009) different forms of pion wave function φ π ( x, s ) was used. q(k+q π ) Also in the paper L. Ametller,L. Bergstrom,A. Bramon, and E. Masso, Nucl. Phys. B228 , 301 (1983); γ * (q) A. E. Dorokhov, (2009), arXiv:0905.4577 was pointed that pion form factor in the frames of the constituent quark model has the double logarithmic asymptotic at large momentum transfer. m 2 1 � 2 arcsin 2 ( m π ) + 1 2 ln 2 β Q + 1 � V ( Q 2 ) = π , (5) 2 arcsin 2 ( m π m 2 π + t 2 M Q ) 2 M Q β Q − 1 � 4 M 2 Q where β Q = 1 + Q 2 . E. A. Kuraev, Yu. M. Bystritskiy, V. V. Bytev, A. N. Ilyichev ( JINR, BLTP, Dubna, Russia National Scientific and Educational Centre of Particle and High Energy Possible explanation of BaBar anomaly with the use of Sudakov vertex September 24, 2010 3 / 7

Our approach: Sudakov vertex We suppose Sudakov type of radiative corrections in one of the vertexes ( V. V. Sudakov, Sov. Phys. JETP 3 , 65 (1956); E. A. Kuraev and V. S. Fadin, Yad. Fiz. 27 , 1107 (1978) ). γ (q 1 ) � = − d 4 k � F � Q 2 /M 2 iπ 2 × q Q 2 R S ( Q 2 , p 2 1 , p 2 2 ) q(k) × q + i 0) , (6) ( k 2 − M 2 q + i 0)( p 2 q + i 0)( p 2 1 − M 2 2 − M 2 π (q π ) q(k+q 1 ) where p 1 = k + q 1 , and p 2 = k + q π and Sudakov vertex function R S (J. J. Carazzone, E. C. q(k+q π ) Poggio, and H. R. Quinn, Phys. Rev. D11 , 2286 (1975); J. M. Cornwall and G. Tiktopoulos, Phys. Rev. D13 , 3370 (1976)) is: γ * (q) ln Q 2 1 | ln Q 2 � − α s C F � R S ( Q 2 , p 2 1 , p 2 2 ) = exp , (7) | p 2 | p 2 2 π 2 | q and C F = � N 2 − 1 � / (2 N ) = 4 / 3 . We use here the the where Q 2 ≫ | p 2 1 , 2 | ≫ M 2 Goldberger-Treiman relation on the quark level F π = M q /g q ¯ qπ = 93 MeV. E. A. Kuraev, Yu. M. Bystritskiy, V. V. Bytev, A. N. Ilyichev ( JINR, BLTP, Dubna, Russia National Scientific and Educational Centre of Particle and High Energy Possible explanation of BaBar anomaly with the use of Sudakov vertex September 24, 2010 4 / 7

Our approach: Results The cross section of process e + e − → e + e − π 0 is 0 . 35 Our model α 4 dσ 4 Q 2 V 2 ( Q 2 ) J ( Q 2 ) , = (8) 0 . 30 BaBar-2009 dQ 2 Q 2 V ( Q 2 ) , GeV 1 J ( Q 2 ) 2 L 2 = s + L s ( L e − 1) − ( L e + 1) , 0 . 25 s where L s = ln Q 2 + M 2 , 0 . 20 L e = ln Q 2 e and V ( Q 2 ) is the Sudakov vertex: m 2 0 . 15 M 2 q V ( Q 2 ) = A Φ( z B ) , (9) 2 πF π α s C F 5 10 15 20 25 30 35 Q 2 , GeV 2 1 Q 2 dx z B = C F α s � � 1 − e − z B x (1 − x ) � ln 2 where Φ( z B ) = , . BM 2 x 2 π q 0 Quantities A and B can be considered as a positive fitting parameters of order of unity. We fixed their values as A = 0 . 49 and B = 0 . 23 (which corresponds to effective quark mass m q ≈ 135 MeV) by fitting the BaBar data (The BABAR, B. Aubert et al. , Phys. Rev. D80 , 052002 (2009)). E. A. Kuraev, Yu. M. Bystritskiy, V. V. Bytev, A. N. Ilyichev ( JINR, BLTP, Dubna, Russia National Scientific and Educational Centre of Particle and High Energy Possible explanation of BaBar anomaly with the use of Sudakov vertex September 24, 2010 5 / 7

Annihilation channel The annihilation channel of e + e − → ℓ + ℓ − π 0 , ℓ = µ, τ process: e – (p – ) l – (q – ) π (q π ) e + ( p + ) + e − ( p − ) → γ ∗ ( q ) → → π 0 ( q π ) ℓ + ( q + ) ℓ − ( q − ) , (10) where p 2 ± = 0 , q 2 ± = m 2 ℓ , q 2 π = M 2 , s = q 2 = ( p + + p − ) 2 , s 1 = q 2 1 = ( q + + q − ) 2 . γ * (q) The matrix element of this process is: γ (q 1 ) (4 πα ) 2 1 q 2 J µ J ν ( ℓ ) V ( s ) ǫ µναβ q α q β M = 1 , q 2 e + (p _ ) l + (q + ) J µ = v ( p + ) γ µ u ( p − ) , ¯ J ν ( ℓ ) = v ℓ ( q + ) γ ν u ℓ ( q − ) , ¯ (11) where quantity V ( s ) describes conversion of two off mass shell photons to the neutral pion (pion M 2 � � Q 2 transition formfactor, V ( Q 2 ) = q 2 π 2 F π Q 2 F ). M 2 q The total cross section have a form: � 3 � � ℓ = πα 4 V ( s ) 2 1 − M 2 � s − 5 e → π 0 ℓ ¯ σ e ¯ ln . (12) m 2 6 s 3 ℓ E. A. Kuraev, Yu. M. Bystritskiy, V. V. Bytev, A. N. Ilyichev ( JINR, BLTP, Dubna, Russia National Scientific and Educational Centre of Particle and High Energy Possible explanation of BaBar anomaly with the use of Sudakov vertex September 24, 2010 6 / 7

Conclusions In conclusion we should emphasize once again that applying Sudakov radiative corrections to quark vertex function we imply rather large value of virtualities of one of the photons (i.e. � ≥ 5 GeV 2 ). Thus our approach differs from the ones based on pion wave function � � q 2 � modification A. V. Radyushkin, Phys. Rev. D80 , 094009 (2009) as well as ones based on instanton model A. E. Dorokhov, Phys. Part. Nucl. Lett. 7 , 229 (2010); A. E. Dorokhov, JETP Lett. 91 , 163 (2010); A. E. Dorokhov, Nucl. Phys. Proc. Suppl. 198 , 190 (2010); A. E. Dorokhov, arXiv:1003.4693 which impose some restriction in loop momentum integration. We remind as well the possibility to measure the transition pion form factor in electro-proton scattering ep → eπ 0 p . The relevant cross section will be � 2 V 2 ( Q 2 ) � � � dσ ep → eπ 0 p Q 2 αg ρqq g ρNN F 2 1 ( Q 2 ) + F 2 2 ( Q 2 ) J ( Q 2 ) , = (13) 8 π ( Q 2 + M 2 dQ 2 Q 2 4 M 2 ρ ) p where F 1 , F 2 – are Dirac and Pauli proton form factors and L e = ln Q 2 1 s J ( Q 2 ) 2 L 2 = s + L s ( L e − 1) − ( L e + 1) , L s = ln Q 2 + M 2 , . m 2 e Here instead of virtual photon the virtual vector meson takes place; g ρqq , g ρNN are the ρ meson couplings with quarks and nucleons correspondingly. In this case a problem with background ( ep → e ∆ + → eπ 0 p ) must be overcomed. E. A. Kuraev, Yu. M. Bystritskiy, V. V. Bytev, A. N. Ilyichev ( JINR, BLTP, Dubna, Russia National Scientific and Educational Centre of Particle and High Energy Possible explanation of BaBar anomaly with the use of Sudakov vertex September 24, 2010 7 / 7