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

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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

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Possible explanation of BaBar anomaly with the use of Sudakov vertex

  • E. A. Kuraev1, Yu. M. Bystritskiy1, V. V. Bytev1, A. N. Ilyichev2

1JINR, BLTP, Dubna, Russia 2National 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

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Process e+e− → π0e+e−

The process e+e− → π0e+e−:

π (qπ ) γ (q1) γ*(q) e +(p+ ) e–(p– ) e +(p+

′ )

e –(p–

′ )

π (qπ ) γ (q1) γ*(q) e +(p_ ) e–(p– ) l +(q+ ) l –(q– )

is described my two diagrams: Msc = 2s(4πα)2 q2q2

1

[q × q1]z V (Q2)N+N−, Mann = (4πα)2 q2

1q2 JµJν(l)V (s) ǫµναβqαqβ 1 ,

(1) where N+ = pµ

−

s ¯ v p′

+

γµv (p+) , Jµ = [¯ v (p+) γµu (p−)] , N− = pµ

+

s

  • ¯

u

  • p′

−

  • γµu (p−)
  • ,

Jν(l) = [¯ ul (q−) γνvl (q+)] , (2) and V (Q2) is the vertex, which related with the neutral pion form factor F (Q2) as: V (Q2) = M2

q

2π2FπQ2 F

  • Q2

M2

q

  • .

(3)

  • 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

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Pion Form Factor

Pion form factor for vertex π0 → 2γ can be parameterized in different manners.

π (qπ ) γ (q1) γ*(q) q(k+q1) q(k) q(k+qπ)

In the approach based on QCD collinear factorization theorem (G. P. Lepage and S. J. Brodsky, Phys. Lett. B87, 359 (1979)) V BL(Q2) = 2Fπ 3

1

  • dx

xQ2 φπ(x, s), (4) 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 φπ(x, s) was used. Also in the paper L. Ametller,L. Bergstrom,A. Bramon, and E. Masso, Nucl. Phys. B228, 301 (1983);

  • 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. V (Q2) = m2

π

m2

π + t

1 2 arcsin2( mπ

2MQ )

  • 2 arcsin2( mπ

2MQ ) + 1 2 ln2 βQ + 1 βQ − 1

  • ,

(5) where βQ =

  • 1 +

4M2

Q

Q2 .

  • 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

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SLIDE 4

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) ).

π (qπ ) γ (q1) γ*(q) q(k+q1) q(k) q(k+qπ)

F Q2/M2

q

= −

  • d4k

iπ2 × × Q2RS(Q2, p2

1, p2 2)

(k2 − M2

q + i0)(p2 1 − M2 q + i0)(p2 2 − M2 q + i0),

(6) where p1 = k + q1, and p2 = k + qπ and Sudakov vertex function RS (J. J. Carazzone, E. C. Poggio, and H. R. Quinn, Phys. Rev. D11, 2286 (1975); J. M. Cornwall and G. Tiktopoulos, Phys. Rev. D13, 3370 (1976)) is: RS(Q2, p2

1, p2 2) = exp

  • − αsCF

2π ln Q2 |p2

1| ln Q2

|p2

2|

  • ,

(7) where Q2 ≫ |p2

1,2| ≫ M2 q and CF = N2 − 1 / (2N) = 4/3. We use here the the

Goldberger-Treiman relation on the quark level Fπ = Mq/gq¯

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

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SLIDE 5

Our approach: Results

0.15 0.20 0.25 0.30 0.35 5 10 15 20 25 30 35 Q2 V (Q2), GeV Q2, GeV2

Our model BaBar-2009

The cross section of process e+e− → e+e−π0 is dσ dQ2 = α4 4Q2 V 2(Q2)J(Q2), (8) J(Q2) = 1 2 L2

s + Ls(Le − 1) − (Le + 1),

where Ls = ln

s Q2+M2 ,

Le = ln Q2

m2

e and V (Q2) is the Sudakov vertex:

V (Q2) = A M2

q

2πFπαsCF Φ(zB), (9) where Φ(zB) =

1

  • dx

x

  • 1 − e−zBx(1−x)

, zB = CF αs 2π ln2 Q2 BM2

q

. 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 mq ≈ 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

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SLIDE 6

Annihilation channel

The annihilation channel of e+e− → ℓ+ℓ−π0, ℓ = µ, τ process:

π (qπ ) γ (q1) γ*(q) e +(p_ ) e–(p– ) l +(q+ ) l –(q– )

e+ (p+) + e− (p−) → γ∗ (q) → → π0 (qπ) ℓ+ (q+) ℓ− (q−) , (10) where p2

± = 0, q2 ± = m2 ℓ, q2 π = M2,

s = q2 = (p+ + p−)2, s1 = q2

1 = (q+ + q−)2.

The matrix element of this process is: M = (4πα)2 q2

1q2 JµJν(ℓ)V (s) ǫµναβqαqβ 1 ,

Jµ = ¯ v (p+) γµu (p−) , Jν(ℓ) = ¯ vℓ (q+) γνuℓ (q−) , (11) where quantity V (s) describes conversion of two off mass shell photons to the neutral pion (pion transition formfactor, V (Q2) =

M2

q

2π2FπQ2 F

  • Q2

M2

q

  • ).

The total cross section have a form: σe¯

e→π0ℓ¯ ℓ = πα4V (s)2

6

  • 1 − M2

s 3 ln s m2

ℓ

− 5 3

  • .

(12)

  • 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

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SLIDE 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.

  • q2

≥ 5 GeV2). Thus our approach differs from the ones based on pion wave function 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 → eπ0p. The relevant cross section will be dσep→eπ0p dQ2 =

  • αgρqqgρNN

8π(Q2 + M2

ρ)

2 V 2(Q2) Q2

  • F 2

1 (Q2) +

Q2 4M2

p

F 2

2 (Q2)

  • J(Q2),

(13) where F1, F2 – are Dirac and Pauli proton form factors and J(Q2) = 1 2L2

s + Ls(Le − 1) − (Le + 1),

Ls = ln s Q2 + M2 , Le = ln Q2 m2

e

. Here instead of virtual photon the virtual vector meson takes place; gρqq, gρNN are the ρ meson couplings with quarks and nucleons correspondingly. In this case a problem with background (ep → e∆+ → eπ0p) 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