Transcript
Final state emission radiative corrections to the process e+ e− → π + π − (γ). Contribution to muon anomalous magnetic moment A. I. Ahmadov, E. A. Kuraev, M. K. Volkov, O. Voskresenskaya, E. Zemlyanaya
June 13, 2013
A. I. Ahmadov, E. A. Kuraev, M. K. Volkov, O. Voskresenskaya, Final stateE. emission Zemlyanaya radiative () corrections to the process e+ e− → π + π − (γ). June Contribution 13, 2013 to muon 1 / 37an
Talk Plan
1 2
Motivation Details of calculation 1 2 3 4 5 6 7 8
General formalism Emission of virtual photons Emission of soft photons Hard real photon emission Results for FSE in point-like pion approximation Insertion of pion form factor. Discussion Acknowledgement Appendix
A. I. Ahmadov, E. A. Kuraev, M. K. Volkov, O. Voskresenskaya, Final stateE. emission Zemlyanaya radiative () corrections to the process e+ e− → π + π − (γ). June Contribution 13, 2013 to muon 2 / 37an
Motivation
Analytic calculation of the contribution to anomalous magnetic moment of muon from the channels of annihilation of an electron-positron pair to a pair of charged pi-meson with radiative correction connected with the final state, as well as corrections to the lowest order kernel are presented. The result with the point-like (1) pi-meson assumptions is apoint = apoint + ∆apoint , (1)
apoint = 7.0866 · 10−9 ; ∆apoint = −2.4 · 10−10 . Taking into account the pion form factor in the frames of the Nambu-Jona-Lasinio (NJL) approach leads to (1) (1) aN JL = aN JL + ∆aN JL , aN JL = 5.48 · 10−8 ; ∆aN JL = −3.43 · 10−9 .
A. I. Ahmadov, E. A. Kuraev, M. K. Volkov, O. Voskresenskaya, Final stateE. emission Zemlyanaya radiative () corrections to the process e+ e− → π + π − (γ). June Contribution 13, 2013 to muon 3 / 37an
General formalism It is known (M. Davier, et al., Eur.Phys.J. C 71(2011),1515; C72(2012),1874) that about seventy three per cent of contribution of hadrons to the anomalous magnetic moment of muon aµ = (g − 2)/2 (B.E. Lautrup, A. Peterman and E. de Rafael, Phys. Rep. 3 (1972),4. S. Brodsky, E. de Rafael, Phys.Rev. v 168, p 1620(1968)) 1 aµ = 3 4π
Z∞
+ −
e dsσB
e →π + π −
(s)K (1) (
s ), M2
(1)
4m3π
with σB (s) being the total cross section in the Born approximation and K
(1)
s ( 2) = M
Z1 0
dx
x2 (1 − x) , x2 + Ms2 (1 − x)
(2)
with M being the muon mass, arises from taking into account the simplest process e+ e− → γ ∗ → π + π − , whereas about sixty per cent of the error arises from the uncertainties associated with the pion pair production from the mechanisms with intermediate states of the lightest vector meson ρ, ω (G.Venanzoni,private communication).
A. I. Ahmadov, E. A. Kuraev, M. K. Volkov, O. Voskresenskaya, Final stateE. emission Zemlyanaya radiative () corrections to the process e+ e− → π + π − (γ). June Contribution 13, 2013 to muon 4 / 37an
General formalism
It seems ”natural” to use the result of experimental measuring of the cross section of the process e+ e− → π + π − . But, unfortunately, the experimentally measured total cross section (omitting the effects of detection of the final particles) includes the emission of both virtual and real photons by the initial electron and positron (ISE) and the final state emission (FSE), and possibly, the interference of amplitudes of the emission of the initial and final particles. Assuming that the contribution of these interference terms to the total cross section is zero (charge-blind set-up), we remain with the problem of including such enhanced factors as the form factor Fπ (s) of the charged pion in the time-like region and the delicate procedure of extracting the effects of the initial state emission (both photons and charged particles). Only part of radiative corrections FSE connected with the final π + π − can be included in the frames of one virtual photon polarization operator used above since one implies + − σ e e →hadrons (s) = ((4πα)2 /s)ImΠ(s).
A. I. Ahmadov, E. A. Kuraev, M. K. Volkov, O. Voskresenskaya, Final stateE. emission Zemlyanaya radiative () corrections to the process e+ e− → π + π − (γ). June Contribution 13, 2013 to muon 5 / 37an
General formalism With the polarization operator defined as a transverse part of the virtual photon self-energy tensor Πµν (q) = (qµ qν − q 2 gµν )Π(q 2 ) and applying the dispersion relation (B.E. Lautrup, A. Peterman and E. de Rafael, Phys. Rep. 3 (1972)4; S. Brodsky, E. de Rafael, Phys.Rev. 168 (1968)1620; B. Krause, arXiv: hep-ph/9607259) q2 Π(q ) = − π 2
Z∞
4m2π
ds ImΠ(s) . s q2 − s
(3)
where m is the pion mass. Replacing the Green function of the virtual photon in the one-loop vertex function by the one containing the polarization operator 1 gµν −i 2 → − q π
Z∞
4m2π
ds −igµν ImΠ(s) 2 , s q −s
(4)
one arrives to the known result of lowest order contribution to aµ from the hadronic intermediate state
A. I. Ahmadov, E. A. Kuraev, M. K. Volkov, O. Voskresenskaya, Final stateE. emission Zemlyanaya radiative () corrections to the process e+ e− → π + π − (γ). June Contribution 13, 2013 to muon 6 / 37an
General formalism
a(1) µ
α2 = 2 3π
Z∞
ds
R(s)K (1) (s/M 2 ) 3s e+ e− →had. , R(s) = σ (s), s 4πα2
(5)
4m2π
and the lowest order kernel is K
(1)
2
(s/M ) =
Z1 0
dx
x2 (1 − x) . x2 + (1 − x)(s/M 2 )
(6)
The problem consist in removing from the experimentally measured cross section the radiative corrections associated with the initial electron-positron state, including the emission of virtual and real photon emission.
A. I. Ahmadov, E. A. Kuraev, M. K. Volkov, O. Voskresenskaya, Final stateE. emission Zemlyanaya radiative () corrections to the process e+ e− → π + π − (γ). June Contribution 13, 2013 to muon 7 / 37an
General formalism
This procedure can be the source of errors and uncertainties. One can include the the pion form factor in the form of the replacing ImΠ(s) → Fπ2 (s)ImΠ(s).
(7)
Below we calculate the contribution to aµ from the processes e+ e− → π + π − and e+ e− → π + π − (γ) assuming pion as a point-like particle, taking into account the emission of virtual and real photons by the charged pions only. To obtain the explicit formulae describing FSE is the motivation of our paper.
A. I. Ahmadov, E. A. Kuraev, M. K. Volkov, O. Voskresenskaya, Final stateE. emission Zemlyanaya radiative () corrections to the process e+ e− → π + π − (γ). June Contribution 13, 2013 to muon 8 / 37an
General formalism The differential (center of mass reference frame (cmf) is implied) and total cross sections of the process e+ (p+ ) + e− (p− ) → π+ (q+ ) + π− (q− )
(8)
in the lowest order of perturbation theory and the assumption of point like pion interaction with the virtual photon have the form dσ πα2 β 3 πα2 β 3 = (1 − c2 ); σ(s) = , dc 4s 3s
(9)
with s = (p+ + p− )2 = 4E 2 is the square of the total energy c = cos θ, and θ is the angle between the directions of the initial electron and the negative charged pion in cmf. Inserting the explicit value of the total cross section we obtain a(1) µ
α2 ρ2 = 6π 2
Z1 0
2
dxx (1 − x)
Z1 0
dββ 4 mµ , ρ= . 2 2 2 4(1 − x) + x ρ (1 − β ) mπ
(10)
(1)
Numeric estimations give aµ = 7.08665 × 10−9 .
A. I. Ahmadov, E. A. Kuraev, M. K. Volkov, O. Voskresenskaya, Final stateE. emission Zemlyanaya radiative () corrections to the process e+ e− → π + π − (γ). June Contribution 13, 2013 to muon 9 / 37an
General formalism
In the next order of perturbation theory we must consider the contribution arising from the correction associated with the emission of virtual and real photons (soft and hard) by the final π + π − state. It results in the replacement σ(s) → σ(s)(1 + δ(s)). Keeping in mind the correction to the kernel we obtain 1 aµ = 3 4π
Z∞
dsσ(s)(1 + δ(s))[K (1) (s/M 2 ) +
α (2) K (s/M 2 )]. π
(11)
4m2
The quantity K (2) (s/M 2 ) was computed in paper (R.Barbieri, E. Remiddi, Nucl Phys. B90 (1975)233). It is presented in Appendix. Radiative correction to the final state π + π − will be considered below.
A. I. Ahmadov, E. A. Kuraev, M. K. Volkov, O. Voskresenskaya, Final stateE. emission Zemlyanaya radiative () corrections to the process e+ e− → π + π − (γ). JuneContribution 13, 2013 to10 muon / 37an
Emission of virtual photons To start with virtual correction we find first the vertex function for scattering of the charged pion in the external field. Than we write it down in the annihilation channel and use to calculate the relevant virtual correction to the cross section. The vertex function of the process π− (p1 ) + γ ∗ (q) → π− (p2 ) has the form Z Nµ dk α , (k) = k 2 − λ2 ; (1) = k 2 − 2p1 k; (2) = k 2 − 2p2 k, Γµ = 4π (k)(1)(2) d4 k dk = 2 , Nµ = (p1 + p2 − 2k)µ (2p1 − k)λ (2p2 − k)λ . (12) iπ Writing Nµ as Nµ = (p1 + p2 − 2k)µ [4p1 p2 + (1) + (2) − (k)] and performing the loop momenta integration, we obtain for the un-renormalized vertex function Γun µ = F
un
2
2
2
(q ) = (2m − q )
Z1 1
α (p1 + p2 )µ F un (q 2 ), 4π
dx m2 qx2 Λ2 [ln + ln − 1] + 3 + ln , qx2 λ2 m2 m2
p21 = p22 = m2 , qx = p1 x + p2 (1 − x), qx2 = m2 − x(1 − x)q 2 , q = p2 − p1 . (13)
A. I. Ahmadov, E. A. Kuraev, M. K. Volkov, O. Voskresenskaya, Final stateE. emission Zemlyanaya radiative () corrections to the process e+ e− → π + π − (γ). JuneContribution 13, 2013 to11 muon / 37an
Emission of virtual photons Here λ, λ are the ultraviolet cut-off parameter and the fictitious photon mass. The regularization consist in the construction F (q 2 ) = F un (q 2 ) − F un (0). So we have Γµ = 2
2
2
F (q ) = (2m − q )
Z1 1
α (p1 + p2 )µ F (q 2 ), 4π
dx m2 qx2 m [ln + ln − 2] + 4[1 − ln ]. qx2 λ2 m2 λ
(14)
Introducing the new variable (1 − θ)2 , θ = −q 2 /m2 and using Z1 1
Z1 1
dx ln qx2
qx2 m2
=
dx 2θ 1 = 2 ln , 2 2 qx m (1 − θ ) θ
2θ 1 2 π2 [ ln θ − 2 ln θ ln(1 + θ) − 2Li (−θ) − ], 2 m2 (1 − θ2 ) 2 6
(15)
A. I. Ahmadov, E. A. Kuraev, M. K. Volkov, O. Voskresenskaya, Final stateE. emission Zemlyanaya radiative () corrections to the process e+ e− → π + π − (γ). JuneContribution 13, 2013 to12 muon / 37an
Emission of virtual photons
we obtain Γµ (p1 , p2 ) = +
α m 1 + θ2 1 (p1 + p2 )µ (ln − 1)( − ln − 1)+ 2 π λ 1−θ θ π2 1 + θ2 2 [ln θ − 4 ln θ ln(1 + θ) − 4Li2 (−θ) − . (16) 4(1 − θ2 ) 3
For the crossing channel γ ∗ (q, µ) → π− (q− ) + π+ (q+ ) we use the substitutions (R. Barbieri, J. A. Mignaco, E. Remiddi, IL Nuovo Cimento, 11A (1972)824) p2 → q− , p1 → −q+ , θ → −x + iǫ, 0 < ǫ << 1, p 1−β x= , β = 1 − (4m2 /s), s = (q+ + q− )2 = 4E 2 . 1+β
(17)
A. I. Ahmadov, E. A. Kuraev, M. K. Volkov, O. Voskresenskaya, Final stateE. emission Zemlyanaya radiative () corrections to the process e+ e− → π + π − (γ). JuneContribution 13, 2013 to13 muon / 37an
Emission of virtual photons
This quantity acquire the imaginary part for s > 4m2 : α (q− − q+ )µ F (x), π 2 1+x λ + 1)( ln x + 1 + iπ) + F (x) = (ln m 1 − x2 1 + x2 4 + ln2 x − π 2 − 4(1 − x2 ) 3 4Li2 (x) − 4 ln x ln(1 − x) + iπ(2 ln x − 4 ln(1 − x)) . Γµ =
A. I. Ahmadov, E. A. Kuraev, M. K. Volkov, O. Voskresenskaya, Final stateE. emission Zemlyanaya radiative () corrections to the process e+ e− → π + π − (γ). JuneContribution 13, 2013 to14 muon / 37an
Emission of virtual photons Writing ReF (x) as ReF (x) = (−1 +
1 + β2 λ L) ln + FV , 2β m 1+β L = ln , 1−β
(18)
we write down the relevant contribution to the total cross section as ∆V σ(s) =
2α3 β 3 1 + β2 λ [(−1 + L) ln + FV (β)], 3s 2β m
(19)
with FV (β) = −1 +
1 + β2 1 1 1−β 2β [L − L2 + π 2 + Li2 ( ) − L ln ]. 2β 4 3 1+β 1+β
(20)
A. I. Ahmadov, E. A. Kuraev, M. K. Volkov, O. Voskresenskaya, Final stateE. emission Zemlyanaya radiative () corrections to the process e+ e− → π + π − (γ). JuneContribution 13, 2013 to15 muon / 37an
Emission of soft photons Consider now the contribution from the emission of the soft real photon channel. It have the form 2 Z ′ 3 α d k q− q+ ∆S σ = − 2 σB (s) − , (21) 4π ω q− k q+ k p where the sign prime means ω = ~k 2 + λ2 < ∆E and it is implied ∆E << E. Using the relations α 4π 2
Z
α 4π 2
′
d3 k m2 α 2∆E 1 = [ln − L], ω (q− k)2 π λ 2β
Z
′
L = ln
1+β , 1−β
d3 k (q+ q− ) α 1 + β2 2∆E = [ln L + J(β)], ω (q+ k)(q− k) π 2β λ
(22)
(23)
A. I. Ahmadov, E. A. Kuraev, M. K. Volkov, O. Voskresenskaya, Final stateE. emission Zemlyanaya radiative () corrections to the process e+ e− → π + π − (γ). JuneContribution 13, 2013 to16 muon / 37an
Emission of soft photons
with J(β) = Li2 (−β) − Li2 (β) + Li2 (
1+β 1 1 1+β 1 ) − L2 + ln2 ( ) − π2 , 2 4 2 2 12
(24)
we obtain the contribution to the cross section ∆S σ(s) =
2α3 β 3 1 + β2 2∆E [(−1 + L) ln + FS (β)], 3s 2β λ 1 1 + β2 + J(β). FS (β) = − 2βL 2β
(25)
Here the prime means ω < ∆E.
A. I. Ahmadov, E. A. Kuraev, M. K. Volkov, O. Voskresenskaya, Final stateE. emission Zemlyanaya radiative () corrections to the process e+ e− → π + π − (γ). JuneContribution 13, 2013 to17 muon / 37an
Hard real photon emission
Consider at least the contribution from the hard photon emission channel ω > ∆E. The matrix element of this process has the form M=
(4πα)3/2 µ J Tµν e(k)ν , s
(26)
with e(k) is the polarization vector of the photon, J µ = v¯(p+ )γ µ u(p− ) is the current associated with the leptons, and Tµν =
1 1 (2q− + k)ν (Q + k)µ + (−2q+ − k)ν (Q − k)µ − 2gµν . (27) 2(q− k) 2(q+ k)
It can be checked that this expression obeys the gauge invariance conditions Tµν q µ = Tµν k ν = 0.
A. I. Ahmadov, E. A. Kuraev, M. K. Volkov, O. Voskresenskaya, Final stateE. emission Zemlyanaya radiative () corrections to the process e+ e− → π + π − (γ). JuneContribution 13, 2013 to18 muon / 37an
Hard real photon emission We use as well the relation (Akhiezer A. I., Berestetskij V. B., Quantum Electrodynamics, Moscow, Science, 1981; J. D. Bjorken, S. Drell ”Relativistic Quantum Fields”, McGraw-Hill (1965)) Z XZ 1 2 µ ν 2 |M | dΓ3 = − T rpˆ+ γ pˆ− γ (gµν − qµ qν /q ) IdΓ3 , 3 spin I = Tρσ T ρσ ,
(28)
with dΓ3 being the element of the phase space of the final particles dΓ3 =
d3 q− d3 q+ d3 k 1 δ 4 (q − q− − q+ − k). 2E− 2E+ 2ω (2π)5
(29)
It can be written as d3 q− d3 q+ d3 k 1 δ(q0 − E− − E+ − ω), 2E− 2E+ 2ω 2E+ (2π)5 q q 2 + 2~ with E+ = (~q− + ~k)2 + m2 = ω 2 + E− q−~k.
(30)
A. I. Ahmadov, E. A. Kuraev, M. K. Volkov, O. Voskresenskaya, Final stateE. emission Zemlyanaya radiative () corrections to the process e+ e− → π + π − (γ). JuneContribution 13, 2013 to19 muon / 37an
Hard real photon emission Performing the integration on cos θ, where θ is the angle in cmf between 3-momenta of pion and photon, we obtain dΓ3 =
π2 s 2ω 2E− dνdν− dν+ δ(ν + ν− + ν+ ), ν = , ν− = , 4(2π)5 q0 q0 2E+ ν= , q0 = 2E. q0
(31)
In terms of energy fractions we obtain I
2(1 − ν) 2(1 − ν) 1 1 + − β 2 (1 − β 2 )[ + ]+ 1 − ν+ 1 − ν− (1 − ν+ )2 (1 − ν− )2 1 1 2 (ν − β 2 )(ν − 1 − β 2 )[ + ]. (32) ν 1 − ν+ 1 − ν−
= 8+ +
The domain of integration D is
∆E m2 ν 2 < ν < β 2 ; ν + ν− + ν+ = 2; (1 − ν− )(1 − ν+ )(1 − ν) > , E r s ν ν β2 − ν (1 − R) < 1 − ν± < (1 + R), R = . 2 2 1−ν
(33)
A. I. Ahmadov, E. A. Kuraev, M. K. Volkov, O. Voskresenskaya, Final stateE. emission Zemlyanaya radiative () corrections to the process e+ e− → π + π − (γ). JuneContribution 13, 2013 to20 muon / 37an
Hard real photon emission Performing the integration over ν± we obtain Z Idν− dν+ δ(2 − ν − ν− − ν+ ) = 4[2R(ν −
β 2 (1 − ν) β 2 (1 + β 2 ) 1+R )+( − 2β 2 ) ln ]. ν ν 1−R
(34)
Performing the further integration we use the substitution t = R, 0 < t < tm , t2m = β 2 − (∆E/E)(1 − β 2 ). The corresponding contribution to the cross section is ∆H σ(s) =
2α3 β 3 1 + β 2 E [( L − 1) ln + FH (β)], 3s 2β ∆E
(35)
with FH (β) = −
1 + β2 1 − β2 1 3 + 7β 2 G(β) + ln − 3 (3 + β 2 )(1 − β 2 )L + , (36) 2 β 4β 8β 4β 2
A. I. Ahmadov, E. A. Kuraev, M. K. Volkov, O. Voskresenskaya, Final stateE. emission Zemlyanaya radiative () corrections to the process e+ e− → π + π − (γ). JuneContribution 13, 2013 to21 muon / 37an
Hard real photon emission
and G(β) =
Zβ 0
dt 1 − t2 ln 2 = 2 1−t β − t2
1−β 1+β Li2 ( ) − Li2 ( ) + Li2 (1 + β) − Li2 (1 − β). 2 2
(37)
The total contribution does not depend on ” photon mass” λ as well as on the auxiliary parameter ∆E: ∆σ =
1 2α3 β 3 1 1 + β 2 [ ( L − 1) ln + FV + FS + FH ]. 3s 2 2β 1 − β2
(38)
A. I. Ahmadov, E. A. Kuraev, M. K. Volkov, O. Voskresenskaya, Final stateE. emission Zemlyanaya radiative () corrections to the process e+ e− → π + π − (γ). JuneContribution 13, 2013 to22 muon / 37an
Hard real photon emission
After some algebra one obtain ∆σ
e¯ e→π π ¯
α πα2 β 3 (s) = 2 σB (s)∆(β), σB (s) = ,β = π 3s
r
1−
4m2π , s
(39)
and 3 4 1 + β2 1 5 ∆(β) = − ln − 2 ln β + [− π 2 + L + 2 2 1−β 2β 12 4 3 1 1−β [1 − L] − L ln β + Li2 ( ) + 3Li2 (−β) − Li2 (β) + 2β 2β 1+β 1−β 1 1+β 1+β 3Li2 ( ) − 2Li2 ( ) + 2 ln β ln(1 + β) − 2Li2 (1 − β) + ln2 ( )]. (40) 2 2 2 2
A. I. Ahmadov, E. A. Kuraev, M. K. Volkov, O. Voskresenskaya, Final stateE. emission Zemlyanaya radiative () corrections to the process e+ e− → π + π − (γ). JuneContribution 13, 2013 to23 muon / 37an
Results for FSE in point-like pion approximation The total contribution to aµ can be obtained from the general formulae (see(9)) by replacement β 4 → β 4 [1 +
2α ∆(β)] = β 4 [1 + δ(s)] π
(41)
Numeric estimation leads to ∆π aµ = −6.923 × 10−11 .
(42)
(1)
A total set of the lowest order RC to aµ takes into account as well the correction to the kernel ker
∆
α3 aµ = 3π 3
Z1 0
s 4 βπ4 dβπ (2) K (b), b = 2 = 2 . 1 − βπ2 M ρ (1 − βπ2 )
(43)
A. I. Ahmadov, E. A. Kuraev, M. K. Volkov, O. Voskresenskaya, Final stateE. emission Zemlyanaya radiative () corrections to the process e+ e− → π + π − (γ). JuneContribution 13, 2013 to24 muon / 37an
Results for FSE in point-like pion approximation Explicit form of the kernel K (2) (b) as well as its expansion in powers b−1 are presented in Appendix. Using the explicit form of K (2) can be successfully applied to the region 1 − βπ2 ∼ 1 and is not convenient for the region 1 − βπ2 << 1. In this region we apply its expansion in powers M 2 /s, which was obtained in (B. Krause, Phys. Lett. B 390, 392 (1997); arXiv: hep-ph/9607259). For this aim we choice an auxiliary parameter β0 ∼ 1 ker
∆
1 aµ = 3 4π
Zβ0 0
βπ4 dβπ (2) K (b)BR + 1 − βπ2
Z1
β0
βπ4 dβπ (2) K (b)Kr . 1 − βπ2
(44)
The result does not depend on β0 and is ∆ker aµ = −1.73 · 10−10 .
(45)
The total contribution of the correction to aµ from RC to both the final π + π − and the kernel is ∆Aµ = ∆π + ∆ker = −2.4 · 10−10 .
(46)
A. I. Ahmadov, E. A. Kuraev, M. K. Volkov, O. Voskresenskaya, Final stateE. emission Zemlyanaya radiative () corrections to the process e+ e− → π + π − (γ). JuneContribution 13, 2013 to25 muon / 37an
Insertion of pion form factor. Discussion The result, obtained in the point-like approximation about an order of magnitude lower than one measured in experiment (M. Davier, et al., Eur.Phys.J. C 71(2011),1515; C72(2012),1874) aµ ≈ 6.974 · 10−8 . The conversion of a virtual photon to the π + π − (γ) state in the time-like region is realized through the intermediate state with vector mesons ρ(775), ω(782), φ(1020) with the following decay to the two pion state. The main contribution arise from ρ(775) meson state. Keeping in mind the resonance nature of this transition it can be taken into account by the replacement in (1) σB (s) → σB (s)Z,
(47)
The contribution of ω(782), Rω arises due to a rather small ρ − ω mixing. It has two sources- one is connected with the quark u, d mass difference md − mu = 3.75M eV, mu = 280M eV , and the other is connected with the transition ω → γ → ρ (M.K. Volkov, Sov. J, Part. Nucl.17 (1986)186 [Fiz. Elem.Chast. Atom. Yadra 17 (1986)433].) Bω = R
s s √ √ . m2ω − s − i sΓω m2ρ − s − i sΓρ
(48)
A. I. Ahmadov, E. A. Kuraev, M. K. Volkov, O. Voskresenskaya, Final stateE. emission Zemlyanaya radiative () corrections to the process e+ e− → π + π − (γ). JuneContribution 13, 2013 to26 muon / 37an
Insertion of pion form factor. Discussion Adding the contribution of the photon and photon-rho meson conversion (M.K. Volkov and D. Kostunin, Phys. Rev. C 86 (2012)025202) √ m2ρ − i sΓρ s √ √ Bγρ = 1 + 2 = 2 , mρ − s − i sΓρ mρ − s − i sΓρ
(49)
we have µ2ρ − µρ + γρ2 + (µ2ρ − 1)2 + γρ2 2 (µ2ω − 1)2 − γρ γω R 2 + [(µω − 1)2 + γω2 ][(µ2ω − 1)2 + γρ2 ] γρ + (µ2ρ − 1)2 + γρ2 2 (µ2ω − 1)(γρ + γω ) R 2 , [(µω − 1)2 + γω2 ][(µ2ω − 1)2 + γρ2 ]
Z(x) = |Bγρ + Bω |2 =
(50)
A. I. Ahmadov, E. A. Kuraev, M. K. Volkov, O. Voskresenskaya, Final stateE. emission Zemlyanaya radiative () corrections to the process e+ e− → π + π − (γ). JuneContribution 13, 2013 to27 muon / 37an
Insertion of pion form factor. Discussion with gρ3 1 m2d 4πα [ ln − ] ≈ 2.27 · 10−4 , 2 2 3gρ 16π mu gρ m2ρ s 30.86 m2 31.43 x = 2 , µ2ρ = = ; µ2ω = ω = , mπ s x s x Γ2ρ 1.07 Γ2 6.08 · 10−2 γρ2 = = ; γω2 = ω = . s x s x R=
(51)
Our final results are a(1) =
α2 12π 2
Z∞ 4
∆a =
α3 6π 3
Z∞ 4
dx 4 Z(x)(1 − )3/2 K(x), x2 x
dx 4 Z(x)(1 − )3/2 [∆(β)K(x) + ∆K(x)]. x2 x
(52)
A. I. Ahmadov, E. A. Kuraev, M. K. Volkov, O. Voskresenskaya, Final stateE. emission Zemlyanaya radiative () corrections to the process e+ e− → π + π − (γ). JuneContribution 13, 2013 to28 muon / 37an
Insertion of pion form factor. Discussion The expression for ∆K(x) is presented in Appendix K(x) = xK
(1)
(x) =
Z1 0
y 2 (1 − y)xdy . y 2 + x(1 − y)ρ2
(53)
The explicit expression for ∆(β) is given above. The result of numerical calculations is (1)
aN JL ≈ 5.48 · 10−8 ; ∆a ≈ −3.43 · 10−9 .
(54)
The contribution of the term of an order of 1/(xρ2 )4 is expected to be on the level of several per cent, which determine the accuracy of our calculations. The relative wight of π + π − hadron state is (1)
aN JL = 0.78. aexp
(55)
A. I. Ahmadov, E. A. Kuraev, M. K. Volkov, O. Voskresenskaya, Final stateE. emission Zemlyanaya radiative () corrections to the process e+ e− → π + π − (γ). JuneContribution 13, 2013 to29 muon / 37an
Insertion of pion form factor. Discussion
Here we do not take into account the contribution of double vacuum polarization – with the two-hadronic insertion and the QED one with the electron-positron intermediate state. Both of them were considered in the recent paper (D. Greynat and E. de Rafael, JHEP 07 (2012) 020). In (A. Hoefer, J. Glusa, F. Jegerlehner, Eur. Phys. J. C24 (2002)59), an attempt to take into account the initial state emission of an additional pair of charged particles from the experimental data was made. In (D. Greynat and E. de Rafael, JHEP 07 (2012) 020), a similar calculation was performed by using the duality approximation (constituent quarks and gluons – hadrons) and applying the result of (G. Kallen and A. Sabry, Danske Videnskab, 29, N 17 (1955)17) for the final state emission of a fermion-anti-fermion pair.
A. I. Ahmadov, E. A. Kuraev, M. K. Volkov, O. Voskresenskaya, Final stateE. emission Zemlyanaya radiative () corrections to the process e+ e− → π + π − (γ). JuneContribution 13, 2013 to30 muon / 37an
Acknowledgement
We are grateful to grant RFBR 11-02-00112 for financial support. We are grateful to Yu.M.Bystritskiy for his attention and help.
A. I. Ahmadov, E. A. Kuraev, M. K. Volkov, O. Voskresenskaya, Final stateE. emission Zemlyanaya radiative () corrections to the process e+ e− → π + π − (γ). JuneContribution 13, 2013 to31 muon / 37an
Appendix The explicit form of the kernel K (2) (b) was obtained in paper of R. Barbieri and E. Remiddi (R.Barbieri , E. Remiddi, Nucl Phys. B 90 (1975)233). The contribution of 14 Feynman diagram was taken into account. It have a form 19 7 23 2 1 139 115 + b+ − b+ b + log(b) + K (2) (b)BR = − 144 72 12 36 144 b−4 4 127 115 2 23 3 log y 9 5 1 2 (− + b− b + b )p + + b − b2 − ξ(2) + 3 36 72 144 4 24 2 b b(b − 4)
5 2 1 17 7 log y b log2 b + (− b + b2 − b3 ) p log b + 96 2 24 48 b(b − 4) 19 53 29 2 1 2 + b− b − + log2 y + 24 48 96 3b b − 4 17 2 7 3 1 13 7 1 Dp (b) + − b + b2 − (−2b + b − b ) p 6 12 3 6 4 b(b − 4) 1 3 4 D (b) 1 7 1 p m b − + ( − b + b2 )T (b), (56) 6 b−4 2 6 2 b(b − 4)
A. I. Ahmadov, E. A. Kuraev, M. K. Volkov, O. Voskresenskaya, Final stateE. emission Zemlyanaya radiative () corrections to the process e+ e− → π + π − (γ). JuneContribution 13, 2013 to32 muon / 37an
Appendix
where √ √ b− b−4 y= √ , √ b+ b−4 1 Dp (b) = Li2 (y) + log(y) log(1 − y) − log2 y − ξ(2), 4 1 1 Dm (b) = Li2 (−y) + log2 y + ξ(2), 4 2 T (b) = −6Li3 (y) − 3Li3(−y) + log2 y log(1 − y) +
1 (log2 y + 6ξ(2)) log(1 + y) + 2 log y(Li2 (−y) + 2Li2 (y)). 2
(57)
A. I. Ahmadov, E. A. Kuraev, M. K. Volkov, O. Voskresenskaya, Final stateE. emission Zemlyanaya radiative () corrections to the process e+ e− → π + π − (γ). JuneContribution 13, 2013 to33 muon / 37an
Appendix The function Li2 (y), Li3 (y) are the dilogarithm and trilogarithm defined through Li2 (y) = −
Zy
Li2 (−y) = −
Zy
Li3 (y) =
Zy Zy
0
Li3 (−y) =
0
0
0
dt ln(1 − t) = − t
Z1
dt ln(1 − ty), t
dt ln(1 + t) = − t
Z1
dt ln(1 + ty); t
0
0
dt [ln t − ln y] ln(1 − t) = t
Z1
dt ln t ln(1 − ty), t
dt [ln t − ln y] ln(1 + t) = t
Z1
dt ln t ln(1 + ty). t
0
(58)
0
A. I. Ahmadov, E. A. Kuraev, M. K. Volkov, O. Voskresenskaya, Final stateE. emission Zemlyanaya radiative () corrections to the process e+ e− → π + π − (γ). JuneContribution 13, 2013 to34 muon / 37an
Appendix In paper of B.Krause (B. Krause, Phys. Lett. B 390, 392 (1997); arXiv: hep-ph/9607259) the expansion on b = s/m2µ was obtained 1 223 23 1 8785 37 367 19 2 K (2) (b)Kr = [ − 2ξ2 − L] + [ − ξ2 − L+ L ]+ b 54 36 b 1152 8 216 144 1 13072841 883 10079 141 2 1 2034703 [ − ξ2 − L+ L ] + 3[ − 2 b 432000 40 3600 80 b 16000 3903 6517 961 2 ξ2 − L+ L ] . (59) 40 1800 80 with ξ2 =
π2 , L = ln b. 6
(60)
In the text above we use 1 1 1 ∆K(x) = 2 [c0 + d0 L + 2 [c1 + d1 L + d2 L2 ] + [c2 + d2 L + e2 L2 ] + ρ xρ (xρ2 )2 1 [c3 + c3 L + d3 L2 ]], L = ln(xρ2 ) (61) (xρ2 )3
A. I. Ahmadov, E. A. Kuraev, M. K. Volkov, O. Voskresenskaya, Final stateE. emission Zemlyanaya radiative () corrections to the process e+ e− → π + π − (γ). JuneContribution 13, 2013 to35 muon / 37an
Appendix
The numeric values are c0 = −0.843; d0 = −0.639; e0 = 0;
c1 = 0.027; d1 = −2.8; e1 = 0.132;
c2 = −6.01; d2 = −2.8; e2 = 1.76; c3 = −33.1; d3 = −3.62; e3 = 12.01.
(62)
In Figure 1 the βπ dependence of the exact integrand (2) FBR (βπ ) = βπ4 KBR /(1 − βπ2 ) and its expansion in powers of M 2 /s (2) FKr (βπ ) = βπ4 KKr /(1 − βπ2 ) are presented. One see the large compensations in FBR take place for β → 1.
A. I. Ahmadov, E. A. Kuraev, M. K. Volkov, O. Voskresenskaya, Final stateE. emission Zemlyanaya radiative () corrections to the process e+ e− → π + π − (γ). JuneContribution 13, 2013 to36 muon / 37an
Figures 0.5 0.4 0.3
K(2)β4/[1−β2]
0.2 0.1 0 −0.1 −0.2 −0.3 −0.4 −0.5 0
0.2
0.4
β
0.6
0.8
1
Figure: Dependence of K (2) (β): solid line - exact formulae, dashed line - power M 2 /s expansion.
A. I. Ahmadov, E. A. Kuraev, M. K. Volkov, O. Voskresenskaya, Final stateE. emission Zemlyanaya radiative () corrections to the process e+ e− → π + π − (γ). JuneContribution 13, 2013 to37 muon / 37an
