NLO radiative corrections for Forward-Backward and Left-Right Asymmetries at a B-Factory
A. Aleksejevs S. Barkanova C. Miller J. M. Roney V. Zykunov
hep-ph




myheadings





aaleksejevs@grenfell.mun.ca



sbarkanova@grenfell.mun.ca
Grenfell Campus of Memorial University, Corner Brook, Canada


calebmiller@uvic.ca
University of Victoria, Victoria, Canada


mroney@uvic.ca
University of Victoria, Victoria, Canada


vladimir.zykunov@cern.ch
Joint Institute for Nuclear Research, Dubna, Russia 
Francisk Skorina Gomel State University,  Gomel, Belarus




This paper presents the first
 calculations of the parity-violating polarization asymmetry and forward-backward asymmetry of the
e^- e^+ →μ^+ μ^- (γ) process at a center-of-mass energy of 10.58 GeV
with up to one-loop electroweak radiative corrections.




The calculations are relevant for future precision electroweak measurements at the Belle II experiment, which is now collecting data at
 the SuperKEKB e^- e^+ collider with a center-of-mass energy at the mass of the Υ(4S) resonance.

In this paper we take under full control the bremsstrahlung process at the conditions of Belle II/SuperKEKB, and the possibilities for a soft photon approach are discussed.
The scale of the obtained relative corrections to the parity-violating and forward-backward asymmetries is significant and the scattering angle dependencies of the asymmetries is non-trivial. 
As an additional validation cross-check using an independent formulation, the calculated asymmetries are compared to results from the  Monte Carlo generator.






 12.15.Lk, 13.88.+e, 25.30.Bf  



NLO radiative corrections for Forward-Backward and Left-Right Asymmetries at a B-Factory
    Vladimir Zykunov
    December 30, 2023
========================================================================================







§ INTRODUCTION





Electroweak measurements can be made at a  high luminosity electron-positron collider   B-factory, such as Belle II/SuperKEKB <cit.> operating at a center-of-mass (CM) energy of  E_cm=√(s)=10.579 GeV (the mass  of the  Υ(4S) meson), via γ-Z interference in the process e^- e^+ → f f̅. In the Standard Model this interference term is parameterized in terms of the axial vector coupling of the fermion f, equal to its third component of weak isospin, g_a(f) = I_3(f), and its vector coupling, g_v(f) = I^3_f - 2 Q_f sin^2θ_W (I^3_e,μ,τ=-1/2, I^3_ν=+1/2, and cosθ_W = m_W/m_Z), where Q_f is its electric charge and θ_W is the weak mixing angle. The precision on the measurement of the effective weak mixing angle, and hence the effective vector couplings of the neutral current, would be comparable to those measured on the Z^0 pole at LEP and SLC, but at a much lower energy, if the  electron beam of the B-factory has at least a 70% spin polarization <cit.> in a left-right asymmetry measurement. Currently, SuperKEKB does not have a polarized beam and the work presented here is a necessary component of the physics justification for installing polarization in that machine in a potential upgrade. Without polarized beam, Belle II/SuperKEKB could still measure the forward-backward asymmetry but with a significantly lower precision on sin^2 θ_W^eff, as shown in this paper. A forward-backward asymmetry measurement would, however, still provide a useful measurement of the axial vector coupling  constant for the final-state fermion, f.


With a polarized beam,  the vector current couplings to electrons, muons, taus, s-quarks, c-quarks, and b-quarks can be measured and would enable a precision comparison with the Standard Model predictions of their running from 10.579 GeV to the Z-pole. Deviations of the
 running would signal the presence of new physics. On the other hand, assuming the
 running holds, these measurements can be used to significantly reduce the uncertainties
 on the Z-pole values of the couplings.
The electroweak fits that now include the measured Higgs boson parameters <cit.>
 show reasonable internal consistency, but there is a  2.5σ deviation
associated with the determination of the Z b b̅ couplings and  sin^2θ_W^eff from the
forward-backward asymmetries for b-quarks at LEP. The tension is even greater, 3.2σ, between this bb̅
 determination of  sin^2 θ_W^eff and that
  from SLD, which provides the single most precise determination of  sin^2 θ_W^eff  using a left-right asymmetry measurement. 
 Therefore, it would be interesting to have additional precision measurements of the Z b b̅ vertex.
Because SuperKEKB produces B mesons just above threshold it would have
 a unique ability to measure the neutral current vector coupling
of b-quarks in a manner that is  free from fragmentation uncertainties <cit.> and would provide
a significant decrease in its uncertainty compared to the value measured at LEP, 
where the dominant systematic error came from fragmentation uncertainties. 


In order to  extract reliable information from the experimental data, it is necessary
to take into account higher order effects of electroweak theory, i.e. electroweak radiative corrections (EWC). 
The procedure for the inclusion of EWC is an indispensable part of any modern experiment, 
but will be of  paramount importance for precision electroweak measurements of Belle II/SuperKEKB.
Consequently, theoretical predictions for the observables 
must include not only full treatment of one-loop radiative corrections (NLO) 
but also leading two-loop corrections (NNLO).


Significant theoretical effort already has been dedicated to NLO EWC to electron-positron annihilation
starting with <cit.>, where EWC for this process with arbitrary polarization are calculated for center-of-mass (CM) energies between 40 and 140 GeV.
For the LEP and SLC colliders the process  e^- e^+ → ff̅
demanded consideration of the EWC at Z-boson pole with new precision.
The following collaborations have performed this task:
BHM and WOH  <cit.>,
LEPTOP <cit.>,
TOPAZ0 <cit.>,
and ZFITTER <cit.>.
More recent results for EWC in “after LEP/SLC” era are provided by   <cit.> and SANC <cit.> codes.


















The main goal of this 
work is to calculate the full set of one-loop (NLO) EWC with the highest precision possible.

 In order to avoid technical errors and to provide a validation cross-check,
 we do the same calculations in two independent and different ways and compare the results – first,
 with a semi-automatic approach (“computer algebra") employing
FeynArts <cit.>, FormCalc <cit.>, LoopTools <cit.> and Form <cit.>, 
 with no simplifications, and then analytically (“by hand”), in a compact asymptotic form.
 Sect. II details the calculated differential cross sections up to one-loop. The bremsstrahlung process at the lower energies of Belle II/SuperKEKB is fully accounted for in Sect. III, with both a soft photon approximation (SPA) and a more exact hard photon approach (HPA). The analysis of the results obtained through the semi-automatic and asymptotic methods is given in Sect. IV, as well as the comparison of the soft-photon and hard-photon approaches. 
In addition, a comparison is made with results from the  Monte Carlo generator. 
 The sensitivity studies of left-right polarization and forward-backward asymmetries are described in Sect. V.
 Our conclusions and future plans are discussed in Sect. VI.







§ NLO ELECTROWEAK CORRECTIONS AT SIMPLEST CASE: GENERAL NOTATIONS AND MATRIX ELEMENTS


In our calculations we will start with the simplest case of e^- e^+ → f^-f^+ (γ) scattering, where f=μ.
First we will disregarded the electron mass m and final-state fermion mass m_f (valid for f=μ)
wherever possible, and second we treat energy in the CM system  of e^- e^+ 
as a small parameter, in comparison to the masses of W/Z bosons:

    m, m_f ≪ E ≪ m_W,Z.

For this case  we can obtain the total NLO EWC in compact and relatively simple form, free from unphysical parameters
and suitable for an analysis of the kinematic behavior for a given reaction.


Let us start by writing the cross section for the scattering of polarized electrons on unpolarized positrons,

    e^-(p_1) + e^+(p_2) → f^-(p_3)+f^+(p_4),

using the Born approximation shown in Fig. <ref>, we find:

    σ≈π^3/2s |M_0|^2.





Here σ is a short notation for the differential cross section

    σ≡dσ/d(cosθ),

θ  is the scattering angle of the detected muon
with 4-momentum p_3 in the CM system of the initial electron and positron. 
The  4-momenta of initial (p_1 and p_2) and final
(p_3 and p_4) fermions generate a standard
set of Mandelstam variables:

    s=(p_1+p_2)^2, t=(p_1-p_3)^2, u=(p_1-p_4)^2.


Defining M_0 as the Born (O(α)) 
amplitude (matrix element),
we describe the structure of M_0:

    M_0 = ∑_j=γ,Z M^j,M^j = iα/π I_μ^j  D^j J^μ,j,

where the electron and muon currents are

    I_μ^j = u̅(-p_2)γ_μ (v_e^j-a_e^jγ_5) u(p_1),J_μ^j = u̅(p_3)γ_μ (v_f^j-a_f^jγ_5) u(-p_4)

and D^j is represented by:

    D^j=1/s-m_j^2+im_jΓ_j  (j=γ,Z),

which depends on the Z-boson mass (m_Z) and width (Γ_Z), or
on the photon mass m_γ≡λ. 
The photon mass  is set to zero everywhere with the exception of specially indicated
cases where it is taken to be an infinitesimal parameter that regularizes the infrared divergence (IRD). 


The squared amplitude M_0  forms the Born cross section:

    σ^0 =π^3/2s |M_0|^2 
    = πα^2/s∑_i,k=γ,Z D^iD^k^* μ^ikik,

where

    μ^ikjl = T_+ λ_+^ikjl - T_- λ_-^ikjl,  T_± = t^2 ± u^2,

and


    λ_+^ikjl = λ_1^ikλ_fV^jl, λ_-^ikjl = λ_2^ikλ_fA^jl,  λ_1^ik = λ_eV^ik-p_B λ_eA^ik,λ_2^ik = λ_eA^ik-p_B λ_eV^ik,

with p_B representing the degree of electron polarization.


The λ-type functions have the following structures (here g=e,f):

    λ_gV^ij=v_g^iv_g^j + a_g^ia_g^j,λ_gA^ij=v_g^ia_g^j + a_g^iv_g^j,

where the vector and axial coupling constants are

    v_g^γ=-Q_g, a_g^γ=0, v_g^Z=(I_g^3-2Q_gs_W^2)/(2s_Wc_W),
     a_g^Z=I_g^3/(2s_Wc_W),

Q_g is the electric charge of particle g in units of the proton's charge.
Let us recall that I_g^3=-1/2, I_ν^3=+1/2 etc. and
s_W  (c_W)
is the sine (cosine) of the Weinberg mixing angle expressed in terms of the Z- and W-boson
masses according to the on-shell definition in the Standard Model:

    c_W=m_W/m_Z,s_W=√(1-c_W^2).










At the Next-to-Leading-Order (NLO), we can introduce the NLO 
 differential cross section(O(α^3)) via an interference term given by the second term of the following expansion:


    σ = π^3/2s |M_0+M_1|^2≈π^3/2s (M_0M_0^† + 2  [M_1M_0^†] ).


Here, the one-loop amplitude M_1 has structure of the sum of boson self-energy (BSE),
vertex (Ver) and box diagrams (see  Fig. <ref>):




    M_1=M_ BSE+M_ Ver+M_ Box.

We use the on-shell renormalization scheme from <cit.>,
so there are no contributions from the electron self-energies.
The infrared-finite BSE term can easily be expressed as:

    M_ BSE= iα/π∑_i,j=γ,Z I_μ^i  D^ij_S J^μ,j,

with 

    D_S^ij=-D^iΣ̂_T^ij(s) D^j,

where Σ̂_T^ij(s)
is the transverse part of the renormalized photon, Z-boson and γ Z self-energies.
The longitudinal parts of the boson self-energy make contributions
that are proportional to  m^2/r (r=s,t,u);
therefore they are very small and are not considered here.


For the Belle II experiment, the CM energy of the electron and positron is √(s)=10.579 GeV. 
Specifically for the Hollik renormalization conditions <cit.>, we have the following numerical results for 
the truncated and renormalized self energies (Σ̂^ij_T):

    [Σ̂_T^γγ(s)] D^γ s = -0.0361, [Σ̂_T^γ Z(s)] D^γ s     = -0.0301, [Σ̂_T^ZZ(s)] D^Z s                = -0.0317,


    [Σ̂_T^γγ(s)] D^γ s =  0.0159, [Σ̂_T^γ Z(s)] D^γ s     = -0.0056, [Σ̂_T^ZZ(s)] D^Z s                = -0.0003.



In order to derive the vertex amplitude (2nd and 3rd diagrams in Fig. <ref>),
we use the form factors notation in the manner similar to the work of <cit.>. 
Here, we will replace the coupling constants v_g^j, a_g^j with the form factors
 v_g^γ (Z)→ v_g^F_γ (Z), a_g^γ (Z)→ a_g^F_γ (Z),
where for the photon

    v_g^F_γ    =    α/4π[ Λ_1^γ
       + ((v_g^Z)^2 + (a_g^Z)^2 ) Λ_2^Z
       + 3/4s_W^2Λ_3^W ],


    a_g^F_γ    =    α/4π[ 2v_g^Za_g^Z Λ_2^Z
       + 3/4s_W^2Λ_3^W ],

and for  Z-boson

    v_g^F_Z    =    α/4π[ v_g^Z Λ_1^γ
       + v_g^Z ((v_g^Z)^2 + 3(a_g^Z)^2 ) Λ_2^Z +
    
    
    
     1/8s_W^3c_WΛ_2^W  -
     3c_W/4s_W^3Λ_3^W ],


    a_g^F_Z    =    α/4π[ a_g^Z Λ_1^γ
       + a_g^Z ( 3(v_g^Z)^2 + (a_g^Z)^2 ) Λ_2^Z +
    
    
    
    1/8s_W^3c_WΛ_2^W  -
       3c_W/4s_W^3Λ_3^W ].


The function Λ_1^γ corresponds to the contribution of triangle diagrams with the photon in the loop,
 Λ_2 corresponds to  the triangle diagrams with the massive boson – Z or W, and Λ_3 corresponds to the  the triangle diagrams with 3-boson vertices  – WWγ or WWZ. These complex functions have been studied in detail
 and presented, for example, in  <cit.>.







Hence,

    M_ Ver = iα/π∑_j=γ,Z( 
     I_μ^F_j  D^js J^μ,j + I_μ^j  D^js J^μ,F_j).

The infrared singularity is regularized by giving the photon a small mass λ
and in the vertex amplitude can be extracted in the form:

    M_ Ver^λ =
    -α/π( lns/mm_f -1 )  lns/λ^2 M_0.

The remaining (infrared-finite) part of the vertex amplitude has a simple form
convenient for further analysis:

    M_ Ver^f =
    M_ Ver - M_ Ver^λ =
    M_ Ver(λ^2 → s).




The box amplitude can be presented as a sum of all two-boson exchange contributions:

    M_ Box= M_γγ +M_γ Z + M_ZZ +M_WW.

We need to account for both direct and crossed γγ, γ Z and ZZ-boxes:

    M_ij= M_ij^D + M_ij^C  (i,j=γ,Z),

but, obviously, for WW-boxes we only need the direct expression.
The infrared parts of the γγ- and γ Z-boxes are similarly given by

    M_γγ (γ Z)^λ =
    α/2πlnu/tlntu/λ^4  M_0.

The finite part of the γγ-box can be found in <cit.>.
Using asymptotic methods, we can significantly simplify the box amplitudes
containing at least one heavy boson (see, for example, <cit.>, where simplifications were done
on the cross-section level).
Finally, we provide the expressions for M_ii^D,C in the low energy approximation:

    M_ii^D
    =  - i (α/π)^2   1/16 m_i^2u̅(-p_2) γ^μγ^αγ^ν (v_e^B-a_e^Bγ_5) u(p_1) 
     ·u̅(p_3) γ_νγ_αγ_μ (v_f^B-a_f^Bγ_5) u(-p_4),



    M_ii^C
    =  i (α/π)^2   1/16 m_i^2u̅(-p_2) γ^μγ^αγ^ν (v_e^B-a_e^Bγ_5) u(p_1) 
     ·u̅(p_3) γ_μγ_αγ_ν (v_f^B-a_f^Bγ_5) u(-p_4),

with the coupling-constants combinations for ZZ- and WW-boxes (B=ZZ, WW)

    v^ZZ=(v^Z_g)^2+(a^Z_g)^2, a^ZZ=2v^Z_ga^Z_g, v^WW=a^WW=1/(4s_W^2).





Now we are ready to present the one-loop  amplitude as the sum of 
IR-divergent (index λ) and IR-finite (index f) parts: M_1 = M_1^λ + M_1^f,
where

    M_1^λ = α/2πΓ_1^λ M_0,Γ_1^λ = 4 B lnλ/√(s).

and the value B can be presented in the form

    B= lnst/m m_f u - 1.




Using (<ref>), it is straightforward to write the expression for the NLO cross section:

    σ_1^V =π^3/s [M_1M_0^†] =  σ^λ_1 + σ^f_1,

where IR-divergent and regularized NLO cross section is given by

    σ^λ_1 = α/πΓ_1^λσ^0.

The IR-finite part can be represented using the notation of the relative correction (Γ_1^f)

    σ^f_1 = α/πΓ_1^f  σ^0 
    = σ_ BSE + σ^f_ Ver + σ^f_ Box,

where at one-loop level the cross sections are written as follows:

    σ_ BSE   =   2 πα^2/s∑_i,j,k=γ,Z D_S^ijD^k^* μ^ikjk,
    
    σ^f_ Ver   =   2πα^2/s∑_i,k=γ,Z D^iD^k^* 
    [ μ^F_ikik + μ^ikF_ik ],
    
    σ^f_ Box   =   π^3/s (M^f_γγ +M^f_γ Z + M_ZZ +M_WW)  M_0^†.

In (<ref>), the IR-finite part of vertex form factors was used according (<ref>).





§ BREMSSTRAHLUNG:  CANCELLATION OF INFRARED DIVERGENCE 



The bremsstrahlung diagrams are illustrated in Fig. <ref>,
where the first two diagrams correspond to initial state radiation (ISR),
whereas the last two correspond to final state radiation (FSR).





We express the full differential cross section for the process

    e^-(p_1) + e^+(p_2) → f^-(p_3)+f^+(p_4)+γ(p),


as

    dσ_R = α^3/π^2 s∑ |R|^2 dΓ_3,

where phase space is defined as

    dΓ_3 = Γ (p_1+p_2-p_3-p_4-p) d^3p_3/2p_3_0d^3p_4/2p_4_0d^3p/2p_0


and

    ∑ |R|^2 = ∑_i,j=γ,Z( Q_e^2R_e^ij+Q_eQ_f R_i^ij + Q_f^2R_f^ij).

where the three terms in the sum are the ISR, interference and FSR parts, respectively.

The ISR part can be written as

    R_e^ij = -Π^i Π^j^* 
     Tr[ (  γ^μ-2p_1^ρ+p̂γ^ρ/z_1 
                  + 2p_2^ρ-γ^ρp̂/v_1γ^μ)
               1/2 (λ_1^ij - λ_2^ijγ_5 ) p̂_1    
    
    (  -2p_1^ρ+γ^ρp̂/z_1γ^ν
                  + γ^ν2p_2^ρ-p̂γ^ρ/v_1)
               1/2p̂_2  ]
     Tr[ γ_μ (λ_f V^ij - λ_f A^ijγ_5) p̂_4 γ_νp̂_3  ].

The FSR part can be found by substitution 
R_f^ij = R_e^ij (Π^j → D^j, p_1,2↔ - p_4,3, 
λ_1,2↔λ_fV,fA),
and for the interference term we have

    R_i^ij = -Π^i D^j^* 
     Tr[ (  γ^μ-2p_1^ρ+p̂γ^ρ/z_1 
                  + 2p_2^ρ-γ^ρp̂/v_1γ^μ)
               1/2 (λ_1^ij - λ_2^ijγ_5 ) p̂_1   
            γ_ν1/2p̂_2  ]
    
     Tr[ γ_μ (λ_f V^ij - λ_f A^ijγ_5) p̂_4
            (  -2p_4^ρ-γ^ρp̂/vγ^ν
                  + γ^ν2p_3^ρ+p̂γ^ρ/z)
          p̂_3  ] -
    
    
     -D^i Π^j^* 
     Tr[ γ_μ1/2 (λ_1^ij - λ_2^ijγ_5 ) p̂_1
    (   -2p_1^ρ+p̂γ^ρ/z_1γ^ν
                  + γ^ν2p_2^ρ-γ^ρp̂/v_1)
            1/2p̂_2  ]
    
     Tr[ 
            ( γ^μ-2p_4^ρ-γ^ρp̂/v
                  + 2p_3^ρ+p̂γ^ρ/zγ^μ)
    (λ_f V^ij - λ_f A^ijγ_5) p̂_4
    γ_νp̂_3  ].

For the radiative case, the truncated propagator has the following form

    Π^j=1/s-z-v-m_j^2+im_jΓ_j  (j=γ,Z).



In the last three equations we have used four radiative invariants (they tend to zero at p → 0):

    z_1=2p_1p, v_1=2p_2p, z=2p_3p, v=2p_4p,

together with three invariants s, t, u, and taking into account the 
momentum conservation, we can write the following identities

    z-z_1=v_1-v,  s+t+u=v+2m^2+2m_f^2.

Here we have five (4+3-2=5) independent variables in the description of bremsstrahlung process.
Phase space of the emitted photon dΓ_3 can be expressed in the basis of these invariants

    dΓ_3 = π/16 sdt dv dz dv_1/√(-Γ_4),

and  -Γ_4  is a usual Gram determinant.


Next we divide the bremsstrahlung cross section into soft and hard parts using a separator ω.
The soft part σ^γ(ω) is  integrated under the condition that the photon energy 
(all energies are in the CM system  of e^- e^+) is less than ω.
The hard part of bremsstrahlung cross section σ^γ(ω,Ω)  corresponds 
to the photon energy  greater than ω and less than Ω.
To evaluate the cross section induced by the emission of a single soft photon,
we follow the methods of Berends et al.  <cit.> (see also  <cit.>, <cit.>).
To obtain the result, we must calculate the 3-dimensional
integral over the phase space of the emitted real soft photon:

    L(λ,ω) =
    -1/4π∫_p_0<ωd^3 p/p_0
    T^α(p) T_α(p)   =  -Γ_1^λ + R_1,

where

    T^α(p)
    =p_1^α/p_1p - p_2^α/p_2p +p_3^α/p_3p - p_4^α/p_2p,

and

    R_1=-4B ln√(s)/2ω 
    - ( lnm^2/s +1/2ln^2m^2/s + π^2/3)
    
    
     		- ( lnm_f^2/s +1/2ln^2m_f^2/s + π^2/3)
    		+2 _2-t/u -2 _2-u/t.


As a result the soft cross section can be factorized in terms of the Born cross section in this soft-photon bremsstrahlung approximation:

    σ^γ(ω)= α/π[ -Γ_1^λ +R_1 ] σ^0.

In the rest of the article we will refer to it as the Soft Photon Approximation (SPA). 
The contribution due to soft photons is evaluated in with our semi-automatic approach, with no further simplifications. 



The hard photon approach (HPA) fully accounts for the photon in the final state, where the HPA emission cross section is calculated with a Monte Carlo integration technique
using the VEGAS routine <cit.> in the region ω≤ p_0 ≤Ω. 
The hard photon bremsstrahlung cross section can be expressed as

    σ^γ(ω,Ω)= 
    α^3/8π s∫_ω≤ p_0 ≤Ωdv  dz  dv_1/√(-Γ_4)s-v/s∑ |R|^2 θ(-Γ_4).

Here we have used the ultra-relativistic form of the Jacobian (s-v)/s, which originates in the transition from radiative t invariant

    t= 1/2( 2m^2+2m^2_f-s+v+cosθ√(s-4m^2/s)√((s-v)^2-4m^2_fs))

to the cosine of the scattering angle: cosθ. The integral in (<ref>) can be evaluated first analytically  over the 
variables v_1 and z (explicit details are given in <cit.>), and then numerically.











Putting it all together at one-loop, we get:

    σ^1 
    = σ_1^V + σ^γ(ω) + σ^γ(ω,Ω).

Obviously σ^1 does not depend on either λ or ω.
The independence on the mass of the photon can be justified by direct analytical cancellations of λ, 
and as a result we get

    σ_1^λ + σ^γ(ω) 
    =   α/π R_1  σ^0.

Independence on ω is obvious by definition.
But since the hard photon bremsstrahlung integration was performed numerically, we verify that and
observe ω independence with a relative numerical uncertainty not exceeding order of 10^-4.








§ NUMERICAL RESULTS


Electroweak input parameters of the on-shell renormalization scheme
(m_W, m_Z and α) are naturally defined as measurable
quantities with fixed values at all orders of perturbation theory.
As a result, the s^2_W=1-m_W^2/m_Z^2 definition of the weak mixing angle 
is also fixed at all orders of perturbation theory.



From muon decay one can establish the relationship between the most
precisely measured quantity, Fermi constant G_μ=1.1663787(6)·10^-5 ^-2,
and the m_W. This can be achieved by comparing muon
lifetimes calculated in Fermi four-fermion interaction theory and
the Standard Model calculations at one-loop level. This gives the following
relationship:

    m_W^2    =    πα/√(2)G_μs^2_W(1-Δ r).

Here Δ r is a radiative correction which is calculated in
the on-shell renormalization scheme <cit.> and has the following structure:

    Δ r     =    Σ̂_WW(0)/m_W^2+α/4π s_W^2(6+7-4s_W^2/2s_W^2ln c_W^2)+c_W^2/m_Z^2s_W^2[Σ̂_γ Z^2(m_Z^2)/m_Z^2+Σ̂_γγ(m_Z^2)].

Here, Σ̂_V_1 V_2 is defined as truncated and renormalized self-energy graph for V_1→ V_2 mixing.

The formulae (<ref>) and (<ref>) gives the effective
m_W value of 80.4628, which we use in our calculations. For the numerical calculations we have used α=1/137.035 999, m_Z=91.1876, and m_H=125  as input parameters  according to <cit.>. 


The electron, muon, and τ-lepton
masses are taken as m_e=0.510 998 910,  m_μ=0.105 658 367, m_τ=1.776 84
and the quark masses for loop contributions as
m_u=0.069 83, m_c=1.2, m_t=174,
m_d=0.069 84, m_s=0.15, and m_b=4.6.
The light quark masses provide a shift in the fine structure constant due to hadronic
vacuum polarization Δα_had^(5)(m_Z^2)=0.02757 <cit.>,
where

    Δα_had^(5)(s)=α/3π∑_q=u,d,s,c,b Q_q^2 (lns/m_q^2-5/3).

Here, we choose to use the light quark masses
as parameters regulated by the hadronic vacuum polarization.






Let us introduce superscript C which corresponds to the specific type of contribution to a cross section or asymmetry.
C can be 0 (Born contribution), 1 (one-loop EWC contribution), or 0+1 (both these types):
C={0, 1, 0+1}.
The relative correction to the unpolarized differential cross section (denoted by subscript 00) is

    δ_00= 
     σ^1_L+σ^1_R/σ^0_L+σ^0_R=
    σ^1_00/σ^0_00,
            
where the subscripts L and R on the cross sections correspond
to the degree of polarization for electron p_B = -1 and p_B = +1, respectively.
The relative correction to the unpolarized total cross section is

    δ_T= 
     Σ^1_F+Σ^1_B/Σ^0_F+Σ^0_B=
    Σ^1_T/Σ^0_T,
            
where forward and backward cross sections are defined as

    Σ^C_F =∫_0^cos aσ^C_00· d(cosθ),  Σ^C_B =∫_-cos a^0 σ^C_00· d(cosθ).

The relative correction to integrated cross section is

    δ_Σ= 
     Σ^1_L+Σ^1_R/Σ^0_L+Σ^0_R,
            

where the left and right integrated  cross sections are given by

    Σ^C_L =∫_cos b^cos aσ^C_L· d(cosθ),  Σ^C_R =∫_cos b^cos aσ^C_R· d(cosθ),

and the integration is over the cosine of the polar angle of the out-going negative fermion.

The parity-violating (left-right) asymmetry is defined in a traditional way

    A^C_LR =
     σ^C_L-σ^C_R/σ^C_L+σ^C_R,

which is at the Born level has the following structure

 
    A^0_LR   =   -s/4m_W^2(y-1)^2/2(y-1)y+11-4s_W^2/s^2_W
    
       =   -2s/m^2_Z[a_e v_μ+a_μ v_e (1-2y)/2(y-1)y+1],

with y=-t/s. The left-right integrated asymmetry is constructed from integrated cross sections 

 
    A^C_LRΣ =
      Σ^C_L-Σ^C_R/Σ^C_L+Σ^C_R.

Born results for the integrated asymmetry can be written in the following form

 
    A^0_LRΣ   =   -s/8m_W^21-4s_W^2/s^2_W2cosacosb+6(cosa+cosb)+cos2a+cos2b+8/2cosacosb+cos2a+cos2b+8
    
       =   -2s/m^2_Z[a_e v_μ+ a_μ v_e 6(cosa+cosb)/2cosacosb+cos2a+cos2b+8].


In the case, when we consider full acceptance (a=0^∘ and b=180^∘), expressions for the integrated asymmetry simplify considerably:

 
    A^0_LRΣ|_0^∘^180^∘=-s/8m_W^21-4s_W^2/s^2_W=-2s/m^2_Za_e v_μ=-√(2)G_μ s/παs_W^2 c_W^2 a_e v_μ =-1/√(2)G_μ s/πα g_a(e) g_v(μ).


The choice of the polarization asymmetry (or integrated asymmetry) as one of the observables is driven by its high sensitivity
to Weinberg mixing angle. In case the physics beyond the Standard Model has a parity violating contributor (as for a  Z^' boson), it would be best to use
A^C_LR and A^C_LRΣ in the study of the properties of new physics particles.
By analogy, the forward-backward asymmetry is defined as
 
    A^C_FB =
      Σ^C_F-Σ^C_B/Σ^C_F+Σ^C_B,


At the Born level A^0_FB is found to be:


    A^0_FB =a_ea_μ6s  cos a/3+cos^2 a s(1+2v_ev_μ)-m_Z^2/(s-m_Z^2)^2+2s v_ev_μ (s-m_Z^2)+s^2(v_ev_μ+a_ea_μ),

here, and in the above formulas, {v_f,a_f}≡{v^Z_f,a^Z_f}. Since A^0_FB is directly proportional to the product a_ea_μ, it is a very useful observable if we would like to search for the candidates beyond the SM, with an axial part of the coupling only.

Finally, we would like to define the NLO absolute corrections to the Born asymmetries: 

    Δ_LR
    = A_ LR^0+1-A_ LR^0, Δ_FB
     = A_ FB^0+1-A_ FB^0, Δ_LRΣ
     = A_ LRΣ^0+1-A_ LRΣ^0.
            


In our analysis we start with comparison between the asymptotic and full semi-automatic calculations.
The results for the relative correction δ_00 using the SPA approach can be found in Table <ref>
 for different μ^- scattering angles in the CM of the e^-e^+ system.
Table <ref> shows the asymptotic and full semi-automatic results, respectively.
For the cut on the maximum energy of emitted soft photon,  we take 
γ_1=ω/√(s).
Here we used γ_1=0.05; this corresponds to the maximum photon energy 
0.05·√(s) = 0.52885 for Belle II conditions.
We also found very good agreement between the two approaches for any reasonable choice of γ_1.






Various numerical results for asymmetries and radiative corrections are presented on Figs. <ref>–<ref>.
Here, for the cut on energy of the emitted hard photon,  
in the center-of-mass system of e^- and e^+, we used Ω = 2.0.


As we can see on Fig.<ref>, the correction to the unpolarized cross section related to the forward/backward  kinematics is not negligible. 
The correction in the region 50^∘≤θ≤ 130^∘ is linearly decreasing with its central value at  ∼5.0%. 
It is important to note that our comparison between asymptotic and full semi-automatic results (see Tbl.<ref>) has used only the soft-photon contribution to the unpolarized cross-section and that obviously disagrees with the values of the correction on Fig.<ref> (left plot), where the hard photon bremsstrahlung contribution was also included. For the L-R polarization asymmetry on Fig.<ref>, we observe a
 standard dependence of the asymmetry on scattering angle. 
Here, as expected, the asymmetry reaches its maximum value at forward angles, which is explained by the short range interaction regime, 
where the parity violating Z-boson exchange dominates the contribution to the numerator of the asymmetry term. At backward angles we observe that the asymmetry
 is trending towards a zero value due to the large range interaction regime, where short range 
Z-boson exchange has a negligible contribution, and hence the entire L-R asymmetry goes to zero. 

The total cross section and NLO correction, as a function of detector acceptance, are shown on Fig.<ref>.

The correction to the total cross section reaches the value of ∼46.8%, for full geometrical acceptance, and is relatively constant.



The integrated L-R asymmetry A^0+1_LRΣ and its NLO correction Δ_LRΣ are shown on Fig.<ref>. The maximum value of A^0+1_LRΣ (for a=10^∘ and b=170^∘) is approximately equal to the average value of differential
 L-R asymmetry, which also corresponds to A_LR^0+1 at θ=90^∘.
 Results for the calculated A^0+1_FB asymmetry are shown in Fig.<ref>.
 
 
Figs.<ref>-<ref> are dedicated to the sensitivity study of calculated observables to the cuts on the energy of emitted soft photons. In these plots we show dependencies of the observables on the photon's energy cut  Ω, where the dashed line was 
obtained using the soft-photon approach only, and the solid line corresponds to the calculation with hard-photon emission. 

As it can be seen, for the asymmetries, either A_LR^0+1, A_FB^0+1 or A_LRΣ^0+1, the two approaches start 
to deviate significantly at Ω≈ 0.5. 
This justifies the importance of inclusion of hard-photon emission calculations when it is required to provide analysis for 
observables such as asymmetries. 

However, for the various cross sections such as dσ^0+1, Σ^0+1_T or Σ^0+1_00 
the discrepancy between two approaches start to become visible only at Ω≈ 4.5, 
which is rather close to the maximum energy of emitted photons, Ω =  5.2885. 
Since the calculations in the soft-photon approach are considerably simpler, we can rely on SPA when dealing with cross section calculations.





Comparisons with  Monte Carlo:

The 𝒦𝒦<cit.> Monte Carlo code is used by a number of particle physics experiments, including BaBar, Belle and Belle II, to simulate
e^+e^-→μ^+μ^- (nγ) and e^+e^-→τ^+τ^- (nγ) events. 
In ,  photon emission effects from the initial beams as well as outgoing fermions are calculated up to second order,
 including interference effects, using Coherent Exclusive Exponentiation (CEEX) <cit.> and
electroweak corrections using the DIZET library, which is based on the on-shell renormalization scheme <cit.>.
The calculations of this work are compared to those provided by  version 4.19, which uses DIZET version 6.05.
 In order to carry out these comparisons the particle masses used in  were changed to match those in Sect. <ref> and
 the Weinberg mixing angle, which is also an input to ,  was set to the value corresponding to the on-shell value of
sin^2θ_W = 0.221392,  as described by (<ref>).   

  Two billion e^+e^-→μ^+μ^- events were generated with  for both a left-handed polarized e^- beam and a right-handed polarized e^- beam. Each simulated event was required to produce both muons within an angular acceptance
 of a=10^∘ and b=170^∘. From the simulated events comparisons were made with each observable in Figs.<ref>-<ref>. 
 For Figs.<ref>-<ref> the  results were binned in cosθ with bins 0.125 in width.
 The mean of each bin was used to determine the cosθ value of the points.
 In both of these figures the  results are in agreement with our calculations.
 In order to obtain the differential cross section in  we calculate the integrated cross section in the bin and then normalize it by the width of the bin. In Figs.<ref>-<ref> the  events are binned by angular acceptance. Note that as one end of the bin is fixed and the other moved to various angular cuts, some  events populate multiple bins and therefore the points are not statistically independent. 
 Using  the forward-backward asymmetry was determined with two separate methods. The
 first method counts the events that fall in an angular acceptance between ± a and 90^∘ with a 2 GeV Ω cut, shown in Fig.<ref>(Left).
 The second method counts events as a function of the Ω cut in an angular acceptance of a=30^∘ and 90^∘, as seen in Fig.<ref>(Right). The forward-backward asymmetry seen in Fig.<ref> shows an offset of a  few percent between our calculations and the  results.
  This is most likely a result of A_FB receiving a substantial contribution  from the IR-finite part
 of the photon bremsstrahlung terms (see Fig.8, left plot). In our case we consider only one-photon emission in initial and final states,
 while  accounts for higher photon multiplicity when the bremsstrahlung contribution to A_FB is calculated. 

 In Fig.<ref> we switch from angular acceptances to cuts on the energy of the emitted photon. As multiple photons are produced in , not just a single photon, we define Ω as Ω_KK =√(s)/2(1-s'/s), where s is the square of the center-of-mass energy and s^' is the square of the invariant mass of the muon pair. In Figs.<ref>-<ref> the  events are binned according to the Ω cut value of the event. As some events populate multiple bins as the Ω cut value is varied, this again leads to statistical correlations between bins on these plots. 
 The level of agreement between the  cross sections and the NLO corrected (0+1) cross sections can be seen in Fig.<ref>.
 Fig.<ref>  compares the   results with our calculations of A_LR as a function of the  Ω cut.
 In order to compare our calculations of A_LR at θ=90^∘ as a function of the Ω cut  to those of  (Fig.<ref>(Left)), 
 the acceptance for charged muons generated by  is set to 70^∘<θ<110^∘, a region over which the A_LR dependence on cosθ is linear
 to a good approximation  (see Fig.<ref>).
 Note that, again, the point-to-point correlations are large.
 From Fig.<ref>(Right), it is evident that in the region 1 GeV<Ω <3 GeV  the  results integrated over 10^∘<θ<170^∘ 
 are in good agreement with the HPA calculation, within the  statistical uncertainties.
  This statistical uncertainty arises from the finite number of events generated by  for each of  the two e^- polarization states.
 Accounting for the numbers of  events from each sample that survive the acceptance and Ω requirements,
the absolute  statistical uncertainty on A_LR is ± 1.9× 10^-5.



 It is evident that at low values of Ω there is significant  disagreement between  and the SPA and HPA treatments. This is a result of the fact that  addresses infrared divergences via exponentiation whereas in the SPA and HPA treatments, the infrared divergences persist.






 




§ SENSITIVITY STUDY


We next study the sensitivities 
of the  observables A^0+1_LRΣ (<ref>) and A^0+1_FB (<ref>)
to the effective weak mixing angle (s̅_W^2≡sin^2θ_W^eff) and vector part
of the Z-boson to fermion coupling (v^Z_eff=I_3 - 2 Q_g s̅_W^2).







In order to represent the e^+e^-→μ^+μ^- matrix
element with the simple effective Born-like amplitude, we can use 
leading order low energy one-loop oblique corrections to the Born matrix element. 
Overall we can write for the QED and electroweak parts <cit.>: 

    M_γ    =    α(s)Q_eQ_μ/s(v̅_eγ_νu_e)(u̅_μγ^νv_μ),
    
    
    M_Z    =    G_μ/√(2)κm_Z^2/s-m_Z^2+is/m_ZΓ_Z(v̅_eγ_ν[I^3_e-2s̅_W^2(s)Q_e-I^3_eγ_5]u_e)(u̅_μγ^ν[I^3_μ-2s̅_W^2(s)Q_μ-I^3_μγ_5]v_μ).

Here α(s) represents the running value of fine structure constant,
defined as 

    α(s)     =    α/1+[Σ̂_γγ(s)]/s,

and s̅_W^2(s) defines
the effective running Weinberg mixing angle through the following
expression 

    s̅_W^2(s)     =     s_W^2-s_Wc_W[Σ̂_γ Z(s)]/s+[Σ̂_γγ(s)].

Parameter κ, is defined based on relationship to expression (<ref>)
in the following way:

    κ    =    1-Δ r/1+[∂/∂ sΣ̂_ZZ(s)].

The effective mixing angle is frequently used as one
of the primary parameters in precision electroweak physics and here we study the dependencies of
 A^0+1_LRΣ and A^0+1_FB on s̅^2_W.


To start with, we show on Tbl.<ref>, s̅_W^2(s)
computed in different renormalization schemes at zero and Z-pole kinematics.
Our calculated on-shell values of s̅_W^2(s) compare favorably

with those calculated in the MS scheme, as reported in the PDG MS.






For the kinematics relevant to the Belle II experiment, √(s)=10.58 GeV,
the on-shell effective value of s̅_W^2(s) is equal to 0.23413.
In order to study the sensitivity of the polarization asymmetry
to the variation of s̅_W^2(s), we can simply vary the value of m_W, then calculating
s̅_W^2(s) and asymmetries, we construct parametric dependencies of the asymmetry on s̅_W^2(s) or v^Z_eff.
It is important to note that in the analysis of the sensitivity of the asymmetries we took the cut on the bremsstrahlung photons at 2.0.



In order to evaluate the experimental asymmetry uncertainties that feed into the sensitivities, we  make the following reasonable assumptions 
regarding  pertinent experimental parameters  that potentially can be achieved at Belle II/SuperKEKB if there is an upgrade that
introduces polarization:

  
  * the  electron beam polarization is p_B= 0.7000 ± 0.0035, the positron beam is unpolarized.
  
  * p_B can measured with 0.5% precision, and this dominates the systematic error on A_LR.
  
  * A_FB can be measured  with an absolute systematic uncertainty of 0.005.
  
  * Belle II collects 20 ab^-1 of data with the electron beam  polarization 
        and selects e^- e^+ →μ^- μ^+(γ)   events with  50% efficiency.

  
  * The average √(s), which has a root-mean-square (RMS) spread of 5 MeV <cit.>, is known to ± 1.2MeV
 of the peak of the Υ(4S) resonance[SuperKEKB operations, following past practice of previous generation e^+e^-  B-factories, will ensure that √(s) is at the peak of the Υ(4S) by scanning the energy of one of the beams in a manner that maximizes the rate of e^+e^-→ hadrons throughout of data-taking runs. As the RMS spread in √(s) is significantly smaller than the Υ(4S) width (20.5±2.5MeV), the average value of √(s) will be known to ± 1.2MeV, the experimental precision on the Υ(4S) mass <cit.>.].



With such parameters we can expect an absolute statistical uncertainty on both A_FB and A_LR of 9.4× 10^-6.
This gives a total uncertainty on A_LR Σ(with b=170^∘) of ±0.0000094(stat)±0.0000030(syst)=±0.0000097(total).
The error is  dominated by the statistical uncertainty and  gives a relative uncertainty on
A_LR Σ of 1.6%.
The total uncertainty on A_FB(with a=10^∘; b=170^∘) is ± 0.0050 (total).
In this case, the uncertainty  is completely dominated by the systematic uncertainty and gives a 
relative error on A_FB of 9.4%.

The reason for this difference in relative uncertainties is that the systematic
error on A_LR scales as the relative error because p_B is a multiplicative correction
needed for the measurement and has no other large systematic error since essentially all
other detector systematic errors cancel. 
On the other hand, for A_FB  the dominant systematic errors arise in the detector and do not fully cancel: it is necessary to
measure the angles and forward and backward acceptances, the boost to transform into the CM frame,
and understand any charge asymmetries in the detector. As these are systematic uncertainties in the 
detector asymmetries,  they are absolute uncertainties on A_FB.















Fig.<ref> shows a linear dependence of A^0+1_LR Σ on s̅_W^2 (s=10.58^2 ^2).
That is evident from the fact that the polarization asymmetry is proportional
to the interference term: 2[M_γM_Z^*],
which is linearly proportional to s̅_W^2(s). 
As it can be seen from Fig.<ref>,  the absolute uncertainty for A^0+1_LRΣ equal to 
± 0.000097, translates into an uncertainty of 0.21% on s̅_W^2(s) at s=10.58^2 ^2.

In general the on-shell extraction of s̅_W^2(s) from an experimental
polarization asymmetry could be done by determining the effective
m_W from the measured A^0+1_LRΣ, and then determine s̅_W^2(s)
for that specific effective m_W from equation (<ref>).


It is known that in the time-like region, the value of s̅_W^2(s) changes  rapidly near resonances. Although we have not included
 the effect of hadronic resonances in our treatment, we can estimate the impact of such an effect on the precision of an asymmetry measurement made
 at the peak of the Υ(4S) resonance using Fig. 1 of reference<cit.>. 
From that figure, sin^2θ_W  changes by approximately 0.003 over the 20.5 MeV full width of the Υ(4S)
 resonance and therefore there is a sensitivity of Δ_sin^2θ_W/Δ_√(s)=0.00015/MeV 
 in the region of the peak of the Υ(4S) resonance. As √(s) is known to ±1.2 MeV, in the 
interpretation of the integrated A_LR measurement in terms of  s̅_W^2(s),
this translates into an uncertainty on  s̅_W^2(s) of approximately 0.00018, or 0.08%, which contributes
a small additional uncertainty:  adding this in quadrature with the 0.21% coming from the other uncertainties yields a total uncertainty on
s̅_W^2(s) of 0.22%. 
We note that this uncertainty will be common to measurements from each fermion species and therefore
will cancel in evaluations of fermion universality of the  weak mixing angle performed with A_LRΣ at Belle II.





In a similar fashion we can study the sensitivity of the forward-backward asymmetry
to the variations of s̅_W^2(s). The Fig.<ref> shows the similar
dependence of A^0+1_FB on s̅_W^2(s), but with a substantially smaller
slope when compared to Fig.<ref>.

Although the numerical value of A^0+1_FB is much larger than A^0+1_LRΣ,

it's sensitivity to s̅_W^2(s) is rather low. This translates
to an uncertainty of 19.8% on s̅_W^2(s), if we consider
± 0.00050 uncertainty in A^0+1_FB.




Clearly A^0+1_FB contains substantial contributions, which are not sensitive to parity-violating physics. At this point we would like 
to determine the most dominant contributions to A^0+1_FB and their nature.
We will start with the basic definition of various QED and Weak contributions
in the forward-backward asymmetry:

    A^0+1_FB    =Σ_F^0+1-Σ_B^0+1/Σ_F^0+1+Σ_B^0+1=Γ_FB^0+1/Σ_T^0+1

The denominator of (<ref>), is defined as a total integrated
unpolarized cross section including one-loop corrections. We will
keep this part of A^0+1_FB unmodified. This way contributions to the asymmetry are additive.
As for the numerator of A^0+1_FB,
it will be divided into Born, various infrared finite NLO, and soft-bremsstrahlung
contributions. More specifically:

    Γ_FB^0+1    =Γ_FB^0+Γ_FB^γ-SE(γ)+Γ_FB^γ-TR(γ)+Γ_FB^γ-BB(γ)+Γ_FB^γ-SE(Z)+Γ_FB^γ-TR(Z)+Γ_FB^γ-BB(Z)+
       Γ_FB^Z-SE(γ)+Γ_FB^Z-TR(γ)+Γ_FB^Z-BB(γ)+Γ_FB^Z-SE(Z)+Γ_FB^Z-TR(Z)+Γ_FB^Z-BB(Z)+Γ_FB^soft

Here, Γ_FB^0 is the forward-backward Born contribution to the numerator
of A^0+1_FB, and Γ_FB^γ-SE(γ) (for example) and
corresponds to the interference term between Born QED and γ-γ
Self-Energies (SE). Furthermore TR and BB stand for Triangle and Box
type graphs, respectively. 

Our starting point is to show Born and fully corrected forward-backward asymmetries.
We do this for both renormalization conditions, based on <cit.> and <cit.>. 

Results on Fig.<ref> are represented by the infrared finite parts
of virtual and soft-bremsstrahlung corrections only. That would also
be true for all partial NLO contributions appearing in (<ref>).
We have observed practically zero contributions coming from Γ_FB^γ-SE(γ),Γ_FB^γ-TR(γ),Γ_FB^γ-SE(Z),Γ_FB^γ-BB(Z),Γ_FB^Z-SE(γ),Γ_FB^Z-SE(Z),Γ_FB^Z-TR(Z),Γ_FB^Z-BB(γ)
and Γ_FB^Z-BB(Z) terms in (<ref>). This implies
that contributions coming from all types of self-energies and electroweak
(γ-Z, Z-Z and W-W) boxes are negligible, and can be disregarded.
It is important to note that generally electroweak self-energies
or vertex correction graphs are not gauge invariant and hence their
independent contributions have no physical meaning. However, for the
forward-backward asymmetry this can be bypassed, since gauge dependent contributions
largely cancel out even for separate parts, such as self-energies
or vertex correction graphs. We have verified this by comparing self-energies
(or triangles) contributions in the different renormalization conditions
(Denner and Hollik) and found that the results are identical.
At this point we only show contributions which are substantial and
can not be avoided in the calculations of A^0+1_FB. 



As we can see from Fig.<ref> (two top graphs), we have identical
(symmetrical) contributions from interference terms, such as: 2[M_0^γM_one-loop^EW-TR]
and [M_0^ZM_one-loop^γ-TR]. The biggest contribution
comes from the interference term between γ-Born and γγ-Box
(see Fig.<ref>, second row, left). Overall, all one-loop contributions
are systematically additive and the result is shown on Fig.<ref>,
second row, right graph. Since it is clear now that the addition of one-loop
contributions (blue, dot-dashed curve) and Born (green, solid curve)
term would not reproduce the full result for A^0+1_FB, we turn our attention
to IR finite terms of soft-photon bremsstrahlung. 



It is clearly visible on Fig.<ref> (left) that the bremsstrahlung
contribution largely cancels out one-loop results and produce the correction,
shown on  Fig.<ref> (right and blue dot-dashed curve). The addition of the one-loop correction (Fig. <ref>, left and blue dot-dashed curve ) 
and Born result (solid green curve on the same plot) produce the final result for A^0+1_FB (dashed yellow curve).
One of the possible explanations for such a large cancellation could be found
in the fact that both of the IR finite parts of virtual one-loop correction and soft-photon
bremsstrahlung contain collinear divergent terms, which cancel out
in the final result.

 

Overall we conclude that A^0+1_LRΣ is the observable most sensitive 
to the effective electroweak parameters. As such, in order to search for physics beyond the Standard Model at the  
 precision frontier of neutral-current measurements,
 it is crucial to have polarized electron beams in Belle II/SuperKEKB in order to measure A^0+1_LRΣ.





§ CONCLUSION


In this paper we compare the results for the full set of
one-loop EWC to parity violating polarization and forward-backward asymmetries
at the Belle II/SuperKEKB CM energy obtained by different methods.
The soft photon approximation using an exact semi-automatic approach is validated by an asymptotic approach with simplifications giving a compact form.
We take under full control the bremsstrahlung process and 
compare results for the soft and hard photon calculations.
We also evaluate the sensitivity to the variation of s̅_W^2 for both polarization and forward-backward asymmetries. We find that the highest sensitivity is achieved for the measurements using A_LRΣ with a polarized electron beam. 
In addition, we have analyzed various NLO contributions to the IR finite part of A_FB^0+1. 
As a result, we found that the large contribution arising from interference terms between {γ,  Z}-Born, {γ-γ}-Box, and {γ,  Z}-Triangle graphs are compensated by the IR finite part of the soft-photon bremsstrahlung contribution and that self-energies, although important for the overall cross-sections, cancel out for the
forward-backward asymmetry and therefore have an overall negligible contribution to that asymmetry. A comparison is also made with the  Monte Carlo generator for the A_LR and A_FB asymmetries.
Where infrared divergencies are small, our current calculations are in good agreement with those of the  Monte Carlo. 


We plan to broaden these studies to include left-right asymmetries in e^+e^- collisions for Bhabha scattering and for massive final-state fermions 
(tau leptons, charm and bottom quarks), where the negligible-mass assumption is not valid.




In order to further reduce the theoretical uncertainties, our next step is 
 to include the two-loop EWC in the on-shell renormalization scheme, and compare these to the calculation in the MS scheme.

Nonetheless, the results of this paper  demonstrate that the Standard Model predictions for A_LR at 10.58 GeV, and consequently the weak mixing angle at that
 energy, 
are already under excellent theoretical control and provide encouragement to upgrade SuperKEKB with a polarized e^- beam in order to provide a new tool in the 
 search for physics beyond the Standard Model.



§ ACKNOWLEDGMENTS

We are grateful to Eduard Alekseevich Kuraev for stimulating discussions. 
V.Z. thanks the Acadia University (Wolfville, Canada) and Memorial University (Corner Brook, Canada)  for hospitality. 
This work was supported by the Belarusian State Program of Scientific Research “Convergence-2020”
and the Natural Sciences and Engineering Research Council of Canada.





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