Two-Loop Effects in Low-Energy Electroweak Measurements

We outline the recent results on the two-loop electroweak contributions to the electron-electron scattering cross sections and asymmetries. Although the two-loop corrections are strongly suppressed relative to the one-loop corrections, they still contribute a few percent to the polarization asymmetry, and even this small contribution cannot be ignored at for ultra-precision experiments such as MOLLER planned at JLab. The NNLO calculation techniques we developed for the electron-electron scattering can be adapted for electron-proton processes, electron-positron collisions, and other low-energy experiments involving leptons.

There are three major ways to look for new physics beyond the Standard Model (SM): the energy frontier (high-energy colliders), the intensity/precision frontier (intense beams) and the cosmic frontier (underground experiments, ground and space-based telescopes). At the precision frontier, one of the most promising processes is polarized electron-electron (Møller) scattering with parity violation, potentially allowing the indirect detection of hypothetical new physics particles coupling to the SM sector through the kinetic mixing. The first measurement of the Møller scattering cross section was done in 1932 [1], but only the recent improvements in precision allowing to measure the parity-violating left-right asymmetry made Møller scattering a candidate for the new-physics sector detection.
The first observation of parity violation in Møller scattering was made by the E-158 experiment at SLAC [2], which studied Møller scattering of 45-to 48-GeV polarized electrons on the unpolarized electrons in a hydrogen target. Its result at low Q 2 = 0.026 GeV 2 , A P V = (1.31±0.14 (stat.)±0.10 (syst.))×10 −7 [3] allowed one of the most important parameters in the Standard Model -the sine of the Weinberg angle (the weak mixing angle) -to be determined with an accuracy of 0.5% (sin 2 θ W = 0.2397 ± 0.0010 ± 0.0008 in the MS scheme).
A recently-completed JLab experiment measuring the electron-proton scattering asymmetry, Q weak [4], aims to determine sin 2 θ W with relative precision of 0.3%. The results of Q weak commissioning run, constituting about 4% of the data collected [5], give the left-right asymmetry of A P V = −279±35(stat)±31(syst) ppb, the smallest and most precise asymmetry ever measured in e-p scattering, and lead to the first determination of the weak charge of the proton, Q p W = 0.064 ± 0.012, in agreement with the SM prediction of Q p W = 0.0710 ± 0.0007. From the theory aspect, the Q weak precision can be improved by the better control of hadronic corrections and accounting for the NNLO contributions to the electron line discussed in this work. With that, and with the full set of data analyzed, Q weak has potential to place tight constraints on the possible SM extensions. Another PV e-p experiment, P2 proposed for the newly-funded MESA facility at Mainz, aims to determine Q p W even more precisely, to 2%.
The next-generation experiment to study e-e scattering, MOLLER (Measurement Of a Lepton Lepton Electroweak Reaction) [6], planned at JLab following the 11 GeV upgrade, will offer a new level of sensitivity and measure the PV asymmetry in the scattering of longitudinally polarized electrons off an unpolarized target to a precision of 0.73 ppb. That would allow a determination of the weak mixing angle with an uncertainty of δ sin 2 θ W (M S) = ±0.00026 (stat.) ± 0.00013 (syst.) [7], or about 0.1%, an improvement of a factor of five in fractional precision when compared with the E-158 measurement. At such precision, any inconsistency with the Standard Model will signal new physics, so the MOLLER experiment, building on the concept of indirect probes, can provide access to physics at multi-TeV scales. The experiment will undoubtedly be more challenging than previous parity-violating electron scattering experiments, but the MOLLER collaboration has extensive experience in the similar experiments such as MIT Bates, SLAC E158, JLab G0, HAPPEX, PREX and Q weak . The major advantage of the Møller scattering is that e-e scattering asymmetry is much less affected by the uncertainties in the hadronic corrections then e-p asymmetry, and the gamma-Z box radiative correction to PV elastic e-p scattering calculated at 11 GeV in [8] has an accuracy sufficient to keep the uncertainty from this background within the limits of the MOLLER experiment. The rest of the electroweak radiative corrections (EWC), although extensive, can in principle be controlled at sub-1% level, with the SM predictions carried out with full treatment of one-loop radiative corrections and at least leading twoloop corrections.
2 It was repeatedly shown in the literature that even oneloop radiative corrections modify the tree-level prediction for the asymmetry quite significantly ( [9], [10], [11], [12], [13]), so it is essential to have them under a very firm control. (Please see [14] for a review of the low-energy measurements of the weak mixing angle and additional references.) In [12], we found the total correction calculated specifically for 11 GeV e-e scattering to be close to ∼ 65%, with no significant theoretical uncertainties. A much larger theoretical uncertainty in the prediction for the asymmetry will come from the two-loop corrections, so, for the new-generation precision measurements, predictions for its scattering asymmetry must include not only a full treatment of one-loop radiative corrections (NLO) but also leading two-loop corrections (NNLO).
We approach the NNLO EWC in stages, by dividing the corrections to the Born (σ B ∼ |M 0 | 2 ) cross section into two classes: the Q-part induced by quadratic oneloop amplitudes (σ Q ∼ |M 1 | 2 ), and the T -part corresponding to the interference of the Born and two-loop diagrams (σ T ∼ 2ReM 2 M + 0 ). The details of our calculations for the quadratic one-loop amplitudes, the Q-part, are shown in [15], where, following the same approach we used for NLO EWC, we performed a tuned step-by-step comparison between different calculation approaches verifying the results obtained by a semi-automatic approach based on FeynArts, FormCalc, LoopTools and Form with the results from the equations derived by hand. As we found in [15], for the MOLLER kinematic conditions, the Q-part of the NNLO EWC can increase the asymmetry by up to 4%, and depends quite significantly on the energy and scattering angles.
In this paper, we discuss a set of contributions corresponding to the interference of the Born and two-loop diagrams (the T-part), including the gauge invariant set of boson self energies and vertices of two-loop amplitude M 2 , and discuss work still to be done in the future.

II. CROSS SECTION AND ASYMMETRY
The cross section of polarized Møller scattering with the Born kinematics: can be expressed as: where σ ≡ dσ/d cos θ and θ is the scattering angle of the detected electron with 4-momentum k 2 in the center-ofmass system of the initial electrons. The 4-momenta of initial (k 1 and p 1 ) and final (k 2 and p 2 ) electrons generate a standard set of Mandelstam variables: M 0 is the Born (O(α)) amplitude shown in Fig.1. The amplitudes M 1 (Fig.2) and M 2 (Figs. [4][5][6][7][8] correspond to one-loop (O(α 2 )) and two-loop (O(α 3 )) matrix elements, respectively. The one-loop amplitude M 1 consists of the boson selfenergy (BSE) (Fig.2a), vertex (Ver) (Fig.2b,c) and box diagrams (Fig.2d,e). We use the on-shell renormalization scheme from [16,17], so there are no contributions from the electron self-energies.
We present the one-loop complex amplitude as the sum of IR and IR-finite parts M 1 = M λ 1 + M f 1 . The IR-finite part M f 1 can be found in [15] and for the IR part we have: where λ is the photon mass and the complex value B can be presented in the following form (see, for example, [18]): Analogously, the two-loop amplitude is the sum Note that the structure of first term in (6) is the same as in (4) in terms of the soft photon factorization.
To cancel the infrared divergences, we split the differential cross sections into λ-dependent (IRD-terms) and λ-independent (infrared-finite) parts: The one-loop cross section is already carefully evaluated with full control of the uncertainties in [12]. The simplest form for IRD-terms are: The imaginary part of the total cross section could be removed in the sum Qand T -parts due to following properties: |δ λ 1 | 2 + Re δ λ 1 2 = 2 Re δ λ 1 2 , and . Thus, in the following sections we can ignore the imaginary part, i. e. δ λ 1 → Reδ λ 1 and B → ReB.

III. BREMSSTRAHLUNG FOR NLO AND NNLO
Bremsstrahlung for both NLO (Fig.3a) and NNLO (Fig.3b,c) is needed to cancel the infrared divergences in the one-loop and two-loop amplitudes, correspondingly.
(Radiation from only one lepton line is shown in Fig.3, but all four lepton lines are accounted for in our calculations, of course.) To evaluate the cross section induced by the emission of one soft photon with energy less then ω, we follow the methods of [19] (see also [20]). Then, this cross section can be expressed as: σ γ = σ γ 1 + σ γ 2 , where σ γ 1,2 have the similar factorized structure based on the factorization of the soft-photon bremsstrahlung: where The first part of the soft-photon cross section, σ γ 1 , cancels the IRD at the one-loop order, while the second part, σ γ 2 , cancels the IRD at the two-loop order, with half of σ γ 2 going to the cancellation of the IRD in the Q-part and the other half going to treat IRD in the T -part. At last, the cross section induced by the emission of two soft photons with a total energy less then ω is calculated in [15] as: where 1 2 is a statistical factor and R 2 = 8 3 π 2 B 2 . Bringing all terms together, we arrive at the result that is free from infrared divergences. For one loop, the logarithms will cancel out: For the second loop, the cancellation proceeds in a more involved way, that is where

IV. NUMERICAL RESULTS
For the numerical calculations at the central kinematic point of MOLLER (E lab =11 GeV, θ = π/2) we use α, m W , m Z and lepton masses as input parameters in accordance with [21]. The effective quark masses which we use for the vector boson self-energy loop contributions are extracted from shifts in the fine structure constant due to hadronic vacuum polarization ∆α (5) had (m 2 Z )=0.02757 [22]. For the mass of the Higgs boson, we take m H = 125 GeV and for the maximum soft photon energy we use ω = 0.05 √ s, according to [12] and [23].
Let us define the relative corrections to the Born cross section due to a specific type of contributions (labeled by C) as In the text below the term "T -part" corresponds to the contributions of a gauge invariant set of the BSE and vertices only. The parity-violating asymmetry is defined in a traditional way: and the relative corrections to the Born asymmetry due to C-contribution are defined as In general, corrections from different diagrams are not additive. Their total contribution is given by where summation is performed over the index i. There are reasons to believe that the correction δ N N LO is small, but we can not say the same about δ N N LO A . In the table below we bring together all contributions derived for relative corrections to the unpolarized cross section to the asymmetry, including contributions that stem from the gauge-invariant set of two-loop vertex and boson-self-energy diagrams. The three dots mean the contribution from the line above, so we progressively add new contributions as we have them ready. So far, as one can see from the table, the Q-part induced by quadratic one-loop amplitudes (∼ M 1 M + 1 ), and the contributions to T -part corresponding to the interference of the Born and two-loop diagrams (∼ 2ReM 2 M + 0 ) considered here shift the result in the same direction. Under the kinematic conditions of the MOLLER experiment, the asymmetry that takes into account the concerted effect of oneand two-loop contributions decreases by about 62.7%. For comparison, the one-loop contribution yield a value of -69.5%. Clearly, there is still a lot to be done, and no definite conclusion can be made until all contributions are accounted for, but it looks like if no major cancellations are introduced by the remaining two-loop contributions, the NNLO effect on the PV asymmetry may be more significant that previously believed. Thus, it is safe to say even now that the research program for the MOLLER experiment must include evaluation of the full set of two-loop corrections. Although our numerical calculations are done for the MOLLER the kinematic conditions, the analytics will be directly applicable for the collider experiments, so we assume the NLO and NNLO contributions will be affecting their cross section asymmetry significantly as well, and will need to be evaluated for these measurements in the future. As one can see from our numerical data, at the MOLLER kinematic conditions, the part of the NNLO EWC we considered in this work can increase the asymmetry by up to ∼ 7%. The Qand T -parts do not cancel each other but, on the contrary, they are adding up to increase the physical PV effect. Clearly, the large size of the investigated parts demands a detailed and consistent consideration of the rest of the T -part, which will be the next task of our group. Since the problem of EWC for the Møller scattering asymmetry is rather involved, a tuned step-by-step comparison between different calculation approaches is essential. To make sure that our calculations are error-free, we control our results by comparing the data obtained from the equations derived by hand with the numerical data obtained with a semi-automatic approach based on FeynArts, FormCalc, LoopTools and Form. These base languages have already been successfully employed in similar projects [12] and [13], so we are highly confident in their reliability.
In the future, we plan to address the remaining twoloop electroweak corrections to match the planned precision of the MOLLER experiment and the possible future experiments at ILC. Clearly, for the electroweak lowenergy experiments briefly outlined in this paper and for other potential future measurements, it is absolutely essential for an excellent control of one-loop and good understanding of two-loop effects [14].
In this paper, we outlined motivation and summarized some of our work on the two-loop electroweak radiative corrections involving the SM particles. Even if the LHC continues to agree with the Standard Model up to 14 TeV, the MOLLER experiment will continue to look for new physics scenarios that could escape LHC detection, like various hidden weak scale scenarios. If the LHC does observe an anomaly, then MOLLER will have enough sensitivity to provide sufficient constraints to distinguish between the possible new physics scenarios (new massive or super-massive Z 0 bosons, for example). To have that kind of sensitivity, the MOLLER aims to measure the PV asymmetry predicted within SM to be about 33 ppb with an overall precision of 0.7 ppb. The advantage of trying to access new physics via such low-energy e-e scattering asymmetry is that a purely leptonic PV asymmetry is one of the few observables whose theoretical uncertainties are well under control. There is no significant contribution from the hadronic sector, the SM Higgs mass, one of our input parameters, is known well enough for our needs, and the full set of NLO (one-loop) electroweak radiative corrections, although large, is now known to better than 0.1%. Just a decade ago, such precision would not be feasible. Now, with the recent development in computer algebra and increased accessibility of computing facilities, we can aim to further improve the SM prediction for PV asymmetry by calculating the radiative corrections at the NNLO (two-loop) level. Since the EWC corrections depend quite significantly on the energy and scattering angles, they would need to be evaluated for each experiment specifically. For example, at the MOLLER kinematic conditions, the part of EWC induced by quadratic one-loop amplitudes (∼ M 1 M + 1 ) will increase the asymmetry up to 4%, but increases dramatically in the higher-energy region [15]. This by itself is not a problem, since the Q-part is now know. However, we still far from making the final conclusion on behavior of the T -part corresponding to the interference of the Born and two-loop diagrams (∼ 2ReM 2 M + 0 ). So far, dominant two-loops contributions to the PV asymmetry are at the order of 1% and they are coming from (Ver + BSE) 2 and double boxes. As far as we know at the moment, the newphysics particles are not going to contribute significantly enough at two loops to warrant full-scale calculations, but they may contribute quite noticeably at the one-loop level, depending on the SM extension employed.

VI. ACKNOWLEDGMENTS
The authors gratefully acknowledge Yu. Bystritskiy and E. Kuraev for their help with this project, and the the Joint Institute for Nuclear Research for hospitality in 2013. This work is supported by the Natural Sciences and Engineering Research Council of Canada and Belarus scientific program "Convergence". AA and SB thank JLab Theory Group for hospitality during their stay in 2014.
[ Wherever possible, we disregard the electron mass m-in particular, m 2 ≪ s, −t, −u. To two-loop terms (whose estimation is the main objective of the present study) inclusive, the differential cross section for the process in (1) has the form , where θ is the scattering angle of the detected electron whose 4-momentum in the c.m. frame of primary electrons is denoted by k 2 . We will now describe all four terms on the right-hand side of Eq. (3): the first term, which is proportional to M 0 M + 0 , determines the cross section in the Born approximation; the second term, which is proportional to ReM 1 M + 0 , determines the cross section to the one-loop electroweak radiative correction inclusive (one can find a detailed analysis of this correction in [20], for example); the third term, which is proportional to M 1 M + 1 , is the socalled Q part (quadratic part) appearing in the twoloop correction and stemming from the square of the one-loop amplitudes (for the first study of the Q part, we refer the interested reader to [27]); and, finally, the fourth term, which is proportional to ReM 2 M + 0 , is the so-called T part of the two-loop correction (two-loop term proper).
We will now spell out the structure of all amplitudes, starting from the Born (O(α)) amplitude M 0 . The diagrams corresponding to this amplitude are shown in Fig. 1. It has the structure of the difference of the t-and u-channel components; that is, where the t-channel currents of the upper and lower electrons in Fig. 1 (that is, beam and target electrons) are J j µ =ū(p 2 )γ µ (v j − a j γ 5 )u(p 1 ). The coupling constants in the currents are given by v γ = 1, a γ = 0, (6) v Z = (I 3 e + 2s 2 W )/(2s W c W ), a Z = I 3 e /(2s W c W ). We recall that I 3 e = −1/2 and that s W and c W are, respectively, the sine and cosine of the Weinberg angle, which are expressed in terms of the Z-and W -boson masses according to the Standard Model rules as Finally, the propagator structure from (4), depends on the square of the momentum transfer in this channel, r, and on the intermediate-boson mass-that is, on the photon mass m γ ≡ λ or the Z-boson mass m Z . The photon mass is set to zero everywhere, with the exception of special cases where it is used to parametrize the infrared divergence. The square of the Born amplitude M 0 gives the well-known Born cross section; that is, Here, use is made of the following condensed notation for coupling constants (vector and axial-vector ones-v γ(Z) and a γ(Z) , respectively) and degrees of electron polarizations: From Eq. (9), one can readily obtain the well-known result for unpolarized Møller scattering in quantum electrodynamics (QED): The one-loop amplitude M 1 is an O(α 2 ) quantity that one can construct by analogy with the Born amplitude. The one-loop amplitude is the sum of PHYSICS OF ATOMIC NUCLEI Vol. 76 No. 7 2013 Fig.1 Feynman diagrams corresponding to radiationfree Møller scattering in the (a) t and (b) u channels.  diagrams belonging to three types (see Fig. 2): boson self-energies (BSE), vertex functions (Ver), and boxes (Box):

One Loop
In the on-shell renormalization scheme [28, 29], which we use here, there are no contributions from the lepton self-energies. It is reasonable to spell out each of the three oneloop contributions. The BSE contribution, which corresponds to the diagram in Fig. 2a, has the form where the structures involve the expressions forΣ ij T (r), which are the transverse components of the γγ, ZZ, and γZ selfenergies. One frequently refers to them as truncated self-energies. In terms of Feynman rules, they formally correspond to an integral along two closed lines in a fermion (or W -boson) loop. The longitudinal components of the BSE are proportional to m 2 /r and are therefore numerically small; in the following, we do not consider them for this reason.
In order to take into account the diagrams involving electron vertex functions (diagrams in Figs. 2b and 2c), we employ the representation in terms of the form factors δF je V,A from [28]. As a matter of fact, the amplitude associated with a vertex can be obtained from the respective Born amplitude by making the substitutions v j → v Fj and a j → a Fj for the coupling constants, where we have for the photon and for the Z boson. We note that, after these substitutions, the coupling constants acquire a dynamical character since Λ γ 1 (contribution of triangle diagrams featuring an extra virtual photon), Λ B 2 (contribution of diagrams involving an extra massive boson B, where B = Z, W ), and Λ W 3 (contribution of three-boson vertices-W W γ and W W Z) are functions of the momentum transfer. For them, we now present expressions calculated on the basis of the onshell renormalization scheme with the renormalization conditions from [28, 30] (in the following, we will use precisely these conditions). These expressions, which are applicable in the kinematical region of the MOLLER experiment, have the form The first term on the right-hand side of Eq. (53), σ γ 1 , cancels the infrared divergence in the first loop, while the second term, σ γ 2 , supplemented with the cross-section component σ γγ corresponding to the emission of two soft photons whose total energy is below ω (see diagrams in Figs. 4b and 4c as an example) cancels the infrared divergence in the second loop; that is, where 1 2 is a statistical factor that stems from the indistinguishability of the two final-state photons and R 2 = 8 3 π 2 B 2 [34]. A detailed calculation of σ γγ can be found in Appendix A of [27].
Bringing all terms together, we arrive at a result that is free from infrared divergences. In the first loop, the removal of infrared divergences is straightforward (the logarithms involved cancel each other): In the second loop, the cancellation proceeds in a more involved way; that is,  We note that our numerical results for the electroweak radiative correction change only slightly in response to a variation of this value over a broad range: for example, the relative error in the self-energies upon the change in m H from 115 GeV (our group used this value earlier) to 125 GeV is not more than 0.3%.
Further, we will consider in detail the central kinematical point (CKP) of the MOLLER experiment at where the quantity contains a complex-valued expression B that factorizes in front of the Born amplitude M0 owing to the symmetry property B(t, u) = B(u, t); that is, The infrared-finite part of the one-loop amplitude can be represented in the form The last term on the right-hand side of Eq. (41) stems from the unfactorable box amplitude in (36); that is, In just the same way as the one-loop amplitude,