PoS(LL2018)010 https://pos.sissa.it/ -right asymmetry and the relative § , Lidia Kalinovskaya, ections. Longitudinal polarization of energy Bhabha with taking 980 Russia hown. The results are relevant for several , 141980 Russia , 141980 Russia ty, Rostov-on-Don, 344090 Russia ive Commons onal License (CC BY-NC-ND 4.0). ∗† ‡ , Yahor Dydyshka, Leonid Rumyantsev [email protected] Speaker. Also at the Dubna University, Dubna,deceased 141980, Russia. Also at the Institute of Physics, Southern Federal Universi In this report we presentinto theoretical account predictions complete for one-loop high- electroweak radiative corr future - collider projects. the initial beams iscorrection assumed. to the distribution Numerical in the results scattering angle for are the s left § † ‡ ∗ Copyright owned by the author(s) under the terms of the Creat c Attribution-NonCommercial-NoDerivatives 4.0 Internati

E-mail: Dmitri Bardin Andrej Arbuzov Bogoliubov Laboratory of Theoretical Physics, JINR, Dubna Renat Sadykov Dzhelepov Laboratory of Nuclear Problems, JINR, Dubna, 141 Electroweak radiative corrections to polarized Bhabha scattering Serge Bondarenko Bogoliubov Laboratory of Theoretical Physics, JINR, Dubna Loops and Legs in Quantum Field29 Theory April (LL2018) 2018 - 04 MaySt. 2018 Goar, Germany PoS(LL2018)010 n − − e e in- + + (2.2) (2.3) (2.1) − e e e + project, e SANC Andrej Arbuzov e straightforward standard cal- . are the electron energy and the for the outgoing ones: ) group on the complete one-loop abha scattering [1] is one of the nergy hadron colliders, 4 4 st cases. The substantially higher ϑ p heoretical support [17, 18]. In par- p larized electron and has Form Factors (FF). ent generators to simulate Bhabha ( ized positron and electron with the to this process with polarized initial + , tion of various effects from both ex- ons (RC) to Bhabha scattering were with the results of alternative systems e ero tree-level amplitudes the FFs have th polarized beams at the future e ratio of the electron mass to the beam SANC S), respectively. ering with polarized beams. In order to and ϑ or the luminosity determination at 2 and 3 in. In the framework of the )+ / p in the final state, and therefore provide the 3 s in general with an index labeling the corre- cos , p d ˜ √ ( 2 F F | − k e A 1 | + s 1 −→ linear colliders [19, 20] always demonstrated a great π 1 ) = − 2 32 e p F + ( = e − e σ d )+ 1 computer system. The amplitude is parameterized by a certai p ( + e SANC for the incoming particles and 2 p and 1 p is the covariant amplitude of the process, A In comparison with processes being studied at modern hing-e The process of electron-positron scattering known as the Bh The complete one-loop covariant amplitude comes out from th Let us consider elastic scattering of longitudinally polar In this report we present the recent results [21] of the the complete one-loop electroweakpresented (EW) in [16] radiative in correcti the form of Helicity Amplitudes (HA) and colliders. Within thebeen Standard thoroughly Model studied for the many scatteringparticles years of were [2–12]. discussed unpo Corrections inscattering [13, were 14]. created, see e.g. Several [15] Monte and references Carlo there ev key processes in particle physics. In particular it is used f EW RC to polarized Bhabha scattering 1. Introduction where angle in the center-of-mass system (CM 2.1 Covariant one-loop amplitude culation by means ofnumber the of form factors (FFs) which are denoted by sponding Lorentz tensor structure. For processes with non z the form teractions have a clean initial state,possibility a to lower multiplicity perform much moreenergy precise range measurements of in the mo futureperimental colliders and also theoretical demands sides. re-estima Precise measurements wi colliders ILC and CLIC definitelyticular, require physical a programs modern of advanced the t future calculation of the EW radiative correctionsverify to our Bhabha results, scatt we performed severalwhere tuned available. comparisons 2. Bhabha Scattering Cross Section four momenta interest to the effects related to the beam polarization. Where possible, we will neglect the effectsenergy. suppressed For by the th differential cross section we get PoS(LL2018)010 ⊗ (2.6) (2.4) (2.7) (2.5) , bution, ) i as: t u t , t ↔ . Symbol , e s s and ( is due to loop a , h Andrej Arbuzov s LL ) t s  − 2 t e F π + in order to get the v 6 γ i ↔ 2 16 µ = k s ( γ ) ) e /  u 2 δ ⊗ ) , g t t 6 , ( , t , , ) γ ) ) Z s 3 ≡ i i µ 4 4 ( ( e χ ) ) γ I k p p 1) and the virtual part by the Z 2 u u LQ , , , , − j j can be written in terms of the  t s = ) F ( ( , , 3 As. There are six non-zero HAs, e  real hard . s v v ( t e , ) δ j j ( ( I QQ -order terms which can be added non-interfering helicity amplitudes µ µ , Z tructures and FFs. The next step is u 2 J J Z  Z LQ QL , LL Γ ) ) proportional to , t  − 3 1 h permuted arguments Z , F F QL ) p p s ) e e s ( , , u δ iM δ ( j j F s , ) ) Z QQ ( ( t t s I + ¯ χ ¯ , ( v ( u s 2 Z F ) , Z Z ) with the factor is defined below ( 2 , and couplings 2 + χ χ µ  Z ˜ e M Z ] p p LL γ  channel. t χ F , , the number of independent HAs is reduced to , − − Q i ) i − t F ⊗ 2 Z ) t ( ) ( QL I ( ↔ u u u u µ t )( γ , , i F s µ i γ µ s  2 W s t [ 2 J J ( ) e , , c F ) t t s ) Z t s = δ 1 ( 3 µ 1 ( 2 ( W χ Z ) ) γ s p p Z + 0 in the LQ Z Z , 4 , , , A i )+ i i ⊗ γ γ ( F ( )+ ) ( = u ( QQ QQ ¯ ¯ µ u v , u u Z t γ , , )+ )= F F , t t t Γ I corrections can be separated into the virtual (loop) contri h s h = = , , ( − ( ( s s t s γ ) Z − j j ) ( ( µ µ c 2 QL ) ) χ s α A J J e Z Z ( 2 (  , , s γ F 2 e γ γ ⊗ ⊗ , the electron charge ( ( 6 O QQ QQ − ) − − γ F i i µ µ ) 5 µ J J F F µ s γ γ = = ( γ Z + h h + ⊗ ⊗ A + + 1 − µ µ c c + γ γ 2 2 ) −−−− −  3 e e = ( )+ ( e s I 6 ( H − − H ( e γ 2 γ δ ; and one should put channels, respectively. The function = = = = t ie 2 A t − + − or = + = + s ++ − −− and A ++++ − + + = s I In this subsection we present analytic expressions for the H We obtained the compact expression for the Born ( The complete result for The covariant amplitude for high-energy Bhabha scattering H H H H HA approach in the form however, since for Bhabha scattering the part due to theOur soft main photon approach emission, and is the to(HA). one calculate In due a the to cross covariant the section amplitude bythe approach squaring projection we to the derive helicity tensor basis. s four. 2.2 Virtual, soft, and hard contributions with EW RC to polarized Bhabha scattering where “1” stands for the Born level and the term pure one-loop approximation withoutlater. any admixture of highe electromagnetic running coupling constant and four FFs wit corrections. After squaring the amplitude we neglect terms where we have is used in the following short-hand notation: in the PoS(LL2018)010 (2.9) (2.8) 2 (2.10) | i j −  + . s u . i H 2 | − ) | is an auxiliary Andrej Arbuzov   i j i j ∑ 2  λ specific contribu- ) , t u −− . . + 1 ) e  2Li c H P dx − | , ) ln − ) b x + i j 2 , ) + ∑ − − ystem; 1 x e a +  1 ) e ( ( P t s )( )  bha scattering obtained by + , P ln Z e − 2 e , ( − QQ − e P z s e − 0 R m P  e P F + ate the helicity amplitudes and ( P − 2  e − ln ference frame. The polarization ( d analytically. It is factorized in 1 d by direct squaring of the matrix δ 1 ln ) − )( loop arameter which separates kinemat- es in the one-loop virtual QED cor- a between the square of the covariant )= − 2 Born − + ( z e abha scattering cross section with the 1 + ( ocedures) and Eq. (2.9). Z  ( tribution contains infrared divergences σ P 1 2 2 χ σ s u | d . + d i j −  1 xpressed as )+  2 = ion of the Bhabha process with hard photon ++ c are defined as , λ ω ln + ( δ δ b H 4 2 | , | + a  i j 2 i j ( 3 ∑ ) + system computes the contributions due to the soft ) γ , ( − QQ 2ln +  ) ) e 2 e 1 1 H F P + s | m − − 1 − , , i j SANC ∑  1 1 1 − )= ) ( ( c ) ln )( + , σ σ 1 e − b ( e d d P + , 2 P a 1 − − ( . The soft photon contribution has the compact form ) −  )+ ) 1 4Li Z 1 1 , 1 ( , , )( γ ( Born 2 e ( + QQ 1 1 − h e σ Q  s for the sake of brevity. − − F P t s ( ( π π α and the relative correction 1 ϑ + σ σ − 1 d d LR  128 Born 2 A cos 1 and = Monte Carlo event generator. Comparisons of our results for σ d + ( = ± − 2Li LR ) ϑ A + = + e P SANC ϑ , cos − soft e = cos P σ is the maximal energy of a soft photon in the center-of-mass s ( ± d c σ ω d The asymmetry The contribution of the hard real photon emission is obtaine To study the case of the longitudinal polarization, we gener In this section, we present numerical results for EW RC to Bha The Bremsstrahlung module of the The soft photon Bremsstrahlung correction can be calculate For the cross check weamplitude got (we analytical introduced the zero spin for density the matrix difference into our pr Here element. Explicit formulae for the differential distribut apply the formalism described in [22].longitudinal In polarization our of notation the the initial Bh particles can be e EW RC to polarized Bhabha scattering where emission are too long to be listed here. 2.3 Longitudinal polarization means of the tions with the ones existing in the literature are also given 3. Numerical Results and Comparisons infinitesimal photon mass. The dilogarithm is defined as Li where we dropped front of the Born crossical section. domains of It the depends softdependence on is and the contained hard in auxiliary photon p emission in a given re which should be compensated by therections. corresponding divergenc and inclusive hard real photon emission. The soft photon con PoS(LL2018)010 , ) 0 , with 8 . SANC 0 correc- rahlung ) − are given , ( CalcHEP α δ , ( , ) , O 0 , Andrej Arbuzov WHIZARD 0 7 GeV ( . 4 and = ction and for the sum 1876 GeV b . m 91 , = (the first rows) program. The sult for the unpolarized hard Z 105658389 GeV AItalc . g cross section with polarized . [12] and references therein. It M 0 re shown in Figs. 1, 2, and 3 for . s) in the comparison of the ges = , attering angles are due to the small- ering and the relative 1 GeV for the photon energy. µ 215 GeV an any problem with the perturbation o not specify the calculation scheme. rees. tained with the help of the system [24]. Table 1 shows the good . beam polarizations: ison was done for the different values m ≥ 0 ons in this comparison was limited by ) 250, 500, and 1000 GeV for the follow- 8 GeV γ merical results, since the authors usually . + WHIZARD ng and the relative correction = E e , = s P . ( 173 m cm ) E = t , WHIZARD 4514958 GeV 9999 m 4 . . 0 80 , + = 083 GeV 51099907 MeV W . . 0 0 M 5 GeV . and the positron 9 up to = 1 = . ) , 0 d e − = 250 and 1000 GeV is shown in Fig. 4. One can see that the EW e m − m c P ( = m , s , results (the second row) for the Born and hard photon Bremsst √ , : from . 9 with the additional condition 03599976 . . ϑ ) 0 6 SANC . 0 < 137 cos , ( [12] results for the unpolarized differential Born cross se | 8 . 77705 GeV 062 GeV 49977 GeV θ . . . 0 )= 1 0 2 0 − ( cos ( = = = 1 | asymmetry at , τ u − Z ) 9 and different CMS energies. . 6 LR Γ α m m . EW scheme is used in all calculations. 0 A 0 ) < 0 − (in percent) as a function of the electron scattering angle a ( | , 8 The numerical comparison for the hard photon Bremsstrahlun The unpolarized differential cross section of Bhabha scatt The huge relative radiative corrections for the backward sc First of all we verified the agreement between our analytic re The There are many studies devoted to the Bhabha process, see e.g All results are obtained for the set of the energy We obtained also a very good agreement (six significant digit δ θ α AItalc-1.4 . 0 − cos range of scattering angles forthe the condition final and positr agreement between the contributions with the ones obtained with the help of the system [23]. initial particles is performed with the help of the tion 3.2 Numerical results for Born and 1-loop cross section ness of the Born cross section intheory. this The domain, integrated that cross does section not of me the Bhabha scatteri The ( ing magnitudes of the electron 3.1 Tuned comparisons with other codes photon Bremsstrahlung process cross section with the one ob EW RC to polarized Bhabha scattering Eventually, we performed comparisons with the modern packa | do not present the complete list of the input parameters and d is highly non-trivial to realize a tuned comparison of the nu in the Table 2 for various energies and beam polarization deg and of the virtual and the softof photon scattering angles contributions. The compar radiative corrections affect the asymmetry very strongly. the following set of the input parameters: PoS(LL2018)010 ) ) ) θ θ θ cos( cos( cos( Andrej Arbuzov (right) [in %] vs. the (right) [in %] vs. the (right) [in %] vs. the δ δ δ 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0.2 0.2 0.2 − − − 0.4 0.4 0.4 − − − 0.6 0.6 0.6 − − − δ δ δ ve correction ve correction ve correction 0.8 0.8 0.8 − − − 1 1 1 − − − 0 0 0

50 80 60 40 20 20 80 60 40 20 20

− − 300 250 200 150 100 120 100 120 100

[%] [%] [%] δ δ δ 5 1000 GeV. 500 GeV. 250 GeV. ) ) ) θ θ θ = = = s s s cos( cos( cos( √ √ √ 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0.2 0.2 0.2 − − − 0.4 0.4 0.4 − − − 0.6 0.6 0.6 − − − Born 1•loop Born 1•loop Born 1•loop The differential cross section (left) [in pb] and the relati The differential cross section (left) [in pb] and the relati The differential cross section (left) [in pb] and the relati 0.8 0.8 0.8 − − − 1 1 1 − − − 1 1 1 2 2 3 2 3 2 1 1 1 − − − − 10 10 10

10 10 10 10 10

10 10 10

10

) [pb] ) /dcos( d ) [pb] ) /dcos( d ) [pb] ) /dcos( d θ σ θ σ θ σ Figure 3: Figure 2: Figure 1: EW RC to polarized Bhabha scattering cosine of the electron scattering angle for cosine of the electron scattering angle for cosine of the electron scattering angle for PoS(LL2018)010 ) θ cos( 250 GeV (left) Andrej Arbuzov = collisions at the s − √ e + 0 0.2 0.4 0.6 0.8 1 e factory [25] planned to τ 0.2 − − c 0.4 − per 0.6 ative corrections is crucial for the − Born 1•loop colliders. Taking into account beam 0.8 − − dictions for the e beam is foreseen. 1 − + 0 e 0.1 results for the Born and hard Bremsstrahlung

0.02 0.12 0.08 0.06 0.04 LR A 250, 500, and 1000 GeV. 6 = s WHIZARD √ ) θ electroweak radiative corrections provide a considerable and cos( ) vs the cosine of the electron scattering angle at α ( LR SANC O A 250 GeV 500 GeV 1000 GeV = = = s s 0 0.2 0.4 0.6 0.8 1 s √ √ √ 0.2 − 0.4 − 0, 0 -0.8, 0 -0.8, -0.6 -0.8, 0.6 0.6 − Born 1•loop 0.8 The left-right asymmetry Tuned comparison of the 1000 GeV (right). − + 1 e = , pb 3.6792(1), pb 3.9057(1) 3.6792(1) 3.0358(1) 3.9057(1) 4.7756(1) 3.0358(1) 4.7755(1) , pb 14.379(1), pb 15.030(1) 14.379(1) 12.706(1) 15.030(1) 17.355(1) 12.706(1) 17.354(1) , pb 56.677(1), pb 57.774(1) 56.677(1) 56.272(1) 57.775(1) 59.276(1) 56.272(1) 59.275(1) , pb, pb 4.693(1) 4.694(1) 4.976(1) 4.975(1) 3.912(1) 3.913(1) 6.041(1) 6.043(2) , pb, pb 15.14(1) 15.12(1) 15.81(1) 15.79(1) 13.54(1) 13.55(1) 18.07(1) 18.11(2) , pb, pb 48.62(1) 48.65(1) 49.58(1) 49.56(1) 48.74(1) 48.78(1) 50.40(1) 50.44(1) − 0 P s 0.1 , We show that the complete The theoretical description of Bhabha scattering with radi

√

0.16 0.12 0.08 0.06 0.02 0.14 0.04 − LR Born Born hard hard Born Born hard hard Born Born hard hard A e σ σ σ σ σ σ σ σ σ P σ σ σ Figure 4: EW RC to polarized Bhabha scattering and be built in Novosibirsk, where polarization of the electron Table 1: 4. Conclusions high-precision measurements at the modern and future polarization is a novel requirement for the theoretical pre contributions to polarized Bhabha scattering for energies of CLIC and ILC. Our results can be used also at the Su PoS(LL2018)010 δ work of State Andrej Arbuzov (1988) 335. 48 orresponding relative corrections nly claims the necessity to com- ns and to Gizo Nanava for collab- (1984) 414. and Science of Russia. those corrections are known in the polarization). We plan to implement 144 asymmetry. Moreover, the corrections . Commun. to our Monte Carlo event generator and the initial particles. s. (1988) 712. (1991) 485. (1988) 687. 304 49 (1986) 905. 304 7 75 (1936) 195. 154 (1979) 208. 250 GeV 500 GeV 1000 GeV = = = 160 s s s √ √ √ system development. Results were obtained within the frame 0, 0 -0.8, 0 -0.8, -0.6 -0.8, 0.6 SANC , pb 3.8637(4) 3.9445(4) 3.2332(3) 4.6542(7) , pb 61.731(6) 62.587(6) 61.878(6), pb 63.287(7) 15.465(2) 15.870(2) 13.861(1) 17.884(2) 250, 500 and 1000 GeV. Born and 1-loop cross sections of Bhabha scattering and the c + e , pb 3.67921(1) 3.90568(1) 3.03577(3) 4.77562(5) , pb 14.3789(1) 15.0305(1) 12.7061(1) 17.3550(2) , pb 56.6763(1) 57.7738(1) 56.2725(4) 59.2753(5) = P − − − − − − loop loop loop s e e e e e e , The observed magnitude of the first order corrections certai We are grateful to Sabine Riemann for stimulating discussio − − − + + + + + + − √ Born 1 Born 1 Born 1 , % 5.02(1) 0.99(1) 6.50(1) -2.54(1) , % 8.92(1), % 8.33(1) 9.96(1) 7.56(1) 6.77(1) 5.59(1) 9.09(1) 3.05(1) e e e e e e e σ σ δ δ σ σ δ P σ σ [8] D.Yu. Bardin, W. Hollik, T. Riemann, Z. Phys. C [5] M. Bohm, A. Denner, W. Hollik, Nucl. Phys. B [6] F.A. Berends, R. Kleiss, W. Hollik, Nucl. Phys. B [7] S. Kuroda, T. Kamitani, K. Tobimatsu at al., Comput. Phys [1] H.J. Bhabha, Proc. Roy. Soc. Lond. A [2] M. Consoli, Nucl. Phys. B [3] M. Bohm, A. Denner, W. Hollik, R. Sommer, Phys. Lett. B [4] K. Tobimatsu, Y. Shimizu, Prog. Theor. Phys. pute also higher-order correctionsliterature to (but this mainly just process. for thethe Some the known higher-order of pure corrections QED to case Bhabha without work scattering further in on calculation of other relevant contribution impact on the differential cross section and the left-right Acknowledgments themselves are very sensitive to the polarization degree of oration on the Table 2: for EW RC to polarized Bhabha scattering Program No. 3.9696.2017/8.9 from the Ministry of Education References PoS(LL2018)010 (2015) 2227. Andrej Arbuzov 46 (1993) 3. (1998) 1. 401 299 (2006), 481 [Erratum: (2013) 1729. (1991) 281. (1991) 323. 174 184 ov 1995) Geneva, Switzerland, (2018) 013001 [arXiv:1801.00125 355 349 ), Phys. Rept. (2006) 35. 98 48 s. B ommun. CERN Workshop on LEP2 Physics ys. B Commun. inini, Acta Phys. Polon. B v. D s. C , in R. Pittau, Nucl. Phys. B 8 (2011) 1742. 71 (2015) 238. (2008) 131. (1982) 485. (2015) 2213. 260 460 (2015) 371. 12 46 75 (2007) 623]. (1983) 259. 177 123 (1996), pp. 229–298, [hep-ph:9602393] Event generators for Bhabha scattering [hep-ph]]. (followed by 2nd meeting, 15-16 Jun 1995 and 3rd meeting 2-3 N February 2-3, 1995 Comput. Phys. Commun. [9] W. Beenakker, F.A. Berends, S.C. van der Marck, Nucl. Phy [12] J. Fleischer, J. Gluza, A. Lorca, T. Riemann, Eur. J. Phy [14] W. Hollik, Phys. Lett. B [15] S. Jadach et al., [21] D. Bardin, Y. Dydyshka, L. Kalinovskaya et al., Phys. Re EW RC to polarized Bhabha scattering [11] G. Montagna, F. Piccinini, O. Nicrosini, G. Passarino, [13] W. Hollik, A. Zepeda, Z. Phys. C [10] W. Beenakker, F.A. Berends, S.C. van der Marck, Nucl. Ph [20] A. Arbey et al., Eur. Phys. J. C [22] G. Moortgat-Pick et al., Phys. Rept. [23] A. Belyaev, N.D. Christensen, A. Pukhov, Comput. Phys. [25] S. Eidelman, Nucl. Part. Phys. Proc. [24] W. Kilian, T. Ohl, J. Reuter, Eur. Phys. J. C [16] A. Andonov, A. Arbuzov, D. Bardin et al., Comput. Phys. C [19] E. Accomando et al. (ECFA/DESY LC Physics Working Group [18] S. Riemann, Acta Phys. Polon. B [17] C.M. Carloni Calame, G. Montagna, O. Nicrosini, F. Picc