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1% 0.1% 1% 1% O() e+e- SANC SANC SANC s t W Z Z W O() e+e- SANC SANC SANC SANC R O() e+e- ZFITTER HECTOR POLHECTOR SANC s t SANC SANC e+e- e+e- ff e+e- ff e+e- e+e- ff t bl+l() e+e- ff ffA W e MW MZ MH W H Z A Z 1/2 j 1/2 j i i f0 = (ZL )ij fL, f0 = (ZR )ij fR, fL L R fR -1/2 e0 = ZA e mf MH s mf MH 24 2mf 33 - 2Nf 3s QCD 0 1 +, 0 = 0 ln , f MH ln 0 H H H H t Z H Z H t H Z MH MZ 2 2 2 g2 33/2 52 19 MH MH 13 9 m2 MH t Das = - + + -ln2 + + - ln.

2 2 2 2 162 4 24 8 MW MW 4 4R 2MW MW H H t MH H H Z H W MH MW 14 D1as = 3 H MH t 130 Z ZZ W W ff tot W W W W W W.10-7.10-3.10-3.10-6.10-7.10-3.10-3.10-5.10-4.10-3.10-5.10-4.10-4.10-4 H MH MH t 174 Z ZZ W W ff tot W W W W W W.10-7.10-3.10-3.10-6.10-7.10-3.10-3.10-5.10-4.10-3.10-5.10-4.10-4.10-4 H MH SANC R SANC mt = 174, 2 mb = 4, 7 mc = 1, 5 ms = 0, 215 m = 1, 77699 MZ = 91, 1876 MW = 80, 403 O() e+e- - + + + - E e-(k1) + e+(k2) -(p1) + +(p2) 0.0.0.0.0 0.2 0.4 0.6 0.8 1 d(3) = ds dV2 d d(cos ), (2)5 16s V2 = 2p0 (p0 - | cos ) p0 p0 p2| 2 s V R = v2 = s s RE - R v2 1 - RE. 2 RE = s 1 - sin 4R(1 - v2)vR =, R =. 4R v2 + (1 - R)1 + sin R = R RE - R 1 - RE Rmin dhard dhard = + - dR dv2 d cos . dRdv2d cos I II III max min v2 v2 A 1 max min v2 = (1 - R)(1 + A), v2 = (1 - R)(1 - A). 2 A = A(R) R0 2 AI = 1 -, RE R 1 - , R s 1 + R - 2RE AII =, Rmin R RE, 1 - R R(1 - R)AIII = 1 -, Rmin R R. R(1 - R) v2 cos R cos + dhard hard T (s) = dR d cos , dRd cos - + dhard hard F B (s) = dR - d cos . dRd cos 0 - cos -c cos +c c cos R v2 cos R Rac +c 2 dfin f dTin(v2) = V(s) Q2 dv2 dR d cos , f s dv2 dR d cos Rmin -c Rac +c 2 dfin f dFin(v2) = A(s) Q2 dv2 dR - d cos . B f s dv2 dR d cos Rmin 0 -c v2(1 - v2) (1 + R) R Rac(v2) = = - 1 + v2 4R 2 2 V(s) = Q2Q2 + 2QeQfvev e(s) + (ve + a2)(v + a2) |(s)|2, e e A(s) = 2QeQaeae(s) + 4veaeva|(s)|2, GF MZ s (s) = , m2 = MZ - iMZZ(s), Z 2 s - mZ vf = af - 2 sin2 Qf, af = If. W f v2 Tin f f Fin Tin B I 0 (s, c) e(k1) + p(p1N ) e(k2) + X(pj). y -2p1N Ql yl =. -2p1N k Ql k1 - k2 x Ql xl =, SN = -2p1N k1. ylSN e(k1) + q(p1) e(k2) + q(p2). |Meq|2 1 dq = d(2), d(2) = (SU - Q2) dSUdQ2, 8 2 S S Meq S = (-2p1 k1)2 - 4p2k1 0 Q2 = -2me + 2k1k2 - 2| k2| cos k1|| 0 SU = S-U = -2p1Q = 2mq(k1-k2) Born d2NC 1 |Meq|= d fq(, Q2) ( - xl) SN. dxl dyl 16 S q k1 k1|, ke = | =m 0 k1, e me me | k1| Born d2NC xlSN M== 22 fq(xl, Q2) 1 + y1 V0(Q2) - 2 xl yl W0(Q2) dxl dyl Q4 SN q +pe pq 1 - y1 A0(Q2). y1 = 1 - yl V0(Q2) = Q2Q2V + 2QeQq(Q2) VI + 2(Q2) VZ, e q W0(Q2) = Q2Q2V + 2QeQq(Q2) VI + 2(Q2) WZ, e q A0(Q2) = Q2Q2A + 2QeQq (Q2) AI + 2(Q2) AZ. e q V0(Q2) W0(Q2) A0(Q2) 2 V = 1, VI = (ve - peae)vq, VZ = (ve + a2 - 2peaeve)(vq + a2), e q A = 0, AI = (ae - peve)aq, AZ = [2aeve - pe(a2 + ve)]2aqvq, e 2 WZ = (ve + a2 - 2peaeve)(vq - a2).

e q pe(q) = +1 pe(q) = -1 (Q2) Q2 GF MZ (Q2) = . Q2 + MZ 2 2 + p + X, + q + q. d2Born G2 xlSN MW e-p F = (1 - ) u(xl) + c(xl) + y1 d(xl) + s(xl), dxldyl 2 (Q2 + MW ) u(xl) = uv(xl) + us(xl) c(xl) = cs(xl) u c d(xl) = ds(xl) s(xl) = ss(xl) d s v s m2 M2 Qe m2 M2 Q2 q Born Meq d2Vertex d2Born = V + V, ee qq dxldyl dxldyl Q2 m2 3 Q2 1 Q2 e V (Q2) = Qe2 1 - ln ln - 2 + ln - ln2 +, ee m2 m2 2 m2 2 m2 e e e mQ2 3 Q2 1 Q2 q V (Q2) = Qq2 1 - ln ln - 2 + ln - ln2 +. qq m2 m2 2 m2 2 m2 q q q m d2Box-IR d2Born Q= eq(S, U), eq(S, U) -2QeQq ln ln(1 - yl). dxldyl dxldyl m n Soft |Meq | dSoft = dxh fq(xh) d(2) d ( - k0) = dBorn Soft. 2 S q Q2 4Q2 QSoft qq =Qq2 ln - 1 ln + 1 + ln - ln2 - Li2(1), m2 m2 m2 mq q q Q2 42 1 Q2 Q2 S2ySoft ee =Qe2 ln - 1 ln + ln2 + 1 - ln ln m2 m2 2 m2 m2 m2me e e e q - ln2 y1 - Li2(1), 42 QSoft eq = 2QeQq ln - ln ln(y1). m2 m q mq sing d2q xlSN = 22 1 + y1 V0(Q2) + pe pq 1 - y1 A0(Q2) l l dxldyl Ql Q2 1 + z2 1 xl l Q2 ln dz (z - xl) fq(, Q2) - fq(xl, Q2). q l l 2 m2 1 - z z z q fq(xl, Q2) l n d2Hard G2 MW 1 - Qe-u F l = dxldyl (Q2 + MW )2 2 xl l 1 S d fu(, Q2) Q2FFQ+Q2FFL+QeQuFFLQ. l e SN SU u u z p = (M, 0), p1 = x p, kp.p = x p.p1 = 0, p 2M p = p k1 - p, SN M R p p y R p x (p1) = us(p1)s(p1) = 1 - 5 p (p1 + mq).

s Born Born-unpol Born-pol d2ep d2ep d2ep = +, dxldyl dxldyl dxldyl Born-unpol d2ep 22 U = SN Si (x, y) Fi(xl, Q2), l dxldyl Ql i=Born-pol d2ep 2L = pSN Si (xl, yl) Gi(xl, Q2). dxl, dyl Ql i= p J1J Fi Gi FiJ J2 gi J1, J2 Z O() QED,1 Born Hard unpol pol d2ep d2ep d2ep d2ep d2ep = Virt + Soft + = +. dxldyl dxldyl dxldyl dxldyl dxldyl 2 Hard dep 1 1 dk = 23SNyl dyhdQ2 L W, h dxldyl Q4 2 lN rad h 2 lN = yl SN + 4M2Q2 L W l rad U L SF i(yl, Q2, yh, Q2) SGi(yl, Q2, yh, Q2) l h l h Hard dep U = 23ylSN dyhdQ2 Si (yl, Q2, yh, Q2)Fi(xh, Q2) h l h h dxldyl Qh i= L +p Si (yl, Q2, yh, Q2)Gi(xh, Q2). l h h Qh i= U L Si (yl, Q2, yh, Q2) Si (yl, Q2, yh, Q2) l h l h Qh yh = xhSN xl Q2 Qh h z = yh = z, dyh = dz, xl z xh xlSN xlSN Q2 SN 105GeV h max h 1 Q Q2 x2M2 Qh l l dyhdQ2 = dz dQ2, Q2 . h h xlSN h 1 - z z xl Q2 min h Q2 z h O() pol pol long = d2rad/d2Born 100% Ee = 27.5 SANC SANC SANC = A f + f + = f + f f + f + = W f + f + = Z f + f + H f + f + = t b + W + = fu + fd fu + fd + = f + f f + f + = t b + f + f + = W q + q + g = Z q + q + g = H q + q + g = t b + W + g = qu + qd + + g = q + q + + g = qu + qd t + b + g = qu + b t + qd + g = t b + qu + qd + g 0 p0 0 p0 m m + 1 dSoft = dm+1( - p0) = dBornd(g)( - p0). m+1 5 m MSoft 2 MBorn 2 MSoft 2 MBorn 2 sFSoft(cos g). = FSoft(cos ) = s dSoft = dBorn Soft dSoft = dBorn Soft, +1 1 Soft = d + ln + ln(1 - 2) FSoft(). 4 - = cos = cos g Hard(...).prc f1(p1) + f2(p2) f3(p3) + f4(p4) + (g)(p5) I. f1(p1) + f2(p2) f4(p4) + X(P ), II. X(P ) f3(p3) + (p5) ( g(p5)). X P = p3 + p5 p2 p4 ds dZ4 d cos d dI(3) =. 32(2)4 s Z4 = -2p4 p5 min max Z4 (s) Z4 (s) s - s max,min Z4 (s) = s - m2 + m2 (s). 3 2s (s) (s) = s2 - 2s(m2 + m2) + (m2 - m2)3 4 3 I. f1(p1) + f2(p2) (p5) ( g(p5)) + X(P ), II. X(P ) f3(p3) + f4(p4). X P = p3 + p4 P = -s ds s - s (s) dI(3) = d cos d cos d. 2 16s 8(2)2 s t t(p2) b(p1) + f3(p3) + f4(p4) + (p5) ( g(p5)) I. f2(p2) f1(p1) + X(Q), II. X(Q) (p5) ( g(p5)) + Y (P ), III. Y (P ) f3(p3) + f4(p4). ds ds (s) s - s (s) dI(4) = d cos d cos d. (2)2 8m2 16 s 8(2)2 s t s (m3 + m4)2 s (mt - mb)2. b Q = 0 Q0 = s (s) = (m2 + m2 - s)2 - 4m2mt b t b e+e- ff f1f1 ff Z t b+ t bud Born t = 10-1 10-2 10-1loop 1l t = 1loop 1l t b + Born t = 10-1 10-2 10-1loop t = 1loop t b u d b + (+, ) t b+() t bud() t b + + + t b + u + d SANC SANC W Z pp W + X (g, ) + X pp W + X (g, ) + X. + d(p1) + u(p2) W (p1 + p2) (p3) + +(p4) 2 | Vud |2 G2 MW m2 F CC = 1 - s + m2, Born(s) 2 6NC (s - MW )2 + 2 MW s W NC = 3 s 42 s s 3 s CC Born(s) cf ln ln + ln - 2 + ln SoftVirt(s) = CC 2 s m2 m2 2 mu u u 3 s + ln - 4 - 2. 2 m2 d s zmax s 2 CC = dz CC Pqq(z) 2 ln + ln + ln - 2. Hard Born(zs) 2 2 m2 mzmin u d z = s/s s = -2(p1.p2) s = -(p3 + p4)2 zmax = 1 - 2/ s zmin = m2/s 1 + zPqq(z) = cf 1 - z + cos Hard() ln(s/m2 ) u,d n MS s zmax 42 (1 - ) CC = dz z (1 - z)-2 CC ) Pqq(z) -. Hard() Born(zs, 2 zs (1 - 2) zmin 2 ln ln m2 mu d s zmax 2 s CC = dz CC Pqq(z) - + 2 ln + 4 ln(1 - z). Hard Born(zs) 2 zmin s zmax 2 CC = dz CC Pqq(z) ln + ln - 2 - 4 ln(1 - z). Subtr Born(zs) 2 m2 mzmin u d dpp(SN )/d cos h = d(x1x2SN ) 4x1x 1 dx1dx2 q1(x1, 2) q2(x1, 2) q1q0 d cos q (x1 + x2 - (x1 - x2) cos h) dz x 1 + z2 q(x, 2) = q(x, 2) - q(, 2) Q2 ln - 2 ln(1 - z) - 1 q z z 2 1 - z mx q + q(x, 2) MS SANC pp Z/ + X +-(, g) + X pp Z/ + X +-(, g) + X. 4 2 1 m2 m NC = 1 - V0(s) + W(s) Born(s) sNC 3 s s s 42 s 3 s NC = NC Cf ln ln - 1 + ln - 2 - 2. SoftVirt(s) Born(s) s m2 2 m2 q q s zmax s NC = dz NC Pqq(z) ln + ln - 1, Hard Born(zs) 2 mzmin q t t s d(p1) + u(p2) t(p3) + b(p4) b0(-m2, 0, mt) p t C0(-m2, -m2, -s, mt, 0, mb) t b t mt2 = m2 - imtt t t b t s Born t (s - m2)Soft-QCD Virt-QCD t fin + fin = cf CC-s -1 - ln 1 - ln mt m2 mt b 1 42 s - m2 s m2 1 s s t t - ln - ln ln - 3 ln 1 - + ln ln 2 s m2 m2 s 2 m2 mt t t b 3 s 5 m2 m2 mt t + ln + - 2 ln 1 - ln 1 - + Li2 t + 3Li2(1) 2 mtmb 2 s s s | Vud |2| Vtb |2 G2 MW m2 (s - m2)2 s m2 s F t t t - cf ln 1 - + ln. 2 8s2 (s - MW )2 + 2 MW s mt W t t t b (s - s) s + m2 - m2 - s - 2ismtt t b J(-m2, s, mt, it) = ln , b (s - s) s + m2 - m2 + s - 2ismtt t b t s b s Born 5 t (s - m2)2 1 4Hard-QCD t fin = cf CC-s + ln 1 - ln + ln 4 mt m2 m2 2 s t b s - m2 s m2 m2 1 s s t t t + ln ln + + 2 ln 1 - + ln ln m2 m2 s - m2 s 2 m2 mt t t b t 3 s 7 m2 m2 m2 mt t t - ln + - + 2 ln 1 - ln 1 - - + Li2 t 2 mtmb 2 s s s - m2 s t | Vud |2| Vtb |2 G2 MW s F -3Li2(1) + s - m2 + 2m2 ln.

t t 2 12 (s - MW )2 + 2 MW mW t t s Born 3 s m2 mNLO-QCD t fin = cf CC-s - ln + 1 ln 1 - + 2Li2 t 4 m2 s s t | Vud |2| Vtb |2 G2 MW m2 s F t + 4sm2 + 5(s2 - m4) ln t t 2 24s2 (s - MW )2 + 2 MW mt W mt -3(s - m2)2 1 + ln 1 - - 4m2(s - m2). t t t s t t b t t SANC-4d Hard-QCD SANC-s Hard-QCD s t s, GeV t = t = Hard-QCD u + d t + b b + u d + t b + d + t t b b + u d + t b + d + t s mq = mu = md = 66 SANC(F) s/ t = 0 mq t = 0 mq Hard-QCD t b + u d + t SANC(S) t s/ t = 0 mq t = 0 mq Hard-QCD t b + d + t t t t b t g t = 0 t = 0 t = 0 t = t b + + + + g t b + u + d + g t Hard-QCD

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