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0.1% 1% 1% ep e ep e W Z TeV MW sin2 eff W ep e ffbb ffff f, b D SANC 2f2b 0 ep eX ep lepton+-N e+e- ff f1f1HA 0 f1f1ZA 0 J(A) JAW,W A e lN e+e- ff Z, W ep ep O() e(k1) + p(p1) e(k2) + (k) + X(p2) Z S = 2(p1k1), Sl = 2(p1Ql) = Syl, Q2, l Sh = 2(p1Qh) = Syh, Q2, z2 = 2(k2k), h Ql = k1 - k2 Qh = p2 - p1 dk2 d dp2 2 k d = dMh(4)(k2 + k + p2 - k1 - p1)dh, 0 2k2 2k0 2p0 2 dpi 2 dh = (4)(p2 - pi), (Mh = p2) 2 2p0 i i i = dQ2dSldQ2dShdkdh, 2 S l h d k 2 dk = (p1 + Qh)2 + Mh Q4 - (p2 - p1).

h 2k0 p1 = 0 max z2 2 1 1 dz2 dk = d =, 8M 8M|Ql| |Ql (k2 p2)| min 0 z2 k1, k2 k, p2 min,max z1,2 = -2k1,2k z1,2 dz2 = dQ2dSldQ2dSh dh. 4 S l h Rz Rz 4 k1, p1, k2, p2 z1,2 Rz -B1,2 Dz max,min z1,2 =. A1,2 z2 z1 I E I = {(Sh, Q2), (Q2, Sl), (Q2, Sl)}, h h l E = {(Q2, Sl), (Q2, Sh), (Q2, Sh)}. l l h Em = (Q2, Sh) Im = (Q2, Sl) l h Im = (Q2, Sl) Q2 Sh h l i) (Q2)min Q2 Q2, ii) Q2 Q2 Sh, h h l l h (Slmin)I Sl Slmax, (Slmin)II Sl Slmax, 1 (Q2)min = Q2 + (Slmax - Sh) Slmax - (Slmax)2 + 4M2Q2 h l l 2M2 1 (Slmin)I,II = Sh(Q2 + Q2) (Q2 - Q2) h.

l h l h 2Q2 h IR d2R d2Born IR IR = soft(Q2, Sh, ) + hard(Q2, Sh, ). l l dQ2dSh dQ2dSh l l 3 F d2R 23 1 = dQ2dSl Ai(Q2, Sh) Si(Q2, Sh, Q2, Sl) h h dQ2dSh S Q4 l h l h i=1 1 B - Ai(Q2, Sh) Si (Q2, Sh) LIR(Q2, Sh, Q2, Sl), l l h Q4 l l B Ai(Q2, Sh) Si (Q2, Sh) h l Si(Q2, Sh, Q2, Sl) l h LIR(Q2, Sh, Q2, Sl) l h Sl Sl Si(Q2, Sh, Q2) l h F d2R/dQ2dSh l d2theor/dE (E) = - 1, d2B/dE d2theor/dE d2B/dE 3 d2theor d2B d2R = 1 + vert +, dE dE dE F d2R d2B (E) = VR(E) + /, dE dE IR IR VR(E) = vert(E) + soft(E, ) + hard(E, ). VR(E) O() O() + O(2) O() yJ B e O(2L2) O(2L2) e-e+ O() + O(2) e l, yJB y x 2 S = 98400 O() 100 Hector Hector 80 60 40 20 0 -20 -40 -60 -80 --100 -0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 % O() +O(2) + e e e BORN d 22 (Y - y) = (1 - yPeP) + (1 - PeP), dy meE y2Y m, me P, Pe y = y = 1-E/E Y = (1 + m/2E)-1 E, E, Ee d Aexp y e d() d() d() d() Aexp = - / +.

e dy dy dy dy ye = Ee/E y = ye = y y ye y O(3) Ee ERC ERC = 35 E - E - Ee EBC EBC = 40 BORN meas e e e - e BORN meas min min min = 1 - BORN BORN meas meas e e O(3) QED 2 l d dk =, dy l=1 k=1 dy k l l = 1 - l = unpol;

2 - PeP l = pol;

k = 1 - k = b;

2 - k = ;

3 - k = amm;

4 - k = ee;

5 - e k = e;

6 - , k = vp.

QED unpol pol d dk dk = + PeP.

dy k dy dy BREM BORN VR dk dk d k = , ee, e = dy k dy + dy VR k y A Aa y e da() da() - QED Ae dy dy A Aa =, a =, y = - 1.

e BORN da() da() Ae + dy dy - 0 ela ela -2 0.-4 0.-6 0.-8 0.-10 -12 -0.-14 -0.-16 -0.-18 -0.-20 -0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 ERC = 35 EBC = 40 e,min = . min = e e+e- ff R A0, B0,... QeQt B A = -i 4(0) , Q (3) (3) B (3) A = -ie2 Ie It + + + eIt + Z 4s2 c2 (Q2 + M2) W W Z (3) +Ie t+ + et , = 1 5 (L1+ + Q1) (L2+ + Q2) = v (p+) (L1+ + Q1) u (p-) (q-) (L2+ + Q2) v (q+) ; f = vf - af = -2Qf s2, f = e, t LL QL LQ QQ W Z e+e- ff LQD e e t t t i+F (s) + iF (s) i+F (s) + iF (s) + mtIDF (s), L Q L Q D D = (q+ - q-) R (1) add F (s) = F (s) + F (s). L,Q,D L,Q,D L,Q,D = 1 = 1 ZZ 4QeQf IBA A = i (s) , s (0) (s) (s) = / 1 - fer(s) - fer(0). LQD 4 Z LL QL LQ QQ LD QD (s) Z IBA A = i e2 + +F (s, t) + +F (s, t) + + F (s, t) Z LL QL LQ s + F (s, t) + + (-imtD) F (s, t) + (-imtD) F (s, t).

QQ LD QD Z/ s Z (s) = 1/(4s2 c2 )s/ s - M2 + i( /M )s. Z Z Z W W Z 0.4% Z had A0 Z FB MS MS G F H = C1u5eu + C2ue5u + C1d5edd + C2ded5d..., P V G F Q W Q (Z, A) = 2 (Z + A) C1u + (2 A - Z) C1d. W MS O ( ) O 2 S S ee tt t ee uu Q2 m2 m Q2 0 s t ZZ Z W W = 1 Q2 0 ee tt Q2 = 0 ee tt (3) (3) A (0) = Ie If + +F (0) + de +F (0) Z LL QL s2 c2 (-M2) W W Z +df+ F (0) + dedf F (0), LQ QQ (0) Q2 = 0 + = 1 + 5, df = -4|Qf |s2 W C1f C2f ee tt (3) C1f = If f + dff - r (1 + df), LL LQ (3) C2f = If f + def - r (1 + de), LL QL f = u, d f = 1 + F (0) LL,QL,LQ LL,QL,LQ 4sW C1u C1d C2u C2d sin2 APV = (0) sin2 (M2 ). PV W M = 91.1875 M = 150 (M2) = 0.119 Z H S Z sin2 APV = 0.231601 (0.232123) Q (Cs) = -72.06 0.28 (exp.) 0.34 (theo.). W : (k1) + qi(p1) l-(k2) + qf (p2) + qi l+ + qf, : (k1) + qi(p1) (k2) + qf (p2) + qi + qf. % Q2 QCD Born Born d2CC 0 d2CC 0 G2 M 0 W = CC, = CC(1 - y)2, CC = |Vif|2 F fi(x, Q2), dxdy dxdy (M2 + Q2)W M W W |Vif| fi(x, Q2) Q2 Q y =, Q2 = -(p2 - p1)2, x =, = (k1 + p1)2 xS, S = (k1 + P )2, yS P m1,2,l (Q1,2,l) Born d2()NC 0 2 2m2y G2 M 1 0 F Z = NC gL(R) + gR(L)(1 - y)2 - gLgR, NC = fi(x, Q2), dxdy (M2 + Q2)Z gL = - + |Qi| sin2 , gR = |Qi| sin2 , M Z W W Z W O () Corr. Born d2i d2i Virt Soft Hard = 1 + i + i + i, dx dy dx dy i q CC q NC W Q2/MW Virt CC,NC Virt CC,NC gL(R) gL(R), gL = - + |Qi| sin2 , gR = |Qi| sin2 , W W Soft Soft CC, CC Hard Hard Hard Hard q CC NC q CC CC m2 MS MS 1 + x2 Q2 4 Q2 MS = Q2 dx x ln - 1 - 2 ln(1 - x) = Q2 - ln -.

1 2 1 - x m2 2 3 m2 1 + ml 0 O (L) O (2L2) e+e- (2)LL O (2L) i O (2L2) O (2L) MS Corr. Born d2i d2i (2)LL = (1 + i + i - MS), dx dy dx dy (2)LL i E = 80 hadr 10 GF (0) (2)LL i + i - MS - - y x -1.x=0.x=0.-x=0.-2.--3.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 y - u y x x=0.x=0.x=0.-0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 y - d y x uNC dCC 2f2b 4f J ffbb 0 2f2b ffbb 0 Root Root SANC SANC QED QED EW EW Precomputation Precomputation Processes Processes 3 legs 4 legs 3 legs 4f 3b Neutral Current H -> A A Bhabha H -> A Z f1 f1 -> f f H -> Z Z Charged Current H -> W W f1 f1 -> f f t -> b l nu Z -> W W 2f2b b2f Neutral Current H -> f f f f -> A A Z -> nu nu A A -> f f Z -> f f f1 f1 -> H A W -> f f H -> f1 f1 A f1 A -> f1 H t -> W b f f -> Z A t -> W b (FF) f1 f1 -> Z Z t -> W b (HA) f1 f1 -> H Z t -> W b (BR) H -> f1 f1 Z 4 legs Charged Current QCD QCD b f(f1) f1 A Z, W, H t b ffbb 0 5 0 R A Z W A0, B0, C0, D0 a0, b0, c0, d dn t W b F F = 1 + F, F 4sW (3) Qf, If, f = vf + af, f = vf - af, s = e/g, c = M /M. W W W Z ffbb A Z i = 1 - 4 F H(Fi){ } {i} i J ffbb 0 f(p2)f(p1) Z(p3)Z(p4) (+) (-) + + A = k0 v (p1) p3+(p1)(p1)F1 (s, t) + p3+(p1)(p2)F2 (s, t) / / f f ZZ + + + + p3+(p1)(p2)F3 (s, t) + p3+(p2)(p2)F4 (s, t) + p3+F5 (s, t) / / / + + + + p3+F6 (s, t) + +(p1)F7 (s, t) + +(p2)F8 (s, t) / + + Z Z + +(p1)F9 (s, t) + +(p2)F10(s, t) u (p2) (p3)(p4) + + -, Fi+ Fi-, ig k0 = -, = I 5 v (p1) u (p2) 8cW Z (p3(4)) Z () p2 = -m2 (p1 + p2)2 = Q2 = -s, (p2 + p3)2 = T = -t, (p2 + p4)2 = U2 = -u. f(p1)f(p2)H(p3)Z(p4) 0 f1f1 HZ + + A = k v (p1) +f F0 (s, t) + p3+(p1)F1 (s, t) / f f HZ + Z + p3+(p2)F2 (s, t) u (p2) (p3) + f f, + -, Fi+(s, t) Fi-(s, t), / ig2 M Z k = - +f F0 (s, t) 4c2 M2 - s W Z Z H f1f1Z p1 -p3, p2 -p4, p4 -p1, p3 p2 f(p1)f(p2)H(p3)A(p4) 0 H H f1f1H 0 Q2, T, U2 Af f1H = v (p1) k1(p2) - k2(p1) + (k1 + k2) p3 Fv1(Q2, T, U2) / (p2) (p1) 2 2 + - 5Fa1(Q2, T, U2) + p3 Fv2(Q2, T, U2) + 5Fa2(Q2, T, U2) / k2 k 2 + i p3(p1) - k2 Fv3(Q2, T, U2) + 5Fa3(Q2, T, U2) / 2 + i p3(p2) - k1 Fv4(Q2, T, U2) + 5Fa4(Q2, T, U2) u (p2) (p3), / 1 k1 = k2 = mf T + m2 U2 + mf f f1f1 H p1 p1, p2 p2, p3 -p3, p4 -p H f1f1 p1 -p3, p2 -p4, p3 -p1, p4 p H e eH p1 -p3, p2 p2, p3 p1, p4 -p ffZA 0 Z Z F 0 0 j j Af f1Z = v (p1) Str0 vf Fv + af5Fa + Strj Fv + 5Fa u (p2) (p3)Z(p4), j= 1 Str0 = i k2 + k1 p3 + k2 (/3 - (p3)) - k2(p1) - k1(p2) / p (p3) Z(p4) Z j Fv,a Fv,a Str0 Str1- Str0 Strj Strj (p3) = 0.

ffZA 0 f1f1 Z p1 p1, p2 p2, p3 -p3, p4 -p Z Z f1f1 p1 -p3, p2 -p4, p3 -p1, p4 p Z f1 f1Z p1 -p3, p2 p2, p3 p1, p4 -p t t t % t % t J D0 J ffbb ff ZZ(ZA, AA) J t bf1f1 s f1f1 t f1 t bu td b pp (Z, ) e+e-X pp (W ) e+eX = (dPW+INT/d{x})/(dLO/d{x}) {x} = pe {x} = |e| T {x} = pe |e| pe T T |e| -0.-0.-0.6 -0.-0.-0.-0.-0.-0.-0.---1.-1.-1.-1.-1.20 40 60 80 100 120 20 40 60 80 100 1pT(e ) [GeV] pT(e+) [GeV] 0.-0.0.-0.0.-0.-0.-0.--0.-1.-0.-1.-0.-1.--1.-1.-0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.|(e-)| |(e+)| 1.0.-0.-1 --1.2.-2.1.1.1.1.1.1.1.1.1 20 0.8 40 0.60 0.60 0.80 0.4 80 0.100 0.2 0.10 120 1 pp (Z, ) e+e-X e- e+ [%] [%] [%] [%] [%] [%] )| + )| (e | (e | + p (e p ) [GeV] (e ) [GeV] T T

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