Electroweak radiative corrections at colliders

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1 Elctrowak radiativ corrctions at collidrs LE Duc Ninh LE Duc Ninh, MPI für Physik, Munich p.1/43

2 Outlin Introduction (SM at th prsnt, why NLO?) Full NLO corrctions to + W + W Z, ZZZ b bh production at th LHC, Landau singularitis. Currnt projcts: Higg production and CP violation in th cmssm. LE Duc Ninh, MPI für Physik, Munich p.2/43

3 Th SM LEPEWWG 2009: August 2009 Thory uncrtainty α had = α (5) ± ± incl. low Q 2 data m Limit = 157 GV χ Excludd Prliminary m H [GV] LEP dirct sarch ( + ZH, s = 209GV): M H > 114GV CDF and D0 p p H W + W : M H / [162,166]GV. Prcision EW masurmnts: M H < 157GV ( χ 2 = 2.7). LE Duc Ninh, MPI für Physik, Munich p.3/43

4 Why NLO calculations? Importanc of multiparticl procsss at th LHC, linar collidrs: Many havy particls (W, Z, t,...) can b simultanously producd. Each havy particl can dcay into jts, lptons, photons; lading to multiparticl final stats. Irrducibl backgrounds to ths signals. LE Duc Ninh, MPI für Physik, Munich p.4/43

5 Why NLO calculations? Importanc of multiparticl procsss at th LHC, linar collidrs: Many havy particls (W, Z, t,...) can b simultanously producd. Each havy particl can dcay into jts, lptons, photons; lading to multiparticl final stats. Irrducibl backgrounds to ths signals. Importanc of NLO corrctions: LO prdictions suffr from larg scal uncrtainty. nd NLO to rduc thortical rrors. NLO QCD corrctions: O(10 100%), NLO EW corrctions: O(5 20%) LE Duc Ninh, MPI für Physik, Munich p.4/43

6 Going byond LO In principl, w know how to do it at 1-loop: dσ NLO = dσ virt + dσ ral At NLO, many divrgncs appar: UV, IR, collinar, Landau singularitis (mor latr). Rnormalisation to rgulariz UV divrgncs. By adding ral radiation, w cancl all soft and som collinar singularitis. Th lft-ovr collinar singularitis can b factorizd. In pp procsss: ths collinar singularitis ar absorbd into PDFs. In + procsss: initial-stat collinar singularitis induc larg corrctions αln(s/m 2 ). LE Duc Ninh, MPI für Physik, Munich p.5/43

7 Structur of 1-loop amplituds (Virtual) On-loop intgrals: follow t Hooft, Passarino, Vltman (1979). Ida: M(z) = a i A 0 i + b ib 0 i + c ic 0 i + d id 0 i + R a i, b i, c i, d i, R ar rational. Qustion: How to gt th cofficints and th rational trm? Fynman diagram approach: do tnsor rduction for ach diagram (in D = 4 2ǫ). Finit trms lik ǫ 1 contribut to th rational trm R (a by-product). ǫ On-shll mthods (multipl cuts). ÖÒ ÜÓÒ ÙÒ Ö ÃÓ ÓÛ Ö ÓÖ Ö ØØÓ ÞÓ Ò ººº Disc(LHS) = Disc(RHS). OPP (Ç ÓÐ È Ô ÓÔÓÙÐÓ È ØØ Ù) mthod (working at th intgrand lvl). Th choic of which mthod dpnds on th problm in qustion. LE Duc Ninh, MPI für Physik, Munich p.6/43

8 Full NLO corrctions to + W + W Z,ZZZ (Fynman diagram approach) Basd on: Fawzi Boudjma, LDN, Sun Hao, Marcus Wbr, arxiv: LE Duc Ninh, MPI für Physik, Munich p.7/43

9 WW production at LEP 20 LEP 02/03/2001 Prliminary σ WW [pb] RacoonWW / YFSWW 1.14 no ZWW vrtx (Gntl 2.1) only ν xchang (Gntl 2.1) E cm [GV] SM trilinar couplings: wll tstd at LEP. LE Duc Ninh, MPI für Physik, Munich p.8/43

10 WW production at LEP 20 LEP 02/03/2001 Prliminary σ WW [pb] RacoonWW / YFSWW 1.14 no ZWW vrtx (Gntl 2.1) only ν xchang (Gntl 2.1) E cm [GV] SM trilinar couplings: wll tstd at LEP. What about th quartic gaug couplings WWV V? Not wll tstd. LE Duc Ninh, MPI für Physik, Munich p.8/43

11 + V V Z : tr diagrams ZZZ: 9 diagrams, no trilinar and quartic couplings in SM WWZ: 20 diagrams, trilinar and quartic couplings contribut in SM V V H V f V Z Z Z Z(γ) W W W Z Z(γ) W Z W Z(γ) W Z W LE Duc Ninh, MPI für Physik, Munich p.9/43

12 + W + W Z : on-loop diagrams t Hooft-Fynman guag, nglcting S couplings: ν W W γ(z) W Z ν W ν W W W Z γ ν W ν W Z Z ν W ν W Z ν W ν W W Z W ν ν ν Z W W ν W ν W W ν Z ν W W γ W G Z Topology ZZZ(1767) W W Z(2736) Loop Amp. (FormCalc-6.0) 6.4MB 6.9MB 4-point point LE Duc Ninh, MPI für Physik, Munich p.10/43

13 On-loop Rnormalisation UV-divrgnc is rgularisd by th mans of rnormalisation. Indpndnt paramtrs (CKM = 1):, m f, M W, M Z, M H Rnormalizd paramtrs: 0 = Z, M 0 = M + δm Fild rnormalisation: φ 0 i = (δ ij + δz φ ij /2)φ i On-shll schm: All physical masss ar th pol positions of th propagator. Fild rnormalisation: th pol rsidu is qual to 1, no mixing btwn on-shll physical filds. Th matrix δz φ ij is, in gnral, ral but not orthogonal (δzφ ij δzφ ji ). For th SM, th OS schm works so wll bcaus all th physical masss ar indpndnt paramtrs and hnc can b rnormalizd as th pol positions of th propagator. This is not tru for th MSSM (M 2 H ± = M 2 A + M2 W ). LE Duc Ninh, MPI für Physik, Munich p.11/43

14 Loop intgrals and numrical instabilitis k i = P i 1 j=1 p j, i = 1, 2,3,... dt(g) = dt(2k i k j ): Gram dtrminant dt(y ) = dt(m 2 i +m2 j (k i k j ) 2 ): Landau dtrminant LE Duc Ninh, MPI für Physik, Munich p.12/43

15 Loop intgrals and numrical instabilitis k i = P i 1 j=1 p j, i = 1, 2,3,... dt(g) = dt(2k i k j ): Gram dtrminant dt(y ) = dt(m 2 i +m2 j (k i k j ) 2 ): Landau dtrminant 5pt intgrals ar rducd to 4pts ÒÒ Ö Ò ØØÑ Ö ¾¼¼¾ E 0 = P 5 i=1 dt(y i ) dt(y ) D 0(i) LE Duc Ninh, MPI für Physik, Munich p.12/43

16 Loop intgrals and numrical instabilitis k i = P i 1 j=1 p j, i = 1, 2,3,... dt(g) = dt(2k i k j ): Gram dtrminant dt(y ) = dt(m 2 i +m2 j (k i k j ) 2 ): Landau dtrminant 5pt intgrals ar rducd to 4pts ÒÒ Ö Ò ØØÑ Ö ¾¼¼¾ E 0 = P 5 i=1 dt(y i ) dt(y ) D 0(i) Tnsor 4pt intgrals up to rank 4: Passarino-Vltman rduction D ijkl = f(p i, m i )/ dt(g) 4 = numrical instabilitis occur whn dt(g) is small (clos to PS boundary). LE Duc Ninh, MPI für Physik, Munich p.12/43

17 Loop intgrals and numrical instabilitis k i = P i 1 j=1 p j, i = 1, 2,3,... dt(g) = dt(2k i k j ): Gram dtrminant dt(y ) = dt(m 2 i +m2 j (k i k j ) 2 ): Landau dtrminant 5pt intgrals ar rducd to 4pts ÒÒ Ö Ò ØØÑ Ö ¾¼¼¾ E 0 = P 5 i=1 dt(y i ) dt(y ) D 0(i) Tnsor 4pt intgrals up to rank 4: Passarino-Vltman rduction D ijkl = f(p i, m i )/ dt(g) 4 = numrical instabilitis occur whn dt(g) is small (clos to PS boundary). Our solutions: small DtG xpansion or using quadrupl prcision (loop library only, th rsults bcom stabl, 6 tims slowr). LE Duc Ninh, MPI für Physik, Munich p.12/43

18 Loop intgrals and numrical instabilitis k i = P i 1 j=1 p j, i = 1, 2,3,... dt(g) = dt(2k i k j ): Gram dtrminant dt(y ) = dt(m 2 i +m2 j (k i k j ) 2 ): Landau dtrminant 5pt intgrals ar rducd to 4pts ÒÒ Ö Ò ØØÑ Ö ¾¼¼¾ E 0 = P 5 i=1 dt(y i ) dt(y ) D 0(i) Tnsor 4pt intgrals up to rank 4: Passarino-Vltman rduction D ijkl = f(p i, m i )/ dt(g) 4 = numrical instabilitis occur whn dt(g) is small (clos to PS boundary). Our solutions: small DtG xpansion or using quadrupl prcision (loop library only, th rsults bcom stabl, 6 tims slowr). Scalar 4pt intgrals: can also hav numrical cancllation (obsrvd in WWZ). LE Duc Ninh, MPI für Physik, Munich p.12/43

19 Ral corrction dσ + V V Z 1 loop = dσ + V V Z virt + dσ + V V Zγ ral Th virtual part contains both soft and collinar divrgncs. All ths singularitis ar canclld by adding th ral photon radiation procss. LE Duc Ninh, MPI für Physik, Munich p.13/43

20 Ral corrction dσ + V V Z 1 loop = dσ + V V Z virt + dσ + V V Zγ ral Th virtual part contains both soft and collinar divrgncs. All ths singularitis ar canclld by adding th ral photon radiation procss. All singularitis in th ral amplitud can b factorisd, P ff (y) = (1 + y 2 )/(1 y): X X M 1 2 Q gk 0 f σ f Q f σ f 2 p fp f (p λ γ f,f f k)(p f k) M 0 2, X λ γ M 1 2 p i k 0 Q 2 i 2 1 p i k X λ γ M 1 2 p a k 0 Q 2 a 2 1 x a (p a k)» P ff (z i ) m2 i M 0 (p i + k) 2, p i k» P ff (x a ) x am 2 a M 0 (x a p a ) 2. p a k LE Duc Ninh, MPI für Physik, Munich p.13/43

21 Ral corrction dσ + V V Z 1 loop = dσ + V V Z virt + dσ + V V Zγ ral Th virtual part contains both soft and collinar divrgncs. All ths singularitis ar canclld by adding th ral photon radiation procss. All singularitis in th ral amplitud can b factorisd, P ff (y) = (1 + y 2 )/(1 y): X X M 1 2 Q gk 0 f σ f Q f σ f 2 p fp f (p λ γ f,f f k)(p f k) M 0 2, X λ γ M 1 2 p i k 0 Q 2 i 2 1 p i k X λ γ M 1 2 p a k 0 Q 2 a 2 1 x a (p a k)» P ff (z i ) m2 i M 0 (p i + k) 2, p i k» P ff (x a ) x am 2 a M 0 (x a p a ) 2. p a k Aftr adding th virtual and ral corrctions th rsult is still collinar singular. This singularity coms from th initial stat radiation part, in th form α ln(s/m 2 ) aftr int. LE Duc Ninh, MPI für Physik, Munich p.13/43

22 Ral corrction dσ + V V Z 1 loop = dσ + V V Z virt + dσ + V V Zγ ral Th virtual part contains both soft and collinar divrgncs. All ths singularitis ar canclld by adding th ral photon radiation procss. All singularitis in th ral amplitud can b factorisd, P ff (y) = (1 + y 2 )/(1 y): X X M 1 2 Q gk 0 f σ f Q f σ f 2 p fp f (p λ γ f,f f k)(p f k) M 0 2, X λ γ M 1 2 p i k 0 Q 2 i 2 1 p i k X λ γ M 1 2 p a k 0 Q 2 a 2 1 x a (p a k)» P ff (z i ) m2 i M 0 (p i + k) 2, p i k» P ff (x a ) x am 2 a M 0 (x a p a ) 2. p a k Aftr adding th virtual and ral corrctions th rsult is still collinar singular. This singularity coms from th initial stat radiation part, in th form α ln(s/m 2 ) aftr int. Two ways to calculat: phas spac slicing and subtraction mthods. LE Duc Ninh, MPI für Physik, Munich p.13/43

23 Ral corrction: phas spac slicing Ral corrction is cutoff-indpndnt. Factorization condition: δ s and δ c ar sufficintly small. And δ c 2m 2 /s to us th collinar intgration formula. σ[fb] σ hard s=500gv M H =120GV -4 δ c =7 10 σ virt+soft σ virt+soft+hard log δ 10 s σ[fb] σ coll σ fin σ hard s=500gv M H =120GV -3 δ s = log δ 10 c LE Duc Ninh, MPI für Physik, Munich p.14/43

24 Z σ ral = Th subtraction function should b: Ral corrction: dipol subtraction 4 Z (dσ ral dσ sub ) + 4 dσ sub. th sam as th ral function dσ ral in th singular limits. simpl nough so that it can b analytically intgratd ovr th singular rgion. Th dipol subtraction Ø Ò Ë ÝÑÓÙÖ ØØÑ Ö mthod ººº: Z dσ sub = α Z 4 2π σ ndpoint = α Z 2π 3 dx X Z Q i Q j G ij (x) i j X dσ Born Q i Q j G ij. i j 3 dσ Born + σ ndpoint, Th subtraction function is a sum of many dipol trms. Th ndpoint contribution contains all th soft and collinar singularitis of th virtual part, with th opposit signs: σ wak = σ virt + σ ndpoint : soft and coll. finit LE Duc Ninh, MPI für Physik, Munich p.15/43

25 Ral corrction: dipol vs. slicing WWZ s=500gv M H =120GV Slicing Dipol σ Ral [fb] log δ 10 s Slicing: simpl, asy to implmnt, larg intgration rror. W us this to cross chck th rsults. Tricky point: whn on dcrass th rror, th cut-offs must also b rducd. Dipol: subtraction function is quit complicatd (not so asy to implmnt), th intgration rror is typically 10 tims smallr than slicing s, no cut-off dpndnc. Tricky point: misbinning ffct in histograms. Calculating ral corrction is mor tim-consuming than gtting th virtual part. LE Duc Ninh, MPI für Physik, Munich p.16/43

26 NLO calculation in practic Many usful public cods to hlp us, but no prfct cod xists. Warning: do not us ths cods blindly. Fynman diagrams and amplitud xprssions: FynArts-3.4(À Ò),... To writ th amplituds in a dsird form (.g. in trms of basic loop intgrals, spinors, xtrnal momnta,...): FormCalc-6.0(À Ò, in Math+FORM), usrs should hav a good control on it. a good way to factoriz th amps can mak your cod 2 3 tims fastr (optimization). Loop intgrals: tricky part, us diffrnt cods to cross chck. LoopTools(Ú Ò À Ò), OnLOop(Ú Ò À Ñ Ö Ò), D0C( Ó Ì Æ ÙÒ Ä Æ ¼ Û Ø ÓÑÔÐ Ü»Ö Ð ÇÐ Ò ÓÖ ÒÐÙ Ò LoopTools-2.4),... Ñ Phas spac intgration: VEGAS, BASES(Ã Û Ø ), CUBA(À Ò),... LE Duc Ninh, MPI für Physik, Munich p.17/43

27 Chcks on th rsults Non-linar gaug (NLG) invarianc chck: tr and on-loop squard amplitud lvl. W us SloopS( ÖÓ ÓÙ Ñ Ò Ë Ñ ÒÓÚ; FA+NLG). Th rsults should b UV and IR finit. Two indpndnt calculations (cods): min in Fortran 77, collaborator s(ï Ö) in C++. LE Duc Ninh, MPI für Physik, Munich p.18/43

28 NLG Chck and numrical instability NLG fixing Lagrangian ( ÓÙ Ñ ÓÔ Ò ½ ): L GF = 1 ξ W ( µ i αa µ igc W βzµ )W µ+ + ξ W g 2 (v + δh + i κχ 3 )χ ξ Z (.Z + ξ Z g 2c W (v + εh)χ 3 ) 2 1 2ξ A (.A) 2. ( α, β) ÏÏ ½µ ÏÏ ¾µ ¼ ¼µ ¹ º ¼ ¼ ¾ ¼ ½ ¹ ¹ º ¾½ ¾¾¼ ¹¾ º ¼ ¾ ½½½½¾ ¼ ½ ¾¼ ¼ ¾ ¾½ ¹¾ ½ ¼µ ¹ º ¼ ¼ ¾ ¼ ½ ¹ ¹ º ¼ ½ ¹¾ º ¼ ¾ ½½½½½¼ ½½ ¾ ¼ ¼½ ¾ ¹¾ ¼ ½µ ¹ º ¼ ¼ ½ ¾ ¹ ¹ º ¾¾ ½½ ¼ ¹¾ º ¼ ¾ ½½½½ ¼ ½¼½ ½¼ ¾ ¾ ½ ¹¾ ZZZ: at last 10 digit agrmnt with doubl prcision (DP). WWZ: 4 digits with DP, 12 digits with quadrupl prcision. This is an indication of numrical instability. LE Duc Ninh, MPI für Physik, Munich p.19/43

29 + ZZZ: Total Xsction Born σ[fb] 0.6 Wak Full NLO δ[%] Wak Full NLO ZZZ M H =120GV ZZZ M H =120GV s[gv] Input paramtrs: α Gµ = 2G µ s 2 W M2 W /π = α(0)(1 + r) Total Xsction pak about 1fb is at s 550GV s[gv] Th wak corrction gos from 12% to 18% whn s incrass from 500GV to 1TV. Comparisons with Su t al. arxiv: : NLO rsults agr to at last 0.1%. LE Duc Ninh, MPI für Physik, Munich p.20/43

30 + W + W Z: Total Xsction σ[fb] Born Wak Full NLO δ[%] Wak Full NLO WWZ M H =120GV WWZ M H =120GV s[gv] s[gv] Total Xsction pak about 50fb (50 tims largr than σ ZZZ ) is at s 900GV. Th wak corrction gos from 7% to 18% whn s incrass from 500GV to 1.5TV. LE Duc Ninh, MPI für Physik, Munich p.21/43

31 + W + W Z: Distributions (I) dσ/dm(ww)[fb/gv] Born Wak Full NLO - + WWZ s=500gv M H =120GV dσ NLO /dσ Born -1[%] Wak Full NLO - + WWZ s=500gv M H =120GV M(WW)[GV] M(WW)[GV] Quit small corrctions (about 10%) at small GV. At larg GV, larg corrctions ( 50%) du to th hard photon ffct [dominant contribution coms from th low-nrgy photon rgion which corrsponds to larg p Z T and larg M WW.] LE Duc Ninh, MPI für Physik, Munich p.22/43

32 + W + W Z: Distributions (II) dσ/dy(ww)[fb] 60 Born Wak Full NLO - + WWZ s=500gv M H =120GV y(ww) dσ NLO /dσ Born -1[%] Wak Full NLO - + WWZ s=500gv M H =120GV y(ww) NLO corrctions show nw structurs, which cannot b xplaind by an ovrall scal factor. LE Duc Ninh, MPI für Physik, Munich p.23/43

33 Yukawa corrctions to pp b bh at th LHC Landau singularitis Basd on: Fawzi Boudjma, LDN, arxiv: , arxiv: LE Duc Ninh, MPI für Physik, Munich p.24/43

34 Why pp b bh? λ bbh =? LE Duc Ninh, MPI für Physik, Munich p.25/43

35 Why pp b bh? λ bbh =? SM: λ bbh = m b /υ=-0.02 with m b = 4.62GV, υ = 246GV. LE Duc Ninh, MPI für Physik, Munich p.25/43

36 Why pp b bh? λ bbh =? SM: λ bbh = m b /υ=-0.02 with m b = 4.62GV, υ = 246GV. MSSM: if tan β υ 1 /υ 2 is larg, th bottom-higgs Yukawa coupling can b nhancd, lading to larg cross sction. λ bbh = m b υ λ bbh = m b υ [sin(β α) tan β cos(β α)], [cos(β α) tan β sin(β α)]. LE Duc Ninh, MPI für Physik, Munich p.25/43

37 Why pp b bh? λ bbh =? SM: λ bbh = m b /υ=-0.02 with m b = 4.62GV, υ = 246GV. MSSM: if tan β υ 1 /υ 2 is larg, th bottom-higgs Yukawa coupling can b nhancd, lading to larg cross sction. λ bbh = m b υ λ bbh = m b υ [sin(β α) tan β cos(β α)], [cos(β α) tan β sin(β α)]. Tagging b-jts with high p T to idntify th procss, QCD background is rducd. LE Duc Ninh, MPI für Physik, Munich p.25/43

38 On-loop EW corrction: diagrams (a) b t H b χ W (b) b t H b χ W H χ W (c) b t b Each group is QCD gaug invariant λ bbh = 0 (a) = 0, (b, c) 0 LE Duc Ninh, MPI für Physik, Munich p.26/43

39 λ bbh xpansion Thr ar 2 contributions: σ(λ bbh ) = σ(λ bbh = 0) + λ 2 bbh σ (λ bbh = 0) + NLO corrctions: ÛÞ ÓÙ Ñ Ä Æ ¾¼¼ µ λ 2 bbh σ (λ bbh = 0) = σ 0 [1 + δ EW (m t, M H )] On-loop squard: ÛÞ ÓÙ Ñ Ä Æ ¾¼¼ µ σ(λ bbh = 0) A 1 2 (M H, m t ) LE Duc Ninh, MPI für Physik, Munich p.27/43

40 σ EW (λ bbh = 0): M H < 2M W σ[fb] pp bbh s=14tv λ bbh =0 =0)/σ LO [%] σ(λ bbh pp bbh s=14tv M H [GV] M H [GV] M H = 120GV: σ(λ bbh = 0) 1fb. it rapidly incrass whn M H incrass. LE Duc Ninh, MPI für Physik, Munich p.28/43

41 σ EW (λ bbh = 0): M H < 2M W σ[fb] pp bbh s=14tv λ bbh =0 =0)/σ LO [%] σ(λ bbh pp bbh s=14tv M H [GV] M H [GV] M H = 120GV: σ(λ bbh = 0) 1fb. it rapidly incrass whn M H incrass. What happns if M H 2M W? Phas spac intgration dos not convrg (no problm at NLO). LE Duc Ninh, MPI für Physik, Munich p.28/43

42 p H T -distributions(λ bbh = 0)@EW [pb/gv] H T dσ/dp pp bbh s=14tv λ bbh =0 M H =120GV M H =150GV p H [GV] T Larg corrction at som rgion of phas spac. dσ(λ =0)/dσ LO [%] bbh pp bbh s=14tv M H =150GV M H =120GV p H [GV] T LE Duc Ninh, MPI für Physik, Munich p.29/43

43 Th problmatic diagram W found that th problm with PS intgration is rlatd to this: p3 p1 q3 q2 p5 q4 q1 p2 p4 LE Duc Ninh, MPI für Physik, Munich p.30/43

44 Th problmatic diagram W found that th problm with PS intgration is rlatd to this: p3 p1 q3 q2 p5 q4 q1 p2 p4 Considring only this diagram, w found: Th problm is rlatd to th scalar loop intgral (it is NOT th problm with th Gram dtrminant). if ŝ 2m t and M H 2M W loop particls ar all on-shll Landau singularitis? LE Duc Ninh, MPI für Physik, Munich p.30/43

45 2 Landau quations L.D. Landau, Nucl. Phys. 13 (1959) 181; Polkinghorn, Oliv, Landshoff, Edn, Th analytic S-Matrix (1966) T N 0 Z 0 Landau: NY Z dx i i=1 d D P q δ( Ni=1 x i 1) (2π) D [ P N i=1 x i(qi 2 m2 i + iǫ)]n Physical rgion: [x i = x i, x i 0, q i = q i ] Singular only for: ǫ < : i x i (qi 2 m2 i P ) = 0 M i=1 x iq i = 0 pinch singularity LE Duc Ninh, MPI für Physik, Munich p.31/43

46 2 Landau quations L.D. Landau, Nucl. Phys. 13 (1959) 181; Polkinghorn, Oliv, Landshoff, Edn, Th analytic S-Matrix (1966) T N 0 Z 0 Landau: NY Z dx i i=1 d D P q δ( Ni=1 x i 1) (2π) D [ P N i=1 x i(qi 2 m2 i + iǫ)]n Physical rgion: [x i = x i, x i 0, q i = q i ] Singular only for: ǫ < : i x i (qi 2 m2 i P ) = 0 M i=1 x iq i = 0 pinch singularity All x i > 0 (all qi 2 = m2 i ): th lading Landau singularity (LLS) Som x i = 0: sub-lls LE Duc Ninh, MPI für Physik, Munich p.31/43

47 2 Landau quations L.D. Landau, Nucl. Phys. 13 (1959) 181; Polkinghorn, Oliv, Landshoff, Edn, Th analytic S-Matrix (1966) T N 0 Z 0 Landau: NY Z dx i i=1 d D P q δ( Ni=1 x i 1) (2π) D [ P N i=1 x i(qi 2 m2 i + iǫ)]n Physical rgion: [x i = x i, x i 0, q i = q i ] Singular only for: ǫ < : i x i (qi 2 m2 i P ) = 0 M i=1 x iq i = 0 pinch singularity All x i > 0 (all qi 2 = m2 i ): th lading Landau singularity (LLS) Som x i = 0: sub-lls Physical intrprtation (Colman and Norton): Each vrtx: ral spac-tim point Spac tim sparation: dx i = x i q i (no sum); P M i=1 dx i = 0 Propr tim: dτ i = m i x i > 0 (no sum) v i < c LE Duc Ninh, MPI für Physik, Munich p.31/43

48 2 Landau quations L.D. Landau, Nucl. Phys. 13 (1959) 181; Polkinghorn, Oliv, Landshoff, Edn, Th analytic S-Matrix (1966) T N 0 Z 0 Landau: NY Z dx i i=1 d D P q δ( Ni=1 x i 1) (2π) D [ P N i=1 x i(qi 2 m2 i + iǫ)]n Physical rgion: [x i = x i, x i 0, q i = q i ] Singular only for: ǫ < : i x i (qi 2 m2 i P ) = 0 M i=1 x iq i = 0 pinch singularity All x i > 0 (all qi 2 = m2 i ): th lading Landau singularity (LLS) Som x i = 0: sub-lls Physical intrprtation (Colman and Norton): Each vrtx: ral spac-tim point Spac tim sparation: dx i = x i q i (no sum); P M i=1 dx i = 0 Propr tim: dτ i = m i x i > 0 (no sum) v i < c How to chck thos conditions in practic? LE Duc Ninh, MPI für Physik, Munich p.31/43

49 2 important conditions Landau quations: Q ij 2q i.q j = m 2 i + m2 j (q i q j ) 2 (Landau matrix), MX x i q i = 0 i=1 8 >< >: Q 11 x 1 + Q 12 x 2 + Q 1M x M = 0, Q 21 x 1 + Q 22 x 2 + Q 2M x M = 0,. Q M1 x 1 + Q M2 x 2 + Q MM x M = 0. LE Duc Ninh, MPI für Physik, Munich p.32/43

50 2 important conditions Landau quations: Q ij 2q i.q j = m 2 i + m2 j (q i q j ) 2 (Landau matrix), MX x i q i = 0 i=1 8 >< >: Q 11 x 1 + Q 12 x 2 + Q 1M x M = 0, Q 21 x 1 + Q 22 x 2 + Q 2M x M = 0,. Q M1 x 1 + Q M2 x 2 + Q MM x M = 0. Landau dtrminant must vanish: dt(q) = 0 LE Duc Ninh, MPI für Physik, Munich p.32/43

51 2 important conditions Landau quations: Q ij 2q i.q j = m 2 i + m2 j (q i q j ) 2 (Landau matrix), MX x i q i = 0 i=1 8 >< >: Q 11 x 1 + Q 12 x 2 + Q 1M x M = 0, Q 21 x 1 + Q 22 x 2 + Q 2M x M = 0,. Q M1 x 1 + Q M2 x 2 + Q MM x M = 0. Landau dtrminant must vanish: dt(q) = 0 Sign condition (occurring in th physical rgion): x i > 0, i = 1,..., M x j = dt( ˆQ jm )/dt( ˆQ MM ) > 0, j = 1,..., M 1 dt( ˆQ MM ) = d[dt(q)]/dq MM, dt( ˆQ 1j ) = 1 2 d[dt(q)]/dq 1j. LE Duc Ninh, MPI für Physik, Munich p.32/43

52 Natur of LLS Th LLSs ar intgrabl or not? LE Duc Ninh, MPI für Physik, Munich p.33/43

53 Natur of LLS Th LLSs ar intgrabl or not? For N = 2, D = 4 2ε: (B 0 ) div [dt(q 2 ) iǫ] 1/2 (finit) For N = 3, D = 4 2ε: (C 0 ) div ln[dt(q 3 ) iǫ] (intgrabl) For N = 4, D = 4: (D 0 ) div 1 dt(q4 ) iǫ (intgrabl, th squar is not intgrabl) For N = 5, D = 4: (E 0 ) div 1 dt(q 5 ) iǫ For N 6: No LLS but svral sub-llss. (not intgrabl) Th xact cofficints ar givn in Ä Æ Ö Ú ¼ ½¼º ¼ È Ø µ. LE Duc Ninh, MPI für Physik, Munich p.33/43

54 4-point LLS: g b bh (I) p 1 p 3 q 3 q 2 p 5 q 4 q 1 p 2 p 4 Qustion: What ar th physical conditions to hav a LLS? LE Duc Ninh, MPI für Physik, Munich p.34/43

55 4-point LLS: g b bh (I) p 1 p 3 q 3 q 2 p 5 q 4 q 1 p 2 p 4 Qustion: What ar th physical conditions to hav a LLS? 8 8 >< >: qi 2 q i = m 2 i = qi x 1 q 1 + x 4 q 4 = x 2 q 2 + x 3 q 3 x i > 0 E-p consrvation >< >: M H s 2M W 2mt s 1,2 (m t + M W ) 2 m t > M W LE Duc Ninh, MPI für Physik, Munich p.34/43

56 4-point LLS: g b bh (I) p 1 p 3 q 3 q 2 p 5 q 4 q 1 p 2 p 4 Qustion: What ar th physical conditions to hav a LLS? 8 8 >< >: qi 2 q i = m 2 i = qi x 1 q 1 + x 4 q 4 = x 2 q 2 + x 3 q 3 x i > 0 E-p consrvation >< >: M H s 2M W 2mt s 1,2 (m t + M W ) 2 m t > M W Physical pictur: th off-shll gluon splits into two on-shll top quarks, ach top quark thn dcays into a bottom quark and an on-shll W gaug boson. Finally, th W gaug bosons fus into th Higgs. Th problm is rlatd to intrnal unstabl particls. LE Duc Ninh, MPI für Physik, Munich p.34/43

57 4-point LLS: g b bh (III) Ral(D0) s 2 [GV] s 1 [GV] Img(D0) s2 [GV] s [GV] D 0 = D 0 (M 2 H, 0, s,0, s 1, s 2, M 2 W, M2 W, m2 t, m 2 t). Input paramtrs: s = 353GV > 2m t, M H = 165GV > 2M W, m b = 0. Rgion of LLS at th cntr of th phas spac. Tak s 1 = q 2(m 2 t + M2 W ) GV LE Duc Ninh, MPI für Physik, Munich p.35/43

58 sub-llss Dt(S )/(3*10 ) 4 0 Im(D ) 0 p 1 p 3 R(D ) 0 q 3 q p 5 s=353gv q 4 q 1-1 M H =165GV s 1 =271.06GV p 2 p s 2 [GV] LE Duc Ninh, MPI für Physik, Munich p.36/43

59 sub-llss Dt(S )/(3*10 ) 4 0 Im(D ) 0 p 1 p 3 R(D ) 0 q 3 q p 5 s=353gv q 4 q 1-1 M H =165GV s 1 =271.06GV p 2 p s 2 [GV] LE Duc Ninh, MPI für Physik, Munich p.36/43

60 Solution Th widths of intrnal unstabl particls (t, W) must b takn into account: m 2 t m 2 t im t Γ t, M 2 W M2 W im WΓ W. Mathmatically, th width ffct is to mov Landau singularitis into th complx plan, so thy do not occur in th physical rgion. LE Duc Ninh, MPI für Physik, Munich p.37/43

61 Solution Th widths of intrnal unstabl particls (t, W) must b takn into account: m 2 t m 2 t im t Γ t, M 2 W M2 W im WΓ W. Mathmatically, th width ffct is to mov Landau singularitis into th complx plan, so thy do not occur in th physical rgion. W nd 4-point intgrals with complx masss, us D0C( Ó Ì Æ ÙÒ Ä Æ ¾¼¼ ): D 0 (Γ t,γ W ) = writtn in trms of 32 Spnc functions. 1 P 2 P R 4 dt(q) i=1 j=1 ( 1)i+j 1 0 dy 1 ln(a y y j y 2 + B j y + C j ) i LE Duc Ninh, MPI für Physik, Munich p.37/43

62 Complx masss ) 0 Ral(D ) 0 Img(D Γ t,w =0-1 Γ t,w =0-1.5 Γ t =1.5GV, Γ W =2.1GV -1.5 Γ t =1.5GV, Γ W =2.1GV s 2 [GV] s 2 [GV] All Landau singularitis ar compltly rgularizd. LE Duc Ninh, MPI für Physik, Munich p.38/43

63 σ(λ bbh = 0): M H 2M W 14 ÛÞ ÓÙ Ñ Ä Æ ¾¼¼ µ σ[fb] pp bbh s=14tv λ bbh =0 Lading Landau Singularity Γ t =1.5GV Γ W =2.1GV Γ t =0 Γ W =0 =0)/σ LO [%] bbh σ(λ pp bbh s=14tv Γ t 1.5GV Γ W =2.1GV M H [GV] M H [GV] Th singular bhaviour is nicly tamd by introducing th widths. LE Duc Ninh, MPI für Physik, Munich p.39/43

64 Currnt projcts LE Duc Ninh, MPI für Physik, Munich p.40/43

65 Higgs production and CPV in th cmssm Ì Æ ÙÒ ÏÓÐ Ò ÀÓÐÐ Ä Æ Ó MSSM is an attractiv xtnsion of th SM. cmssm: nw sourcs for CPV (µ, M i and A f can hav phass) can hlp to xplain th obsrvd abundanc of mattr ovr antimattr. CP asymmtry: δ CP = σ(pp X+ Y ) σ(pp X Y + ) σ(pp X + Y ) + σ(pp X Y + ) Higgs propagators: important ffcts in Higgs production and CP asymmtris. Rnormalisation: DR schm. LE Duc Ninh, MPI für Physik, Munich p.41/43

66 Thank you LE Duc Ninh, MPI für Physik, Munich p.42/43

Electroweak corrections to tri-boson production at the ILC

Electroweak corrections to tri-boson production at the ILC Elctrowak corrctions to tri-boson production at th ILC Duc Ninh LE ldninh@mppmu.mpg.d ork in collaboration with Fawzi BOUDJEMA, Hao SUN and Marcus EBER Corfu, 2009 Sp 4th Duc Ninh LE, Gaug boson production