Helicity Conservation (HC) in SUSY: Application to the 1loop EW corrections to + +

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1 Helicity Conservation (HC) in SUSY: Application to the 1loop EW corrections to dw χ + + and dl i with F.M. Renard, J. Layssac Smmary What is HC? HC is exact in MSSM and approximate in SM. + + Example: The 1loop EW amplitdes for dw and d Lχi and their helicity amplitdes at hih eneries. HC helps derivin SUSY asymptotic relations amon sch processes, which may be relevant at LHC. HC provides a sefl test of calclations. It shold be possible to extend HC to any NMSSM model. 1

2 What is HC? Renard+G: PRL 94:131601(2005), PR D73:097301(2006) In SUSY it is exact. It claims that at sám² SUSY and fixed anles, all non-vanishin helicity amplitdes in a two-to-two processes mst satisfy the HC rle Fab ( λ λ cλ dλ ) λ + λ = λ + λ All amplitdes violatin HC, mst vanish in SUSY, in this limit. In SM, HC is approximate. It has been proved, p to the 1-loop Leadin Lo level, provided (s, t, ) m² w. This means that in SM, the HC violatin amplitdes o asymptotically to (sally non-vanishin bt small) constants. Anomalos coplins violate HC in all cases I know. 2

3 HC at the Born level (s, t, ) á all masses, s/t=fixed If no external ae bosons appear, the validity of HC is almost trivial. µ λ2 e λ1 γ µ γ,z µ λ2 e λ1 µ R e L P R P L B µ R ẽ L In the case of external ae bosons, lare ae cancellations amon the diarams are needed. The coplins mst be the standard renormalizable ones. At Born level, HC is valid, in both SM and MSSM 3

4 HC at the 1-loop level with Layssac, Porfyriadis, Renard, Diakonidis (s, t, ) á all masses s/t=fixed γγ Æ γγ, γγ Æ ZZ, γγæ γ Z The helicity conservin amplitdes are mch larer than the helicity violatin ones and predominantly imainary in both, SM and MSSM. InSM, the helicity violatin amplitdes o to (small) constants. In MSSM, the helicity violatin amplitdes vanish exactly. (Particlarly tricky for lonitdinal Z s) We have also observed (and sed) the MSSM vanishin of the asymptotic helicity violatin amplitdes in the 1loop calclation of χ χ γγ χ χ γ i j, i j Z 4

5 Proof of HC, to all orders in MSSM (R-parity was assmed). Nelect all dimensional parameters: soft terms and the µ term. A new U(1) is identified, respected by all vertices, except the f-fscalar Ykawa terms. This leads to HC for all two-to-two processes, with external fermions or scalars only. It is then observed that the SUSY transformation, when projected on the one-particle states, relates ae aino with helicities of the same sin This then leads to HC for all two-to-two processes. For mltiparticle final states, the sitation is more complicated and HC is not enerally valid. Maybe 5

6 Application to 1loop EW for + ( λ ) + ( λ ) d( λ ) + W ( λ ) F d W W ( λ =± 1), ( λ =± 1, 0) d HC F λλλλ λ λ λ W = λ W, W HC F ----, F -+-+ HV1 F ---+, F ---0 HV2 F -+--, F -+-0 Born exists lare No Born very small Born exists small 6

7 + Renormalization of ( λ ) + ( λ ) d( λ ) + W ( λ ) Dimensional relarization always sed. d W Renormalization is done by insertin to the Born contribtions the conter terms de to bbbles involvin two external les and determinin the wave fnction renormalization of the external, d and W fields, the conter term to the Wd coplin and the self-enery corrections to the internal - and d-propaators in the two tree diarams. 7

8 We need benchmarks coverin a wide rane of SUSY masses. BSSW, FLN msp4 and SPS1a' are consistent with everythin known; (Baer, Nath, SPA). GeV BBSSW FLN msp4 SPS1a' liht m 1/ m A tanβ M SUSY

9 The most important Æ d W+ amplitdes in SM and MSSM are the HC ones Im parts are mch smaller than the Real parts.. Loop effects become very important above 1TeV. 9

10 The HV1 Æ d W + amplitdes are very small. (No Born contribtions for them.) 10

11 For s d0.3tev, the HV2 Æ d W + amplitdes may be comparable to the HC ones. HC HV2 Im parts are mch smaller than the Real parts. They become neliible above ~2TeV. Born approximation shold be adeqate for them. 11

12 At hiher eneries, HV2áHC for Æ d W + HC HV2 12

13 Anlar dependence for the HV2 and HC Æ d W + amplitdes at s=0.5tev HV2 HC At s=0.5tev, the 1-loop SM and MSSSM effects are barely visible. HC t HV2 bt comparable. 13

14 Anlar dependence for the HV2 and HC Æ d W + amplitdes at s=4 TeV HV2 HC At s=4 TeV, the 1-loop SM and MSSSM effects are enhanced. HC p HV1, HV2 14

15 The HC ÆdW amplitdes in the LL+ constant approximation, are qite accrate above ~0.5 TeV SM: η=0, MSSM: η=1, C SM +C SUSY =C MSSM C SM and C MSSM are dimensionless constants which enerally depend on the helicity amplitde and the anle. 15

16 Application to 1loop EW corrections for ( λ ) + ( λ ) d + χ ( λ ) 1 1 λ =, λ =± 1, λi =± 2 2 F F, F, F, λλλ L i i i + + F ++ χ + i (a) (b) For independent helicity amplitdes. Only one respects HC dl dl dl dl dl dl dl dl dl dl dl dl dl dl dl dl (c) 16

17 Renormalization Aain done by insertin to the Born contribtions the conter terms (c.t.) de to the wave fnction renormalization of the external, and χ + L i fields; (charino self eneries rather tricky) the d χ + coplin, L i and the self-enery corrections to the internal - and d L -propaators of the tree diarams. Only bbble diarams havin two external les are needed, always calclated in dimensional relarization.. d 17

18 Enery dependence of the helicity amplitdes for SPS1a' at θ=60 ο For s t 5TeV, HC is approximately established in SPS1a' 18

19 Anlar dependence of the helicity amplitdes for SPS1a' at 1 and 4 TeV HC is faster established away from the forward and backward reions for SPS1a' 19

20 Enery dependence of the helicity amplitdes for FLN msp4 models at θ=60 ο For msp4 also, HC is approximately established for s t 5TeV. 20

21 Anlar dependence of the helicity amplitdes for FLN msp4s1a' at 2 and 4 TeV 21

22 At hih eneries, the 1-loop corrections to the dominant HC amplitde are described by loarithmic and constant term corrections to the Born amplitdes. 22

23 SUSY relations amon the dominant HC amplitdes at asymptotic eneries a χ i 2 2 α (1+ 26 c ) w M 2 2 ln SUSY = 2 4π 72cs w w mz ( θ ) cos / 2 F + dw dw F + + = = cos / 2 + ( θ ) ( θ ) d + L χi ++ Z1 i F sin / 2 (1 + a ) χ i SUSY F-relation dσ ( dw ) d cosθ 1 R iw dσ( d Lχi ) dcosθ + + SUSY σ-relation R iw = s ( m + + m ) s ( ) χi d L i L s m 1/2 1/ 2 2 m + m χ d 2 w 2 } Z 1i 2 (1 + ) sin a χ cosθ + cos i θ 2 θ At finite eneries these relations are violated by the constant and masssppressed terms. In SUSY σ-relation, violation comes also from the HC-violatin amplitdes. 23

24 SUSY F-relations ( θ ) ( θ ) ( θ ) d + L χi ++ Z1 i F sin / 2 (1 + a ) cos / 2 + dw F + + cos / 2 F + dw χ i F-relations affect only the asymptotically dominant HC amplitdes. They are almost perfect for the liht MSSM model, and worsen as we move to hiher SUSY masses in SPS1a' and FLN msp4. 24

25 The SUSY σ-relation for SPS1a'. M SUSY =0.35TeV + + dσ ( dw ) 1 dσ ( d ), Lχi d cs o θ R d cosθ iw In principle, the SUSY σ-relations cold be mch worse than the F-relations, at non-asymptotic eneries, since they also involve the sqares of the HV amplitdes. Actally, they are considerably better! 25

26 The SUSY σ-relation for FLN msp4. M SUSY =1.5TeV + + dσ ( dw ) 1 dσ ( d ), Lχi d cs o θ R d cosθ iw In these models, the SUSY σ-relations at non-asymptotic eneries, are mch better than the SUSY F-relations. For some reason, the sb-dominant HV amplitdes redce the discrepancies. 26

27 Conclsions SUSY : All HC violatin two-to-two amplitdes mst vanish exactly at asymptotic eneries and fixed anles. SM : In all 1loop processes we have looked at, the HC violatin amplitdes o asymptotically to small constants. No eneral proof is known. SM approximately respects HC. For ØdW, HC effects may be visible at LHC. For d χ +, HC is delayed by the lare L, χ + i L i masses. The SUSY σ-relation amon these two processes may be valid at LHC. HC is sefl for checkin SUSY loop calclations. Codes for the ØdW and d Lχ + i helicity amplitdes at 1loop EW in MSSM, are available in FORTRANcodes. All anomalos coplins we know of, violate HC. d 27

28 Back p paes slides 28

29 We next trn to EW corrections to the W+jet and d Lχ + i prodction at LHC, sin the exact 1-loop reslts. 29

30 LHC prodction: W ± +jet The relevant sbprocesses are W W + dw, d W, d W + + dw, d W, d W + They are calclated by applyin crossin and CP to the ØdW + sqared amplitdes throh R s t 2 I (,, ) = Fλλλλ, d W λλ λλ W d + d ˆ( σ dw ) pt = I θ + dp 768 π s t T [ R R ] I π θ 30

31 + d ˆ( σ d W ) pt = [ R ] II θ + RII π θ, dp 768 π s t R II T R (, t, s), I + d ˆ( σ d W ) pt = [ R ] III θ + RIII π θ, dp 288 π s t R III = = T R (, t s, ), I 31

32 W+jet at LHC EW 1-loop corrections to the p T distribtion in W+jet prodction in SM and MSSM, at LHC. Infrared diverences are relarized by imposin m γ =m Z Infrared and QCD corrections mst be inclded in a real measrement. 32

33 D = SM + MSSM + dσ ( W )/ dpt dσ ( W )/ dp SM + dσ ( W )/ dp SM MSSM dσ ( W )/ dpt dσ ( W )/ dp SM dσ ( W )/ dp T T T T W+jet at LHC The SUSY redction, compared to the SM contribtions, is of the order of 10%. Similar effects may be expected also in models involvin extra ae bosons or extra lare dimensions 33

34 W+jet at LHC EW 1-loop corrections to the anlar distribtion in W+jet prodction in SM and MSSM, at LHC. 34

35 D an SM + MSSM + dσ ( W )/ dsdcos ϑ dσ ( W )/ dsdcosϑ = SM + dσ ( W )/ dsdcosϑ SM MSSM dσ ( W )/ dsdcos ϑ dσ ( W )/ dsdcosϑ SM dσ ( W )/ dsd cosϑ W+jet at LHC Aain the virtal SUSY effect is at the 10% level. 35

36 W+jet at LHC At s=4tev, where HC is larely established, the anlar dependence is stroner. At sch eneries, only the two HC amplitdes F -±-± are relevant. 36

37 d χ + L i at LHC + d ˆ( σ d Lχi ) β i = [ R ] I θ, dcosθ 3072πs + d ˆ( σ d Lχi ) pt = [ R ] I θ + RI π θ, dp 768 π s t T RI (,, s t ) = Fλµµ λµµ 2 37

38 d χ + L i at LHC EW 1loop corrections 38

39 d χ + L i at LHC EW 1loop corrections 39

40 d χ + L i at LHC EW 1loop corrections 40