Regularization of supersymmetric theories. New results on dimensional reduction

Transcription

1 Regularization of supersymmetric teories New results on dimensional reduction IPPP Duram Loopfest, June 2006

2 Outline Introduction 1 Introduction 2 Consistency of DRED 3 Factorization in DRED 4 Supersymmetry and M -calculations 5 Conclusions

3 Introduction Properties of DREG/DRED (status Jan. 2005) DREG: Dim. Regularization (DREG) D dimensions D Gluon/poton-components 4 Gluino/potino-components DRED: Dim. Reduction (DRED) D dimensions 4 Gluon/poton-components 4 Gluino/potino-components

4 Introduction Motivation: some important observables/calculations... H t, (g 2) b µ 1-Loop processes no problem wit regularization DRED preserves SUSY!! M t DRED SUSY-preserving?? LHC g q g q DRED violates factorization!?

5 Introduction Summary: Properties of DREG and DRED DREG: consistent SUSY-violation factorization + + DRED: consistent SUSY factorization + (+) (+)

6 Introduction Tere is no consistent SUSY regularization teoretical question: SUSY renormalizable? Anomalies? practical question: Wic sceme is best in practical computations?

7 Introduction Tere is no consistent SUSY regularization teoretical question: SUSY renormalizable? Anomalies? Renormalizability, no anomalies: proven indep. of reg. SUSY [Piguet, Sibold 1985] [Piguet et al], MSSM [Hollik, Kraus, Rot, Rupp, Sibold, DS 2002] practical question: Wic sceme is best in practical computations?

8 Introduction Tere is no consistent SUSY regularization teoretical question: SUSY renormalizable? Anomalies? Renormalizability, no anomalies: proven indep. of reg. SUSY [Piguet, Sibold 1985] [Piguet et al], MSSM [Hollik, Kraus, Rot, Rupp, Sibold, DS 2002] practical question: Wic sceme is best in practical computations? Tis talk

9 Outline Introduction 1 Introduction 2 Consistency of DRED 3 Factorization in DRED 4 Supersymmetry and M -calculations 5 Conclusions

10 Outline Consistency of DRED 1 Introduction 2 Consistency of DRED 3 Factorization in DRED 4 Supersymmetry and M -calculations 5 Conclusions

11 Consistency of DRED Were does te inconsistency come from? DREG: D-dimensional space can be consistently defined no inconsistency like 1=0: [Wilson 73],[Collins] DRED: in original form: problem 1 algebraic id.: g (4) µνg (D) ρ ν = g (D) µ ρ etc 2 4-dim id.: det (1)+(2) inconsistent, 1=0 g µ 1ν 1... g µ 1ν 5.. g µ 5ν 1... g µ 5ν 5 = 0, Fierz,...

12 Consistency of DRED Consistent DRED Idea: Use only algebraic id. (1) but no 4-dim id. (2) sould be consistent [Avdeev, Cocia, Vladimirov 1981] matematical construction of quantities satisfying (1) possible proof: DRED is matematically consistent if only (1) is used [DS 2005] Consequences in practice: algebraic id. of DRED as usual one cannot rely on index counting or Fierz identities for many SUSY loop calculations, tis doesn t make a difference

13 Consistency of DRED Quantum Action Principle in DRED Using te consistent formulation of DRED, one can prove te quantum action principle in DRED i δ SUSY T φ 1... φ n = T φ 1... φ n Useful to study symmetry-properties of regularizations Proof as to be carried out for eac regularization, BPHZ DREG DRED [Lowenstein et al 71] [Breitenloner, Maison 77] [DS 2005]

14 Outline Factorization in DRED 1 Introduction 2 Consistency of DRED 3 Factorization in DRED 4 Supersymmetry and M -calculations 5 Conclusions

15 Factorization in DRED Factorization-problem Problem: DRED, m 0 σ DRED (GG t tg) k 2 k 3 P g gg σ DRED (GG t t) + 1 k 2 k 3 K g σ puzzle [Beenakker, Kuijf, van Neerven, Smit 88] [van Neerven, Smit 04] [Beenakker, Höpker, Spira, Zerwas 96] One solution in practice (unsatisfactory complication): resort to DREG SUSY-restoring cts necessary Fundamental question: were does te seemingly non-factorizing term σ puzzle come from?

16 Factorization in DRED Factorization-problem Problem: DRED, m 0 σ DRED (GG t tg) k 2 k 3 P g gg σ DRED (GG t t) + 1 k 2 k 3 K g σ puzzle [Beenakker, Kuijf, van Neerven, Smit 88] [van Neerven, Smit 04] [Beenakker, Höpker, Spira, Zerwas 96] clue: mismatc! Dim. Reduction (DRED) D dimensions 4 Gluon/poton-components 4 Gluino/potino-components

17 DRED and te gluon Factorization in DRED 4-component Gluon in DRED G = D-component gauge field g + ɛ-scalars φ For polarization sums: g µν (4) = g µν (D) + g µν (ɛ) g and φ ave to be treated seperately!

18 DRED and te gluon Factorization in DRED 4-component Gluon in DRED G = D-component gauge field g + ɛ-scalars φ For polarization sums: g µν (4) = g µν (D) + g µν (ɛ) g and φ ave to be treated seperately! Simple kinematics: e.g. GG q q (massless) in general / ere: GG t t (massive) σ GG q q = σ Gg q q = σ Gφ q q σ GG q q σ Gg q q σ Gφ q q

19 Factorization in DRED Factorization result Main result: m reconciled DRED and factorization [Signer, DS 05] σ DRED (GG t tg) P G gg σ Gg + P G φg σ Gφ understood origin of non-factorizing term K g σ puzzle P φ gφ [σ Gg σ Gφ ]

20 Factorization in DRED Factorization result Main result: m reconciled DRED and factorization [Signer, DS 05] σ DRED (GG t tg) P G gg σ Gg + P G φg σ Gφ understood origin of non-factorizing term K g σ puzzle P φ gφ [σ Gg σ Gφ ] Practical consequences adron processes can be computed using DRED

21 Outline Supersymmetry and M -calculations 1 Introduction 2 Consistency of DRED 3 Factorization in DRED 4 Supersymmetry and M -calculations 5 Conclusions

22 Supersymmetry and M -calculations Symmetries of regularizations In principle, we don t ave to boter weter a regularization preserves symmetries

23 Supersymmetry and M -calculations Symmetries of regularizations In practice, life is easier wit a symmetry-preserving regularization!

24 Supersymmetry and M -calculations Symmetries of regularizations In practice, life is easier wit a symmetry-preserving regularization! counterterms Γ ct also preserve symmetries: g g + δg, m m + δm multiplicative renormalization most common situation, often assumed witout proof

25 Supersymmetry and M -calculations Problem: SUSY of DRED DRED preserves SUSY in simple cases Does DRED preserve SUSY in general? Or at least in cases tat are relevant in practice?

26 Supersymmetry and M -calculations DRED preserves SUSY Wat does it mean? SUSY ST-identities 0 = δ SUSY T φ 1... φ n ST-identities must be satisfied after renormalization DRED preserves SUSY if te ST-identities are already satisfied on te regularized level

27 Supersymmetry and M -calculations Quantum action principle as a tool Quantum action principle: STI δ SUSY T φ 1... φ n = 0 valid in DRED T φ 1... φ n = 0 = δ SUSY L Sample application: QCD-gauge invariance in DREG δ gauge L DREG QCD = = 0 δ gauge T φ 1... φ n = 0 DREG preserves all QCD Slavnov-Taylor identities at all orders

28 Supersymmetry and M -calculations Quantum action principle as a tool Quantum action principle: STI δ SUSY T φ 1... φ n = 0 valid in DRED T φ 1... φ n = 0 = δ SUSY L application ere: SUSY of DRED: δ SUSY L DRED = 0 gives rise to Feynman rules [DS 05] DRED probably does not preserve all SUSY-identities but cecking particular ST-identities is simplified using te Q.A.P.

29 Supersymmetry and M -calculations Higgs boson mass and quartic coupling Higgs mass M governed by quartic Higgs self coupling λ λ g 2 in SUSY

30 Supersymmetry and M -calculations Quartic coupling and SUSY Slavnov-Taylor identity H g W λ + finite H... H expresses λ g 2 If it is satisfied by DRED multiplicative renormalization o.k. Needs to be verified at 2-loop level 0? = δ SUSY H

31 Supersymmetry and M -calculations Quartic coupling and SUSY λ Metod: Use quantum action principle replace ST-identity by H = 0 W + H... H g finite H 0? = δ SUSY H H

32 Supersymmetry and M -calculations Quartic coupling and SUSY STI valid if λ H = 0 ɛ q H H g W + finite H... H q g, H q +... =0 Explicit computation STI valid in DRED at two-loop level [Hollik, DS 05]

33 Supersymmetry and M -calculations Quartic coupling and SUSY H g W λ + finite H... H Results: Two-loop STI valid in DRED (in Yukawa-approximation, O(α 2 t,b, α t,bα s )) for M -calculation at tis order, multiplicative renormalization correct Previous calculations sufficient Explicit computation STI valid in DRED at two-loop level [Hollik, DS 05]

34 Supersymmetry and M -calculations How muc do we know now? old: many SUSY identities cecked in DRED: 1-Loop Ward identities [Capper,Jones,van Nieuvenuizen 80] β-functions [Martin, Vaugn 93] [Jack, Jones, Nort 96] 1-Loop S-matrix relation 1-Loop Slavnov-Taylor identities new: furter 2-loop ST-identities Status: sufficient for one-loop SUSY processes [Beenakker,Höpker,Zerwas 96] [Hollik,Kraus,DS 99] [Hollik,DS 01] [Fiscer,Hollik,Rot,DS 03] [DS 05] [Hollik, DS 05] sufficient for two-loop Higgs masses and furter mass relations multiplicative renormalization o.k. no SUSY-restoring counterterms

35 Supersymmetry and M -calculations Summary: Properties of DREG and DRED DREG: consistent SUSY-violation factorization + + DRED: consistent SUSY factorization + (+) (+)

36 Outline Conclusions 1 Introduction 2 Consistency of DRED 3 Factorization in DRED 4 Supersymmetry and M -calculations 5 Conclusions

37 Summary & Outlook Conclusions Comparison of DREG and DRED: Factorization: olds in DREG and DRED, sligtly more complicated in DRED due to different partons g, φ streamlined prescription for adron processes in DRED? Consistency, quantum action principle: ok in DREG and DRED SUSY: DREG breaks SUSY already in simplest cases, DRED preserves SUSY in many cases up to 2-Loop, but not at all orders furter cecks of e.g. RG-running at 3-Loops?