Renormalisation of Unstable Particles in Quantum Field Theories

Home , Pole mass

Renormalisation of Georg Weiglein, DESY unstable particles in W. Zimmermann Memorial Symposium, quantum ﬁeld theories MPI Munich, 05 / 2017 Introduction: the Quantum Universe

Particle Astronomy Physics Experiments Experiments Telescopes Accelerators Satellites Underground

Quantum Field Standard Theory Cosmology (Standard Model Model)

10–18 m 1026 m

Renormalisation of unstable particles in quantum ﬁeld theories, Georg Weiglein, W. Zimmermann Memorial Symposium, MPI Munich, 05 / 2017 2

What is the quantum structure of the vacuum?

• The recent discovery of a Higgs boson hints at a non-trivial structure of the vacuum, i.e. of the lowest-energy state in our universeHiggs physics at Linear Colliders

• The discovered particle providesHiggs accessphysics at ILC K. Desch - Higgs physicsto at ILC studying2 the quantum structure of the vacuum!

• How can a Higgs boson be as light as 125 GeV?

• A new symmetry of nature ⟶ Supersymmetry?

• A new fundamental interaction of nature ⟶ composite Higgs?

• Extra dimensions of space ⟶ impact on gravity on small scales?

• Multiverses ⟶ anthropic principle? Renormalisation of unstable particles in quantum ﬁeld theories, Georg Weiglein, W. Zimmermann Memorial Symposium, MPI Munich, 05 / 2017 3 DescriptionDescription of fundamental of fundamental interactions interactions: via quantum ﬁeldQuantum theories Field Theory (QFT)

internal particles

∆(p)

externalparticles externalparticles

External particles: “on-shell”, fulﬁll relativistic relation between energy, 3-momentum and mass of a free particle 2 µ 2 2 2 p p pµ = E p⃗ = m [units: ! = c =1] ≡ − | |

Internal particles: off-shell, p2 = m2,describedbypropagator ̸ i ∆(p) (in lowest order) ∼ p2 m2 Renormalisation of unstable particlesThe in quantum concept of massfield− intheories, particle physics, Georg Georg Weiglein, Weiglein, W. SymZimmermannposium Begriff Memorial der Masse, Symposium, DPG Frühjahrstagung MPI Munich, 2013, Jena, 05 02/ 201 / 20137 – p.54 Perturbative evaluation of quantum field theories (gaugePerturbative theories) evaluation of quantum field theories

Expansion in coupling constant: α 1 1 ≈ 137 ≪ expansion about theory without interaction ⇔

lowest order, 1 α 137 classical limit ≈ √α √α quantum corrections: loop diagrams √α √α √α √α (α) relative to lowest order O Renormalisation of unstable particles in quantum ﬁeld theories, Georg Weiglein, W.Lectures Zimmermann on SM and Memorial SUSY Phenomenology, Symposium, Georg MPI Weiglein, Munich, Prag 05ue, / 09/2007 2017 – p.385 WhatWhat can can one one learn learn from from quantum quantum corrections corrections? ?

Inclusion of quantum effects more accurate theoretical predictions ⇔ Large loop corrections: QCD corrections are often of (100%) O 2 4 EW enhancement factors: mt , mt ,..., large logarithms (involving two very different scales) Per mille level corrections needed to match EW precision measurements

Quantum effects provide sensitivity to the underlying structure of the theory Renormalisation of unstable particles in quantum ﬁeld theories, Georg Weiglein, W.Lectures Zimmermann on SM and Memorial SUSY Phenomenology, Symposium, Georg MPI Weiglein, Munich, Prag 05ue, / 09/20072017 – p.396 Comparison of electroweak precision data with Electroweak precision physics: high-precision data vs. theory predictionstheory predictions

EW precision data: Theory: 2 lept MZ,MW, sin θeﬀ ,... SM, MSSM, . . .

Test of theory at quantum level:⇓ sensitivity to loop corrections

Indirect constraints on unknown⇓ parameters: MH,...

Effects of “new physics”? Lectures on SM and SUSY Phenomenology, Georg Weiglein, Prague, 09/2007 – p.111 Renormalisation of unstable particles in quantum ﬁeld theories, Georg Weiglein, W. Zimmermann Memorial Symposium, MPI Munich, 05 / 2017 7 High-precisionPower physics of precision

1978 Precise measurement of sinθW @ SLAC via polarized electrons eD eX  Prediction of W and Z mass

1983 Discovery of W and Z bosons at SppS

1989- Precise measurement of W and Z @ SLC/LEP  Prediction of top mass

1995 Discovery of top quark at Tevatron

Precise measurement of W, Z, top @ SLC/ LEP/Tevatron  Prediction of Higgs mass

2012 Discovery of Higgs boson

Precise measurement of W, Z, top, Higgs @ LHC/ILC  Prediction of ??? Planck Renormalisation of unstable particles in quantum ﬁeld theories, Georg Weiglein, W. Zimmermann Memorial Symposium, MPI Munich, 05 / 2017 8 2013-10-14 Higgs Couplings 2013 “Prospects for measuring Higgs boson couplings at the ILC" (T. Tanabe) Theoretical foundation: renormalisability of gauge theories withGauge-invariant spontaneous Lagrangian symmetry of breaking the SM Standard Model Lagrangian as an example: (g ,g ,v, λ ,g)+ (α ) LEW 2 1 f LQCD s

MW, MZ, α , MH, mf ! "# $ !"#$ !"#$

Gauge invariance theory is renormalisable ⇒ [G. ’t Hooft ’71][G. ’t Hooft, M. Veltman ’72] Nobel prize ’99

theory can consistently been treated as a quantised ⇒ field theory: quantum effects can be evaluated ⇒ For non-renormalisable theory: need additional parameters in each loop order to compensate divergenciesLectures on SM and SUSY Phenomenology, Georg Weiglein, Prague, 09/2007 – p.37 Renormalisation of unstable particles in quantum field theories, Georg Weiglein, W. Zimmermann Memorial Symposium, MPI Munich, 05 / 2017 9 QuantumQuantum corrections: corrections: regularisation regularisation and and renormalisation (withrenormalisation a broad brush) q pp 1 d4q ∼ (q2 m2 + iε) (q + k)2 m2 + iε ! − 1 − 2 q + p " # 3 ∞ q dq ∞ dq q : = →∞ ∼ q4 q →∞ !0 !0 integral diverges for large q! ⇒ theory in this form not physically meaningful ⇒ further concept needed: renormalisation ⇒ Renormalisable theories: infinities can consistently be absorbed into parameters of theory Lectures on SM and SUSY Phenomenology, Georg Weiglein, Prague, 09/2007 – p.40

Renormalisation of unstable particles in quantum ﬁeld theories, Georg Weiglein, W. Zimmermann Memorial Symposium, MPI Munich, 05 / 2017 10 Two step procedure:Two step procedure: regularisation regularisation

Regularisation: theory modiﬁed such that expressions become mathematically meaningful “regulator” introduced, removed at the end ⇒ e.g. cut-off in loop integral

Λ ∞ d4q d4q; Λ at the end → →∞ !0 !0 technically more convenient: dimensional regularisation

d4q dDq, D =4 ε; D 4 at the end → − → ! ! Lectures on SM and SUSY Phenomenology, Georg Weiglein, Prague, 09/2007 – p.41 Renormalisation of unstable particles in quantum ﬁeld theories, Georg Weiglein, W. Zimmermann Memorial Symposium, MPI Munich, 05 / 2017 11 Two step Twoprocedure: step procedure: renormalisationrenormalisation

Renormalisation: original “bare” parameters replaced by renormalised parameters + counterterms reparametrization: g0 = g + δg

bare renormalised counterterm parameter!"#$ parameter!"#$ !"#$

Renormalisable theory: divergences compensated by counterterms

Renormalisation of unstable particles in quantum ﬁeld theories, Georg Weiglein, W. ZimmermannLectures on SM Memorial and SUSY Symposium, Phenomenology, MPI Georg Munich, Weiglein, 05 / Prag 201ue,7 09/200712 – p.42 Two aspects of renormalisation: Two aspects of renormalisation

Absorption of divergences

Determination of physical meaning of parameters order by order in perturbation theory

Lectures on SM and SUSY Phenomenology, Georg Weiglein, Prague, 09/2007 – p.43

Renormalisation of unstable particles in quantum ﬁeld theories, Georg Weiglein, W. Zimmermann Memorial Symposium, MPI Munich, 05 / 2017 13 Higher-orderHigher-order contributions contributions to the to propagator the propagator

+ + + ···

i i 2 i i 2 i 2 i p2−m2 p2−m2 iΣ(p ) p2−m2 p2−m2 iΣ(p ) p2−m2 iΣ(p ) p2−m2

i = p2 m2 + Σ(p2) − Renormalised self-energy Pole of the propagator: 2 (denoted below by ⌃ˆ ) ⇒ M 2 m2 + Σ( 2)=0 M − M For a stable particle: Σ( 2) is real M If Σ( 2) =0 Pole shifted by higher-order contributions RenormalisationM ̸ of unstable⇒ particles inThe quantum concept ﬁeld of mass theories, in particle Georg physics, Weiglein, Georg Weiglein, W. Zimmermann Symposium Begriff Memorial der Masse, Symposium, DPG Fr¨uhjahrstagung MPI Munich, 2013, 05 Jena, / 20102 /7 201314 – p.7 Example: mass renormalisation (stable particle) Example: mass renormalisation

2 2 2 Renormalisation of the mass parameter: m0 = m + δm Physical mass: pole of propagator

inverse propagator up to 1-loop order:

+ + x + ···

2 2 2 2 2 2 p m Σ(p ) m +(pδm2m )Z − −

2 2 2 2 2 2 Pole of the propagator: p m + Σ(p ) δmm +(=0p m )Z =0 − − Lectures on SM and SUSY Phenomenology, Georg Weiglein, Prague, 09/2007 – p.46 Renormalisation of unstable particles in quantum ﬁeld theories, Georg Weiglein, W. Zimmermann Memorial Symposium, MPI Munich, 05 / 2017 15 On-shell andOn-shell MS renormalisation and MS renormalisation

2 2 “On-shell renormalisation”: δm = Σ(p ) p2=m2 pole of propagator for p2 = m2 m:“polemass”! ⇒ ⇒ ! Counterterm contains divergence: expansion in ε 4 D,1-loop: ≡ − 1 δm2 = a + b ε0 + c ε + ... ε

divergent ﬁnite"#$% "#$%0 "#$% → for D 4 for D 4 → →

Other renormalisation prescription: 1 δm2 = a “minimal subtraction” (MS) ε Lectures on SM and SUSY Phenomenology, Georg Weiglein, Prague, 09/2007 – p.47 Renormalisation of unstable particles in quantum ﬁeld theories, Georg Weiglein, W. Zimmermann Memorial Symposium, MPI Munich, 05 / 2017 16 On-shell and MS renormalisation

Slight variant of MS renormalisation: ``modiﬁed minimal subtraction’’, MS

MS and MS quantities depend on renormalisation scale μ (needs to be introduced for dimensional reasons)

MS top quark mass: mt (μ), ``running mass’’

The diﬀerence between the pole mass and mt (mt ) from QCD corrections amounts to about 10 GeV!

The strong coupling is usually given as MS quantity: ↵ s ( µ ) , ``running coupling’’

Renormalisation of unstable particles in quantum ﬁeld theories, Georg Weiglein, W. Zimmermann Memorial Symposium, MPI Munich, 05 / 2017 17 Ground-breaking contributions

IL NUOVO CIMENTO VOL. l, N. 1 1° Gennaio 1955

Zur Formulierung quantisierter Feldtheorien.

H. LEHMANN, K. SYMA~ZIK und ~¥. ZI:~[MERMA~N

Ma.r- t)la~tc]~: - l~stitut ]iir P/~ysik - (:iitti~l~le~ (De~d~'chland)

(ricevuto il 22 Novembre 1954)

Summary. - A new formulation of quantized field theories is proposed. Starting from some general requiremertts we derive a set of equations which determine the matrix-elements of field operators and the S-Matrix. These equations contain no renormalization constants, but only experimental masses and coupling parameters. The main advantage over the conventional formulation is thus the elimina.tion of all divergent terms in the basic equations. This means that no renormalization problem arises. The formulation is here restricted to theories which do not involve ,*table bound states. For simplicity we derive the equations for spin 0 particles, however the extension to other cases (e.,o.. quantum electro- dynamics) is obvious. The solutions of the equations are discussed in a power-series expansion. They are then identical with the renormalized expressions of the conventional formulation. However, the equations set up here are not restricted to the application of perturbation theory.

Renormalisation of unstable particles in quantum ﬁeld theories, Georg Weiglein, W. Zimmermann Memorial Symposium, MPI Munich, 05 / 2017 18

1. - Einleitung.

Um die Problemstellung dieser Arbeit zu verdeutlichen, sei zun~chst auf einige unbefriedigende Zfige der gebr~iuchlichen Formulierung yon Feldtheorien hingewiesen: 1) Wiihrend es m6gli(.h ist, fiir die sog. renormierbt~ren Theorien im Rahmen einer Entwicklung nach Kopplungsparametern endliche Resultate fill" die Mutrixelemente der (renormierten) Feldoperatoren und der S-Matrix zu erhalten, ist es bisher nicht gelungen, die Grundgleichungen derartiger Theorien konvergent zu formnlieren. Wit meinen damit die Tatsache, dal~ sowohl in den Feldgleichungen als aueh in denVertauschungsrelationen der renormierten Operatoren divergente Renormierungskonstanten auftreten. IL NUOV0 CIMENT0 VoL. VI, N. 2 lO Agosto 1957

Ground-breaking contributions

On the Formulation of Quantized Field Theories - II.

H. LEHMAN~ and K. SY1VIAI~ZIK Institut ]i~r Theoretische Physik dev Universitdt - Hamburg

W. ZI~IMER~A~ 2gax-Planck.Institut ]i~r Physik - GSttinge~

(ricevuto il 9 Maggio 1957)

Summary. -- We discuss the concept of causal scattering matrices using retarded multiple commutators of field operators.

1. - ScatteringRenormalisation matrix of unstable andparticles fieldin quantum operators. ﬁeld theories, Georg Weiglein, W. Zimmermann Memorial Symposium, MPI Munich, 05 / 2017 19

One of the main problems of relativistic quantum theory is to obtain in- formation about scattering matrix elements. To this end it is customary to start from a quantized field theory; i.e. operator fields which obey certain equations and c~ommutation relations. On the other hand one can try -- following Heisenberg's original suggestion -- to base a theory of interacting particles entirely on requirements imposed on the scattering matrix. In what follows we intend to discuss the connection between such an S-matrix formalism and conventional quantized field theories. Our aim is

a) to clarify some concepts employed in recent work; in particular to define rather precisely what is usually considered a (( causal ~ scattering matrix;

b) to investigate some expressions which seem to be useful if one wants to derive causality conditions for a scattering matrix.

We begin with a pure S-matrix formalism (1). No interacting field operators are considered but only an operator S which describes the scattering,

(1) W. HEISENBERG: Zeits. ]. Phys., 120, 513, 673 (1943). BPHZ renormalisation and LSZ formalism Bogoliubov-Parasiuk-Hepp-Zimmermann renormalization scheme - Scholarpedia 16/05/17 02:38

Bogoliubov-Parasiuk-Hepp-Zimmermann renormalization schemeAuf Google empfehlen Klaus Sibold (2010), Scholarpedia, 5(5):7306. doi:10.4249/scholarpedia.7306.5< TGFWEVKQPrevision #137544 [link to/cite this article] HQTOWNC Dr. Klaus Sibold, Institut für Theoretische Physik Fakultät für Physik und Geowissenschaften Universität Leipzig

The Bogoliubov, Parasiuk, Hepp, Zimmermann (abbreviated BPHZ) renormalization scheme is a mathematically consistent method of rendering Feynman amplitudes finite while maintaining the fundamental postulates of relativistic quantum field theory (Lorentz invariance, unitarity, causality). Technically it is based on the systematic subtraction of momentum space integrals. This distinguishes it from other methods of renormalization. For massless particles the scheme has been enlarged by Lowenstein and is then called BPHZL.

Contents LSZ reduction formula 1 The problem+P SWCPVWO ិGNF VJGQT[ VJG KU 2 Diagrammatics 3 ApplicationC OGVJQF VQ ECNEWNCVG 5OCVTKZ GNGOGPVU VJG UECVVGTKPI 1 µ 1 2 2 4 General remarks = ∂ ϕ∂ ϕ m ϕ + 5 ReferencesCORNKVWFGU HTQO VJG VKOGQTFGTGF EQTTGNCVKQP HWPEVKQPU µ 0 KPV 6 Further Reading L 2 − 2 L 7 See alsoQH C SWCPVWO ិGNF VJGQT[ +V KU C UVGR QH VJG RCVJ VJCV UVCTVU HTQO VJG .CITCPIKCP QH UQOG SWCPVWO ិGNF VJGQT[ CPF Hͳ3 The problem KPV OC[ EQPVCKP C UGNH KPVGTCEVKQP QT KPVGTCEVKQP YKVJ NGCFU VQ RTGFKEVKQP QH OGCUWTCDNG SWCPVKVKGU +V KU PCOGF L g ϕψψ¯ For elucidating the problem let us have a look at an intuitive representation of processes involving particles at the subatomic level. ElementaryQVJGT ិGNFU NKMG C ;WMCYC KPVGTCEVKQP (TQO VJKU particles likeCHVGT electrons, VJG quarks, VJTGG photons and )GTOCP gluons interact with RJ[UKEKUVU each other: in scattering*CTT[ processes incoming .GJOCPP particles collide -WTV and give rise to outgoing particles, the transition from such an initial state to a final state obeying the rules of quantum mechanics. Pictorially this is described.CITCPIKCP in WUKPI 'WNGTཊ.CITCPIG GSWCVKQPU VJG GSWC terms of Feynman5[OCP\KM diagrams. CPF 9QNHJCTV

Figure 4: External lines: physical fermion and 9G OC[ GZRGEV VJG JO ិGNF VQ TGUGODNG VJG CU[ORVQVKE physical antifermion. 0 Figure 3: Propagator: virtual photon. DGJCXKQWT QH VJG HTGG ិGNF CU Y ႂ ᄂᄎ OCMKPI VJG CU 1 In and out ﬁelds UWORVKQP VJCV KP VJG HCT CYC[ RCUV KPVGTCEVKQP FGUETKDGF D[ VJG EWTTGPV K0 KU PGINKIKDNG CU RCTVKENGU CTG HCT HTQO GCEJ QVJGT 6JKU J[RQVJGUKU KU PCOGF VJG CFKCDCVKE J[ http://www.scholarpedia.org/article/Bogoliubov-Parasiuk-Hepp-Zimmermann_renormalization_scheme5OCVTKZ GNGOGPVU CTG CORNKVWFGU QH VTCPUKVKQPU DGVYGGP Page 1 of 9 JO PVU JO p RQVJGUKU *QYGXGT UGNH KPVGTCEVKQP PGXGT HCFGU CYC[ CPF UVCVGU CPF UVCVGU #P UVCVG |{ } KP⟩ FGUETKDGU VJG UVCVG QH C U[UVGO QH RCTVKENGU YJKEJ KP C HCT CYC[ RCUV DGUKFGU OCP[ QVJGT GាGEVU KV ECWUGU C FKាGTGPEG DGVYGGP N DGHQTG KPVGTCEVKPI YGTG OQXKPI HTGGN[ YKVJ FGិPKVG OQ VJG .CITCPIKCP OCUU 0 CPF VJG RJ[UKECN OCUU O QH VJG ͽ Q PVU p DQUQP 6JKU HCEV OWUV DG VCMGP KPVQ CEEQWPV D[ TGYTKVKPI OGPVC ] _ CPF EQPXGTUGN[ CP UVCVG |{ } QWV⟩ FG UETKDGU VJG UVCVG QH C U[UVGO QH RCTVKENGU YJKEJ NQPI CHVGT VJG GSWCVKQP QH OQVKQP CU HQNNQYU KPVGTCEVKQP YKNN DG OQXKPI HTGGN[ YKVJ FGិPKVG OQOGPVC ]Q_ 2 2 2 2 ∂ + m ϕ(x)=j0(x)+ m m ϕ(x)=j(x) *O CPF PVU UVCVGU CTG UVCVGU KP *GKUGPDGTI RKEVWTG UQ VJG[ − 0 UJQWNF PQV DG VJQWIJV VQ FGUETKDG RCTVKENGU CV C FGិPKVG ! " ! " 6JKU GSWCVKQP ECP DG UQNXGF HQTOCNN[ WUKPI VJG TGVCTFGF VKOG DWV TCVJGT VQ FGUETKDG VJG U[UVGO QH RCTVKENGU KP KVU 2 2 )TGGPཐU HWPEVKQP QH VJG -NGKPཊ)QTFQP QRGTCVQT ∂ + m GPVKTG GXQNWVKQP UQ VJCV VJG 5OCVTKZ GNGOGPV

S = q p 3 fi QWV KP 0 F k ik x ik x 2 2 ⟨{ } |{ } ⟩ ∆ (x)=iθ x e− · e · 0 ωk = k + m TGV (2π)32ω − k =ωk # k KU VJG RTQDCDKNKV[ CORNKVWFG HQT C UGV QH RCTVKENGU YJKEJ ! " ! " $ YGTG RTGRCTGF YKVJ FGិPKVG OQOGPVC ]Q_ VQ KPVGTCEV CPF CNNQYKPI WU VQ URNKV KPVGTCEVKQP HTQO CU[ORVQVKE DG DG OGCUWTGF NCVGT CU C PGY UGV QH RCTVKENGU YKVJ OQOGPVC JCXKQWT 6JG UQNWVKQP KU ]R_ 6JG GCU[ YC[ VQ DWKNF JO CPF PVU UVCVGU KU VQ UGGM CRRTQRTK ϕ(x)=√Zϕ (x)+ 4y∆ (x y)j(y) CVG ិGNF QRGTCVQTU VJCV RTQXKFG VJG TKIJV ETGCVKQP CPF CP KP F TGV − PKJKNCVKQP QRGTCVQTU 6JGUG ិGNFU CTG ECNNGF TGURGEVKXGN[ # JO PVU CPF ិGNFU 6JG HCEVQT ᄊ; KU C PQTOCNK\CVKQP HCEVQT VJCV YKNN EQOG ,WUV VQ ិZ KFGCU UWRRQUG YG FGCN YKVJ C -NGKPཊ)QTFQP JCPF[ NCVGT VJG ិGNF ͳ೏⛼ KU C UQNWVKQP QH VJG JQOQIGPGQWU ិGNF VJCV KPVGTCEVU KP UQOG YC[ YJKEJ FQGUP V EQPEGTP WU GSWCVKQP CUUQEKCVGF YKVJ VJG GSWCVKQP QH OQVKQP

And some interesting predictions

Nuclear Physics B259 (1985) 331-350 © North-Holland Publishing Company

HIGGS AND TOP MASS FROM REDUCTION OF COUPLINGS

J. KUBO*, K. SIBOLD** and W. ZIMMERMANN Max-Planck-lnstitut fiir Physik und Astrophysik Werner-Heisenberg-[nstitut fiir Physik F6hringer Ring 6, 8000 Mfinchen 40, Federal Republic of Germany

Received 15 April 1985

We reduce the couplings in the standard model with one Higgs doublet and n generations and obtain for three generations 61 GeV and 81 GeV for the mass of the Higgs particle and the top quark respectively. The error is estimated to be about 10-15%.

Did not quite work for the absolute mass values, but the ratio is remarkably close! Renormalisation of unstable particles in quantum I.ﬁeld Introduction theories, Georg Weiglein, W. Zimmermann Memorial Symposium, MPI Munich, 05 / 2017 21

The standard model for the electroweak and strong interactions is phenomenologically very successful, the price for this success being the relatively large number of free parameters. In the gauge field sector there are the three gauge couplings associated with SU(3), SU(2) and U(1) respectively, in the matter sector there are Higgs and Yukawa couplings [1, 2]. The main aim of model building is to reduce this number of free parameters and not to lose the good agreement with experiment. Grand unified theories permit to replace the three gauge couplings by one, but do not substantially help in the matter sector [3]. Supersymmetric theories may relate Higgs and Yukawa couplings (even to gauge couplings), but they introduce many new particles which are not yet observed [4]. Composite model building, as the third possibility, has not yet led to a realistic alternative [5]. In a less ambitious attempt one may therefore look for other relations amongst couplings within a given model. Estimates of the masses of heavy quarks and/or higgs(es) fall into this category since within a class of models the values of the masses reflect the values of the couplings via the Higgs effect. The main idea put forward thus far is the use of renormalization group equations. In [6] e.g. one exploits a fixed point structure of the Yukawa couplings in first order perturbation theory, in [7] one argues with consistency limits for effective couplings again in first order. Since the reliability of first order approximations is doubtful we apply in the present paper a concept which leads to

* Address after September 1, 1985: Institute of Theoretical Physics, SUNY at Stony Brook, USA. ** Heisenberg Fellow.

331 Asymptotic in- and out-states

QFT is formulated in terms of asymptotic in- and -out states for t → ± ∞: stable particles

LSZ formalism: correct normalisation of S-matrix elements; external particles on-shell + wave-function normalisation factors

Unstable particles:

• Cannot be expressed in terms of asymptotic states

• Decay of an unstable particle: non-zero total width

Renormalisation of unstable particles in quantum ﬁeld theories, Georg Weiglein, W. Zimmermann Memorial Symposium, MPI Munich, 05 / 2017 22 The problem of unstable particles in QFTs

LHC: production of Higgs, top, W, Z, … ; precise predictions needed for production and decay of unstable particles

One could in principle treat the full process of production and decay and have only stable particles (e, �, g, u, d, …) as external states: technically inconvenient, diﬃcult to obtain higher-order corrections for 2 → n process if n is large

But: how to treat the resonance region? Lowest-order propagator would diverge on-shell, p2 → m2 i for p2 m2 p2 m2 !1 ! Need to introduce width of the unstable particle into the propagator, mixes orders of perturbation theory Renormalisation of unstable particles in quantum ﬁeld theories, Georg Weiglein, W. Zimmermann Memorial Symposium, MPI Munich, 05 / 2017 23 Treatment of unstable particles

Very important for practical applications

Long-standing problem (gauge invariance, unitarity, …), still not rigorously solved in QFTs

Some examples:

• Mass and ﬁeld renormalisation of unstable particles

• Wave function normalisation factors for unstable particles

• Resonance and interference eﬀects Renormalisation of unstable particles in quantum ﬁeld theories, Georg Weiglein, W. Zimmermann Memorial Symposium, MPI Munich, 05 / 2017 24 What isWhat the mass is the of mass an unstable of an unstable particle? particle?

Particle masses are not directly physical observables Can only measure cross sections, branching ratios, kinematical distributions, . . . masses are “pseudo-observables” ⇒

Need to deﬁne what is meant by MZ, MW, mt,...: MS mass, pole mass (real pole, real part of complex pole, Breit–Wigner shape with running or constant width), . . .

Determination of MZ, MW, mt,...involvesdeconvolution ⇒ procedure (unfolding) Mass obtained from comparison data – Monte Carlo

MZ, MW, mt,...arenotstrictlymodel-independent ⇒ Renormalisation of unstable particles in quantum ﬁeld theories, Georg Weiglein, W. ZimmermannLectures on SM andMemorial SUSY Phenomenology, Symposium, GeorgMPI Munich, Weiglein, 05 Prag /ue, 201 09/20077 25 – p.53 Mass of an unstable (elementary) particle Mass of an unstable (elementary) particle For an unstable particle: Σ( 2) is complex Pole in the complex plane M ⇒ 2 m2 + Σ( 2)=0, 2 = M 2 iMΓ M − M M − M:physicalmass, Γ:decaywidthoftheunstableparticle The mass of an unstable (elementary) particle is deﬁned ⇒ according to the real part of the complex pole

10 5 Z

Example: 10 4 e+e−→hadrons Cross-section (pb) resonant production 10 3

2 CESR + - 10 DORIS W W of the Z boson and its decay PEP PETRA TRISTAN KEKB SLC PEP-II (point-like particle!) 10 LEP I LEP II 0 20 40 60 80 100 120 140 160 180 200 220 Centre-of-mass energy (GeV) Renormalisation of unstable particles in quantumThe concept field of mass theories, in particle Georg physics, Weiglein, Georg Weiglein, W. Zimmermann Symposium Begriff Memorial der Masse, Symposium, DPG Frühjahrstagung MPI Munich, 2013, 05 Jena, / 20102 /7 201326 – p.8 Expansion around the complex pole for a single resonance d ⌃ˆ p2 m2 + ⌃ˆ(p2)= p2 2 1+ + ... M dp2 ( ) 2 2 p = M ⟶ Breit-Wigner factor with| fixed{z width} ⟶ Field renormalisation | and wave{z function} normalisation factor Note: of unstable particle

Wave-function normalisation factor needs to be evaluated at the complex pole @⌃(p2) One-loop ﬁeld renormalisation: Z (1) = @p2 Complex quantity, no restriction to Re(…) 2 2 p =m Renormalisation of unstable particles in quantum ﬁeld theories, Georg Weiglein, W. Zimmermann Memorial Symposium, MPI Munich, 05 / 2017 27 Expansion around the complex pole (example: MZ) Renormalisation of MW, MZ at the two-loop level

Expansion of amplitude around complex pole: R (e+e− ff¯)= + S +(s 2 ) S′ + A → s 2 − MZ ··· − MZ 2 2 = M iM Γ MZ Z − Z Z Expanding up to (α2) using (Γ /M )= (α) O O Z Z O From 2-loop order on: real part of complex pole, M = pole of real part, M 2 Z ̸ Z 2 2 ′ 2 2 δM (2) = δM(2) +Im ΣT,(1)(M ) Im ΣT,(1)!(M ) " # " # ! gauge-parameter dependent! Renormalisation of unstable particles in quantum ﬁeld$ theories, Georg Weiglein, W. ZimmermannLectures%& on SM Memorial and SUSY Symposium, Phenomenology, MPI Georg Munich, Weiglein,' 05 Prag / 201ue,7 09/200728 – p.55 Physical mass of unstable particles: real part of complexRenormalisation pole of MW, MZ at the two-loop level

Only the complex pole is gauge-invariant ⇒

Expansion around the complex pole leads to a Breit–Wigner shape with constant width

For historical reasons, the experimental values of MZ, MW are deﬁned according to a Breit–Wigner shape with running width

Need to correct for the difference in deﬁnition when ⇒ comparing theory with experiment

Lectures on SM and SUSY Phenomenology, Georg Weiglein, Prague, 09/2007 – p.56 Renormalisation of unstable particles in quantum ﬁeld theories, Georg Weiglein, W. Zimmermann Memorial Symposium, MPI Munich, 05 / 2017 29 Masses and wave function normalisation factors for mixed states Extended Higgs sectors: several Higgs states which can mix with each other

Higgs-mass prediction from propagator matrix

Example:

MSSM with complex parameters; two Higgs doublets, three neutral states

Lowest order: h, H (CP-even) and A (CP-odd) CP violation induced by loop corrections: Mixing of h, H, A ⟶ mass eigenstates h1, h2, h3

Renormalisation of unstable particles in quantum ﬁeld theories, Georg Weiglein, W. Zimmermann Memorial Symposium, MPI Munich, 05 / 2017 30 Higgs physics in the MSSM with Higgs physics in thecomplex MSSM parameters with complex parameters

Five physical states; tree level: h0,H0,A0,H±

Complex parameters enter via (often large) loop corrections: − µ: Higgsino mass parameter

− At,b,τ :trilinearcouplings

− M1,2: gaugino mass parameter (one phase can be eliminated)

− M3: gluino mass m˜g + complex phase

⇒ CP-violating mixing between neutral Higgs bosons h1, h2, h3

Lowest-order Higgs sector has two free parameters

v2 ⇒ choose tan β ≡ , M ± as input parameters v1 H Renormalisation of unstable particles in quantum ﬁeld theories, Georg Weiglein,Beyond W. the Zimmermann Standard Model Memorial (Higgs), Georg Symposium, Weiglein, IMFP13, MPI Munich, Santander, 05 05 / /201 20137 – p.31 40 CP-violating Higgs phenomenology: mixing between allHiggs three propagator-typeneutral Higgs bosons corrections Mixing between h, H, A loop-corrected masses obtained from propagator matrix ⇒ 1 ∆ (p2)= Γˆ (p2) − , Γˆ (p2)=i p21l M (p2) hHA − hHA hHA − n ! " # $ where (up to sub-leading two-loop corrections)

2 ˆ 2 ˆ 2 ˆ 2 mh Σhh(p ) ΣhH (p ) ΣhA(p ) 2 − 2 2− 2 − 2 Mn(p )= Σˆ (p ) m Σˆ (p ) Σˆ (p ) ⎛ − hH H − HH − HA ⎞ Σˆ (p2) Σˆ (p2) m2 Σˆ (p2) ⎜ − hA − HA A − AA ⎟ ⎝ ⎠

2 i Higgs propagators: ∆ii(p )= ⇒ 2 2 ˆ eﬀ 2 p mi + Σii (p ) Renormalisation of unstable particles in quantum ﬁeldSUSY theories, Higgs Production Georg Weiglein, and Decays W. at Zimmermann the LHC, Georg Weiglein,−MemorialHiggs Symposium, Days at Santander MPI Munich, 2010, Santander, 05 / 201 10 /7 201032 – p.12 DeterminationDetermination of the Higgs of masses the masses from the from complex poles the complex poles

ˆ 2 ˆ 2 ˆ 2 ˆ2 2 ˆ 2 ˆ2 2 ˆ 2 eﬀ 2 2 2Γij (p )Γjk(p )Γki(p ) − Γ (p )Γjj(p ) − Γij (p )Γkk(p ) Σˆ (p )=Σˆ (p ) − i ki ii ii ˆ 2 ˆ 2 ˆ2 2 Γjj(p )Γkk(p ) − Γjk(p )

Complex pole 2 of each propagator is determined from M 2 m2 + Σˆ eﬀ ( 2)=0, Mi − i ii Mi where 2 = M 2 iMΓ, M − Expansion around the real part of the complex pole:

2 2 2 2 Σˆ ( ) Σˆ (M )+i Im Σˆ ′ (M ) jk Mha ≈ jk ha Mha jk ha j, k = h, H, A, a =1, 2, 3 ! " Renormalisation of unstable particles in quantum ﬁeldSUSY theories, Higgs Georg Production Weiglein, and Decays W. Zimmermann at the LHC, Georg Memorial Weiglein, HiggsSymposium, Days at Santander MPI Munich, 2010, Santander,05 / 2017 10 /33 2010 – p.13 so that only the two -even states h and H would mix with each other. Our treatment corresponds to the caseCP where non-zero phases from complex parameters are taken into 2 account. Hence all renormalised self-energies ⌃ˆ ij(p ) of the Higgs bosons i, j = h, H, A are in general non-vanishing, so that the matrix M of mass squares consists of the tree-level 2 masses mi on the diagonal and renormalised self-energies on the diagonal and o↵-diagonal entries. Expressed in terms of the lowest-order mass eigenstates h, H, A, which are also eigenstates, the matrix takes the form CP 2 ˆ 2 ˆ 2 ˆ 2 mh ⌃hh(p ) ⌃hH (p ) ⌃hA(p ) 2 ˆ 2 2 ˆ 2 ˆ 2 M(p )= ⌃Hh(p ) mH ⌃HH(p ) ⌃HA(p ) . (1) 0 ˆ 2 ˆ 2 2 ˆ 2 1 ⌃Ah(p ) ⌃AH (p ) mA ⌃AA(p ) @ A The renormalised irreducible 2-point vertex functions

ˆ (p2)=i (p2 m2) + ⌃ˆ (p2) (2) ij i ij ij h i can be collected in the 3 3matrixˆ in terms of M as ⇥ hHA ˆ (p2)=i p21 M(p2) . (3) hHA ⇥ ⇤ Finally, the propagator matrix hHA equals, up to the sign, the inverse of ˆ hHA,

1 (p2)= ˆ (p2) . (4) hHA hHA h i 2 Accordingly, the matrix inversion yields the individual propagators ij(p )asthethe(ij) Propagator matrix2 elements of the 3 3matrixhHA(p ), ⇥ [E. Fuchs, G. W. ’16] hh hH hA = . (5) hHA 0 Hh HH HA1 Ah AH AA

Without mixing: Δhh, …@ has a single pole A The o↵-diagonal entries (for i = j)resultin: 6 For (n x n) mixing: each of the entries Δhh, … in general has n poles ˆ ˆ ˆ ˆ 2 ijkk jkki ij(p )= h h h H h A . (6) ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ2 ˆ ˆ2 ˆ ˆ2 iijjkk +2ijjkki iijk jjki kkij hh hH hA H h H H H A All 2-point vertex functions ˆ(p2)dependonp2 via Eq. (2). Here we do not write the Hh HH HA p2-dependence explicitly for the purpose of a simpler notation, but the full p2-dependence A h A H A A is implied also below. The solutions of the diagonal propagators, ii, can be expressed in Ah AH AA the following compactRenormalisation way: of unstable particles in quantum ﬁeld theories, Georg Weiglein, W. Zimmermann Memorial Symposium, MPI Munich, 05 / 2017 34 2 Figure 3: Contributions from the full mixing propagators ij(p ) for i, j = h, H, A to a generic amplitude (cf. Ref. [13]). If the Zˆ-factorˆ approachˆ is applied,ˆ2 each of the 9 full propagators 2 needs to be approximated by the sum of thejj threekk correspondingjk Breit–Wigner diagrams as ii(p )=shown in Fig. 2. (7) ˆ ˆ ˆ ˆ ˆ2 ˆ ˆ ˆ ˆ ˆ2 ˆ ˆ2 iijjkk + iijk 2ijjkki + jjki + kkij Applying Eq. (76), the amplitude in Eq. (77) can be approximated by the sum over Breit– Wigner propagatorsi multiplied by on-shell Z-factors, in agreement with Ref. [13],

= 3, (8) 2 2 ˆ e↵ ˆX 2 ˆ BW 2 ˆ ˆY p m + ⌃ (p ) Zai (p ) Zaj (78) iA ' ii i a j "a=1 # i,j=Xh,H,A X 3 ˆ ˆX BW 2 ˆ ˆY = Zaii a (p ) Zajj (79) a=1 ! ! X i=Xh,H,A j=Xh,H,A 3 = ˆX BW(p2) ˆY . (80) ha a 5ha a=1 X ˆX The ﬁrst bracket in Eq. (79)representsha , i.e., the vertex X connected to the mass eigenstate ha as for an external Higgs boson in Eq.(30). Subsequently, the second bracket ˆY is equal to the coupling of ha at vertex Y , ha . As opposed to Sect. 2.3,theha is not on-shell here, but a propagator with momentum p2 between the vertices X and Y , rep- BW 2 ˆ resented by the Breit–Wigner propagator a (p ). So the Z-factors are not only useful for the on-shell properties of external Higgs bosons, but they can also be used as an on- shell approximation of the mixing between Higgs propagators. This will be investigated numerically in Sect. 5.

4.3 Calculation of interference terms in the Breit– Wigner formulation In Eq. (76), the Breit–Wigner propagators are combined such that they approximate a given full propagator. Conversely, we will now separate the ha part from the contribution of the other mass eigenstates in the amplitude with Higgs exchange between the vertices

19 Finite waveWave function function normalisation normalisation factors (ﬁnite) for for amplitudesamplitudes with external with external Higgs bosons Higgs bosons Finite wave-function normalisation factors ensure the correct on-shell properties of the S matrix

1 1 Zh = ZH = ∂ i ∂ i 2 2 2 2 ∂p ∆hh(p ) 2 2 ∂p ∆HH(p ) 2 2 p =Mh p =Mh ! "# a ! "# b 1 # # ZA = # Complex quantities, # ∂ i 2 2 ∂p ∆AA(p ) 2 2 evaluated at complex pole p =Mh ! "# c # ∆hH # ∆hH ∆hA ZhH = ZHh = ZAh = ∆hh p2=M2 ∆HH p2=M2 ∆AA p2=M2 # ha # h # hc # # b # ∆hA # ∆HA # ∆HA # ZhA = # ZHA = # ZAH = # ∆hh p2=M2 ∆HH p2=M2 ∆AA p2=M2 # ha # h # hc # # b # Renormalisation# of unstable particles in quantumSUSY ﬁeld Higgs theories, Production Georg and Decays Weiglein, at the W. LHC, Zimmermann Georg# Weiglein, MemorialHiggs Days Symposium, at Santander 2010, MPI Santander,Munich, 10#05 / 2010/ 201 –7 p.1435 # # # FiniteWave wave function function normalisation normalisation for factors amplitudes for with amplitudes with externalexternal Higgs bosons bosons WF constants can be written as (non-unitary) matrix Zˆ,

ˆ ˆ √Zh √ZhZhH √ZhZhA Γha Γh ˆ ˆ Z = √Z Z √Z √Z Z , Γˆ = Z Γˆ ⎛ H Hh H H HA⎞ ⎛ hb ⎞ · ⎛ H ⎞ √Z Z √Z Z √Z Γˆ Γˆ ⎜ A Ah A AH A ⎟ ⎜ hc ⎟ ⎜ A ⎟ ⎝ Every assignment⎠ ⎝ between⎠ ha⎝, hb, ⎠hc, Fulfills the conditions and h, H, A is possible! i lim Zˆ Γˆ ZˆT =1 Unit residues 2 2 2 2 2 p h −p · · hh →M a − Mha ' ( i Off-diagonal lim Zˆ Γˆ ZˆT =1 2 2 2 2 2 p h −p · · HH contributions →M b − Mhb ' ( vanish on- i ˆ ˆ ˆT lim Z Γ2 Z =1 shell p2 2 −p2 2 · · AA →Mhc hc Renormalisation of unstable particles in quantum fieldSUSY− theories, HiggsM Production Georg' andWeiglein, Decays W. at the Zimmermann LHC, Georg( Weiglein, MemorialHiggs Symposium, Days at Santander MPI 2010,Munich, Santander, 05 / 201 10 /7 201036 – p.15 Finite wave function normalisation factors for amplitudes with external Higgs bosons

ha ha h ha H ha A ˆ = Zâ ˆ + ˆ + ˆ + ... 2 2 ha h H A p = ! 2 2 a Zˆ Zˆ Zˆ p = a M p ah aH aA M ˆ ˆ Figure 1: Z-factors for external Higgs bosons: The vertex function ha is constructed from vertex functions î,i= h, H, A, the transition factors Zâi and the overall normalisation factor Loop-correctedZâ. The ellipsis refers to contributions fromLowest-order mixing with Goldstone states and gauge bosons. massp eigenstate In this way, propagator corrections at external legs are e↵ectively absorbed into the vertices of neutral Higgs bosons. If Zˆ-factors are applied to supplement the Born result, only non-Higgs propagator type corrections (such as mixing with the Goldstone and Z-bosons) ⇒Mixingas well as e vertex,ffects boxtaken and realinto corrections account need (via to non-unitary be calculated individually.matrix)

Correct2.4 E normalisation↵ective couplings of the S matrix Since the Zˆ-matrix is not unitary, it does not represent a unitary transformation between the h, H, A and the h1,h2,h3 basis. However, it is not necessary to diagonalise the mass{ matrix} for the determination{ } of the poles of the propagators. Hence there is a prioriRenormalisation no need of unstable to introduce particles in quantum a unitaryﬁeld theories, Georg transformation. Weiglein, W. Zimmermann Though, Memorial Symposium, if a unitaryMPI Munich, 05 matrix / 2017 37 U is desired for the deﬁnition of e↵ective couplings, an approximation of the momentum dependence of Zˆ is required. There is no unique prescription of how to achieve a unitary mixing matrix as an approximation of the Zˆ-matrix, but a possible choice is the p2 =0 approximation [15, 32]. As in the e↵ective potential approach, the external momentum 2 2 p is set to zero in the renormalised self-energies ⌃ˆ ij(p ) ⌃ˆ ij(0) so that they become real. Then U diagonalises the real matrix M(0). U can be! chosen real and it transforms the -eigenstates into the mass eigenstates, CP

h1 h U1h U1H U1A h = U H , U = U U U , (32) 0 21 0 1 0 2h 2H 2A1 h3 A U3h U3H U3A @ A @ A @ A so that U 2 quantiﬁes the admixture of a -odd component inside h [15]. The elements aA CP a of U can then be used to introduce e↵ective couplings of the loop-corrected states ha to any other particles X in terms of the couplings of the unmixed states i by the relation

U ChaX = UaiCiX . (33) i=Xh,H,A absorbing some higher-order corrections, but neglecting imaginary parts and the full momentum dependence of the self-energies. Hence, the application of U resembles the use of Zˆ-factors in Eq. (29) for the purpose of implementing partial higher-order e↵ects into an improved Born result. Yet, the rotation matrix U introduced for e↵ective couplings as a unitary approximation is conceptually di↵erent from the Zˆ-matrix arising from propagator corrections and introduced for the correct normalisation of the S-matrix.

10 expanded around the third complex pole, 2, yielding Mc

2 i ii(p ) (69) ' (p2 2) 1+⌃ˆ e↵0 ( 2) Mc · ii Mc h 1 i 2 BW(p2) ki (70) c e↵ ˆ 0 2 2 2 ' · 1+⌃kk ( c ) kk p = ✓ ◆ Mc 2 M = BW(p2) Zˆ . (71) c · ci Thus, close to one of the complex poles (e.g. 2), the dominant contribution to the Ma full propagator ii can be approximated by the corresponding Breit–Wigner propagator BW ˆ ˆ 2 (a )multipliedbythesquareoftherespectiveZ-factor, (Zai). However, close-by poles may cause overlapping resonances. In order to include this possibility and to extend the range of validity of the Breit–Wigner approximation to a more general case, we take the sum of all three Breit–Wigner contributions into account:

4 Breit–Wigner approximation of3 the full 2 BW 2 2 BW 2 2 BW 2 2 BW 2 2 ii(ppropagators) (p ) Zˆ + (p ) Zˆ + (p ) Zˆ = (p ) Zˆ . (72) ' a ai b bi c ci a ai a=1 The Zˆ-factors depend on the momentum p2 in a twofold way. OnX the one hand, the 2 2 2 2 propagator factors Di(p )=p mi give rise to an explicit p -dependence. On the other 2 4.2.2hand, Expansion the self-energies of⌃ˆ ij( thep ) depend o↵-diagonal on the momentum propagators as well, but away from thresholds their p2 dependence is not particularly pronounced. In this chapter, we will develop We proceedan approximation similarly forof the the full o mixing↵-diagonal propagators propagators, with the aim which to maintain also have the leading three complex poles somomentum that we dependence, can expand but the to greatly propagators simplify thearound mixing them. contributions Note by that makingZˆ ai use= Zâ and ˆ ˆ ofˆ theˆ on-shell Z-factors. 2 2 ˆ Zaj = ZaZaj as defined in Eq. (24). Starting at p a, we express the Z-factorsp in scheme I, ' M p4.1 Unstable particles and the total decay width 2 (p ) 2 In the context2 of determiningij complex2 polesˆ ofˆ propagators,BW 2 we nowˆ ˆ brieflyBW discuss2 reso- ij(p )= 2 ii(p ) ZajZai a (p )=ZajZai a (p ), (73) nances and unstable particles,ii(p ) see e.g. Refs.' [33–37]. Stable particles are associated with a real pole of the S-matrix, whereas the self-energies of unstable particles develop an imag- 2 2 Next, weinary approximate part, so that theij polenear ofp the= propagatorb : is located within the complex momentum plane o↵ the real axis. For a single pole,M the scattering amplitude can be schematically A written near the complex pole2 2 in a gauge-invariant way as 2 ji(p ) Ma 2 ˆ ˆ 2 BW 2 ˆ ˆ BW 2 ij(p )= 2 jj(p ) ZbiZbj b (p )=ZbiZbj b (p ). (74) jj(p ) ' R (s)= 2 + F (s), (59) A s a For p2 2, we switch to a scheme where M the indices i and c belong together. Thereby whereApplicationcs is the squared of centre-of-mass wave function energy, R denotesnormalisation the residue, while factorsF represents we can' writeM non-resonantfor internal contributions. particles The mass Mha of the unstable particle ha is obtained from the real part of the complex pole 2 = M 2 iM , while the imaginary part gives 2 a ha ha ha [E. Fuchs, G. W. ’16] rise to the total2 width.ij Accordingly,(p ) M2 the expansionˆ ˆ 2 aroundBW 2 the complexˆ ˆ poleBW 2 leads2 to Expansionij(p )= of the full propagatorii(p ) Zcj aroundZci c a( pcomplex)=Zcj Zpole:ci c (ap ), (75) a Breit–Wigner propagator (p2) with a constant' width, M dominant contributionii from Breit-Wigner factor, i i BW 2 ˆ which is expressed in termsa ( ofp ):= scheme-invariant2 2 = 2 Z2-factors. Finally,. we take(60) the sum of p p M + iMh h Eqs. (73)-( 75 )toobtain M a h a a a Intimes the following, wave wefunction will use normalisation a Breit–Wigner propagatorfactors: of this form to describe the

contribution of the unstable scalar ha with3 mass Mha and total width ha in the resonance region. (p2) Zˆ BW(p2) Zˆ . (76) ij ' ai a aj a=1 X 4.2 Expansion of the full propagators around the com- This sum isi illustratedj diagrammaticallyi h1 inj Fig.i2. h2 j i h3 j plex poles + + ' ˆ ˆ ˆ Zˆ 1i Z1j Zˆ 2i Z2j Zˆ 3i Z3j Eqs. ( 52 )and( 53 ) imply for 3 3mixingthateachpropagator 17 ii , ij has a pole at 2, 2 and 2. Because of this⇥ structure, an expansion of the full propagators near one ⇒MFigure1ApproximationM2 2: DiagrammaticM3 of the illustration full oﬀ of-shell the full propagator mixing Higgs propagatorsin terms of compared the to the single pole is not expected to yield a suˆ cient approximation. Instead, we will perform an Breit–Wigneron-shell contributions propagators where of the allZ-factors complex encode poles the transition between the interaction expansionand the mass of the eigenstates. full propagators around all of their complex poles. The ﬁnal expression obtained from combining the contributions from the di↵erent poles will constitute a main

result ofRenormalisation the present of unstable paper. particles in quantum field theories, Georg Weiglein, W. Zimmermann Memorial Symposium, MPI Munich, 05 / 2017 38 Eq. (76)representsthecentralresultofthissection,coveringalsothediagonalprop- agators in the special case of i = j. It shows15 how the full propagator can be approximated by the contributions of the three resonance regions, expressed by the Breit–Wigner propa- 2 gators a(p ),a=1, 2, 3, reflecting the main momentum dependence. The mixing among the Higgs bosons is comprised in the Zˆ-factors which are evaluated on-shell. Nonetheless, even a part of the momentum dependence of the self-energies is accounted for because the derivation of Eq. (72) is based on a first-order expansion of the momentum-dependent e↵ective self-energies. Furthermore, the Zˆ-factors serve as transition factors between the loop-corrected mass eigenstates ha and the lowest-order states i (although Zˆ is not a unitary matrix transforming the states into each other). Pictorially, ij is a propagator that begins on the state i and ends on j while mixing occurs in between. Also on the RHS of Fig. 2 and Eq. (76)thepropagatorintheha-basis begins with i and ends on j. Thus, the coupling to the rest of any diagram connected to the propagator is well-defined. In between, each of the ha can propagate, and the correct transition is ensured by Zˆ ai and Zˆ aj. All three combinations are visualised in Fig. 2. If is conserved and only h and H mix, or if two states are nearly degenerate and their resonancesCP widely separated from the remaining complex pole, the full 3 3mixing is (exactly or approximately) reduced to the 2 2mixingcase.Thentheo⇥↵-diagonal Zˆ-factors involving the unmixed state vanish or⇥ become negligible so that some terms in Eq. (76)approachzero. Beyond that, if no mixing occurs among the neutral Higgs bosons, all o↵-diagonal full propagators as well as the o↵-diagonal Zˆ-factors vanish and each diagonal full propagator consists of only a single Breit–Wigner term where the Zˆ-factor is based on the diagonal self-energy instead of the e↵ective self-energy. Thus, Eq. (76)coversallspecialcasesof the a priori 3 3 mixing among the neutral Higgs bosons. ⇥ 4.2.3 Amplitude with mixing based on full or Breit–Wigner propagators In a physical process where neutral Higgs bosons can appear as intermediate particles, all of them need to be included in the prediction, see Fig. 3 and Ref. [13]. The Higgs ˆX part of the amplitude then contains a sum over the irreducible vertex functions i (for a ˆY coupling of Higgs i at the first vertex X)andj (for a coupling of Higgs j at the second vertex Y )timesthefullymomentum-dependentmixingpropagators,

= ˆX (p2) ˆY . (77) A i ij j i,j=Xh,H,A

18 X and Y : ˆX BW(p2) ˆY = ˆX Zˆ BW(p2) Zˆ ˆY , (81) Aha ⌘ ha a ha i ai a aj j i,j=Xh,H,A ˆ Applicationi.e. the exchange of of thethe state propagatorha coupling with the mixed approximation vertices ha from Eq. (30)asfor an external Higgs.

h [E. Fuchs, G. W. ’16] ˆX a ˆY ha ha

h ha h h ha H h ha A = ˆX ˆY ˆX ˆY ˆX ˆY h h + h H + h A Zˆ ah Zˆ ah Zˆ ah Zˆ aH Zˆ ah Zˆ aA

H ha h H ha H H ha A ˆX ˆY ˆX ˆY ˆX ˆY + H h + H H + H A Zˆ aH Zˆ ah Zˆ aH Zˆ aH Zˆ aH Zˆ aA

A ha h A ha H A ha A ˆX ˆY ˆX ˆY ˆX ˆY + A h + A H + A A Zˆ aA Zˆ ah Zˆ aA Zˆ aH Zˆ aA Zˆ aA

i ha j = ˆX ˆY i j i,j=h,H,A ˆ ˆ X Zai Zaj ⇒ Expression of the full process in terms of the on-shell production Figure 4: Diagrammatic representation of the contribution from Eq. (81) of h (a =1, 2, 3) Aha a and decayto the amplitude of the. The intermediate blue lines labelled by statesh denote the Breit–Wigner propagator BW(p2), A a a and the green lines labelled by i, j = h, H, A denote lowest order propagators of h, H, A.

⇒ ConvenientIn order incorporation to calculate the squared of amplitudehigher-order as a coherent contributions,sum, all contributions mixing of and interferenceh1,h2,h3 are summed eff upects first before taking the absolute square, Renormalisation of unstable particles in quantum field theories, Georg Weiglein,3 W.2 Zimmermann Memorial Symposium, MPI Munich, 05 / 2017 39 2 = . (82) |A|coh Aha a=1 X On the contrary, the incoherent sum is the sum of the squared individual amplitudes, which misses the interference contribution,

3 2 2 = . (83) |A|incoh Aha a=1 X Thus, an advantage of the Breit–Wigner propagators is also the possibility to conveniently discern the interference of several resonances from their individual contributions in a squared amplitude

2 2 2 = = 2Re h ⇤ . (84) |A|int |A|coh |A|inccoh A a Ahb at + t - with propagator mixing Without 400000 interference max modified M h scenario: contribution tan b = 50H L M h =126.20 GeV M H + = 153 GeV M H =127.55 GeV 300000 G h = 0.94 GeV G H =1.21 GeV Individual D contribution Dij 3 â 3 pb @ 200000 DBW ◊ Z

` Propagator s DBW ◊ Z no In t DBW ◊ Z h approximation DBW ◊ Z H 100000 DBW ◊ Z A Full result 0 122 124 126 128 130 ` s GeV

Figure⇒ Very 11: Thegood partonic agreement cross section of ˆ( bpropagatorb ⌧ +⌧ )inamodified approximationM max-scenario with with full tan result= ! h 50 andIncorporationM = 153 GeV. Theof interference cross section is calculated effects with is thecrucial full mixing propagators (blue, H± @ D solid), approximatedRenormalisation of by unstable the particles coherent in quantum sum field of theories, Breit–Wigner Georg Weiglein, W. propagators Zimmermann Memorial times Symposium,Zˆ-factors MPI Munich, with 05 the / 2017 40 interference term (red, dashed) and the incoherent sum without the interference term (grey, dot- dashed). The individual contributions mediated by h1 (light blue), h2 (green) and h3 (purple) are shown as dotted lines.

+ Fig. 11 shows the partonic cross section ˆ(bb h, H, A ⌧ ⌧ ) as a function of !2 ! the centre-of-mass energy psˆ, wheres ˆ =(pb + pb) is the squared sum of the momenta of the b-andb-quarks in the initial state. The calculation based on the full propagators

31 Phenomenology of extended Higgs sectors

Most obvious possibility: SM-like state at 125 GeV, corresponds to the lightest state of an extended Higgs sector, other Higgs states are heavy (``decoupling region’’)

Heavy additional Higgs states ⇒ decoupling behaviour

Interference eﬀects can be important for heavy Higgs phenomenology

Examples:

• Interference eﬀects of the heavy Higgs signal with the background and with the state at 125 GeV

• MSSM with CP-violation: h2, h3 are typically nearly mass- degenerate, have a large mixing; resonance-type behaviour Renormalisation of unstable particles in quantum ﬁeld theories, Georg Weiglein, W. Zimmermann Memorial Symposium, MPI Munich, 05 / 2017 41 2RelationoftheHiggsmassandwidthtothecomplexpole of the propagator

Before we start our discussion of oﬀ-shell eﬀects in H VV(∗) in the subsequent section, → we shortly elaborate on the relation between the mass and total width of the Higgs boson and the complex pole of the propagator. Denoting with m0 the tree-level Higgs mass and with Σˆ the renormalized self-energy of the Higgs propagator, the complex pole is obtained 2 2 2 103 ˆ 103 103 103 through the relation+ m0¯ + Σ( )=0,wherethecomplexpolecanbewritteninthe+ ¯ + ¯ + ¯ 2 N 2 e eM− ν−ν¯uud¯ d, √s = 1M TeV N e e− νν¯N uud¯ d, √s e=e 1− TeVνν¯uud¯ d, √s = 1 TeV N e e− νν¯uud¯ d, √s = 1 TeV H +→H H +→ H +→ +→ form = mH imPol(eΓ,e−.Therein)=(0.3, 0.8)m is the physical HiggsPol(e mass,e−)=(0 and.3, Γ0Pol(.8)thee ,e− total)=(0.3 width, 0.8) of Pol(e ,e−)=(0.3, 0.8) M − − − − − the Higgs boson. Expanding2HDM, sβ α =0 the.95 inverse propagator around2HDM th, sβecomplexpoleyieldsα =0.95 2HDM, sβ α =0.95 2HDM, sβ α =0.95 − − − − Events mh = 125 GeV, mH = 400 GeV Events mh = 125 GeV,Events mH =m 600h = GeV 125 GeV, mH = 400 GeV Events mh = 125 GeV, mH = 600 GeV 2 2 ˆ 2 2 2 ˆ ′ 2 102 p m0 + Σ(p ) (p 102 ) 1+Σ ( 10)2 (1) 102 − ≃ − M ! M " in the vicinity of the complex pole. Accordingly, the Higgs propagator in the vicinity of the complex pole can be expressed in the well-known form of a Breit-Wigner propagator with constant width ΓH , 101 101 101 101

2 i i ∆H (p )= 2 2 = 2 2 . (2) p p m + imH ΓH 200 400 600 800 − M1000 − H200 400 600 200800 4001000 600 800 1000 200 400 600 800 1000 m ¯ [GeV] m ¯ [GeV] m ¯ [GeV] m ¯ [GeV] (a)uud¯ d 2 (b)2 uud¯ d (a)uud¯ d (b) uud¯ d Away from the pole, i.e. in the far off-shell region with p mH ,theHiggswidthisnotof 103 103 103 103 + ¯ + ≫ ¯ + ¯ + ¯ relevance. ForN the specifice e− νν¯ processesuud¯ d, √s = 1 that TeV are consideredN ine ethis− ν paperν¯N uud¯ d, √ ours e= choicee 1− TeVνν¯u isud¯ equivalentd, √s = 1 TeV N e e− νν¯uud¯ d, √s = 1 TeV +→ +→ +→ +→ Pol(e ,e−)=(0.3, 0.8) Pol(e ,e−)=(0.3, 0Pol(.8) e ,e−)=(0.3, 0.8) Pol(e ,e−)=(0.3, 0.8) to the complex-mass scheme [41,42],− which is known to providegauge-independentresults.− − − 2HDM, sβ α =0.98 2HDM, sβ α =0.98 2HDM, sβ α =0.98 2HDM, sβ α =0.98 Differences with respect to− the scheme defined in Refs. [43–45]−are expected to− be small, in − Events mh = 125 GeV, mH = 400 GeV Events mh = 125 GeV,Events mH =m 600h = GeV 125 GeV, mH = 400 GeV Events mh = 125 GeV, mH = 600 GeV particular since the constant width ΓH is close to the width therein [45]. For our subsequent 102 SM 102 −3 102 102 discussion we fix mH =125GeVandΓ =4.07 10 GeV, the latter in accordance with H · the prescription of the LHC Higgs cross section working group (LHC-HXSWG)[9–11].

( ) ( ) 3Oﬀ-shell1 contributions in H ZZ1 ∗ and H1 W ±W ∓ ∗ 1 10 → 10 10 → 10 Given the two dominant production processes for a Higgs boson H at a linear collider, e+e− + − → ZH and e e 200νν¯H,wediscussthevalidityofthezero-widthapproximation(400 600 800 1000 200 400 600 200800 4001000 600ZWA)for800 1000 200 400 600 800 1000 → (∗) (∗) the Higgs decays H WW andmuuddH¯ [GeV]ZZ within this section. Themuudd¯ relevant[GeV] Feynmanmuudd¯ [GeV] muudd¯ [GeV] → (c)¯ → (d) ¯ (c)¯ (d) ¯ diagrams are10 presented3 in Fig.Sensitivity 1. Our discussion follows10 3to Refthes. [12–14], 10small3 which signal are speciﬁc toof an additional103 heavy + ¯ + ¯ + ¯ + ¯ N e e− νν¯uud¯ d, √s = 1 TeV N e e− νν¯N uud¯ d, √s e=e 1− TeVνν¯uud¯ d, √s = 1 TeV N e e− νν¯uud¯ d, √s = 1 TeV the dominant production+→ process at the LHC,gluonfusion. +→ +→ +→ Pol(e ,e−)=(0.3, 0.8) Pol(e ,e−)=(0.3, 0Pol(.8) e ,e−)=(0.3, 0.8) Pol(e ,e−)=(0.3, 0.8) − − − − 2HDM, sβ α =0.Higgs99 boson2HDM in, s βaα =0 Two-Higgs-Doublet.99 2HDM, sβ α =0.99 Higgs model couplings2HDM, sβ α =0. 99(2HDM) − − Higgs− couplings − Events mh = 125 GeV, mH = 400 GeV ( ) Events mh = 125 GeV,Events mH ν=m 600h = GeV 125 GeV, mH = 400 GeV Events mh = 125 GeV, mH = 600 GeV e− V ∗ e− [S. Liebler, G. Moortgat-Pick, G. W. ’15] 102 H 102 W H 102 V Higgs couplings, tree level:102 V ( ) SM SM ± W V ∗ ghVV = sin(β − α) gHVV,gHVV = cos(β − α) gHVV,V= W ,Z + + e Z e ν¯ ′ ′ (a)3 (b) 3 g −g 10 ghAZ = cos(β10− α) ,gHAZ = sin(β − α) 101 +101 ¯101 2cos+θ101 ¯ 2cosθ N + − e e− ν(ν¯∗u) ud¯ d, √+s −= 1 TeV (∗N ) e eW− νν¯uud¯ d, √s = 1 TeV W Figure 1: Feynman diagrams for (a) e e ZH ZV+→ V ;(b)e e νν¯H νν¯VV . +→ → Pol(→ e ,e−)=(0.3, 0.8)→ → Pol(e ,e−)=(0.3, 0.8) − SM , , , , cannot− all be small 2HDM, sβ α =0.95 ⇒ ghVV ≤ gHVV ghVV g2HDMHVV g, hAZsβ αgHAZ=0.95 Supplementing200 the ZWA400 for600 the800 production1000 and the200− decay400 part600 of200 the800 400 process1000 600 with800 a 1000 200− 400 600 800 1000

Events + − Events Breit-Wigner propagator, the diffmerentialuud¯ d¯ [GeV] cross sectionmh = 125e e GeV,ZHmH =mZV 400uud¯ d¯ V GeV[GeV]can be writtenmuud¯ d¯ [GeV] mh = 125 GeV, mH =m 600uud¯ d¯ GeV[GeV] (e)→ (f) → (e) (f) In decoupling limit, MA ≫ MZ (already realized for MA > 150 GeV): − − − ∼− Figure 15: Event rates for e+e νν¯u10ud¯ 2d¯ for √s =1TeVandFigure 15: Eventdt =500fb rates for1 eafter+e theνν¯u10ud¯ 2d¯ for √s =1TeVand dt =500fb 1 after the → L cos(β − α→) → 0 L cut pT,4j > 75 GeV as a function of the invariant3 masscut of thepT,4 4j jets>!75m GeVuud¯ d¯ asin athe function context of of the invariant mass of the 4 jets! muud¯ d¯ in the context of atypeII2HDM with tan β =1fordifferent values of (a,b)atypeIIsβ−α 2HDM:= sin(withβ α tan)=0⇒βh=1fordi.95;is SM-like, (c,d) fferentH and valuesA decouple of (a,b) fromsβ−α gauge:= sin( bosonsβ α)=0.95; (c,d) − − s =0.98 and (e,f) s =0.99 and the two mass scenarioss =0. (a,c,e)98 andm (e,f)=400GeVands =0.99 and the two mass scenarios (a,c,e) m =400GeVand β−α β−α β−α H β⇒−αCannot use WBF channels for production of heavyH SUSY (b,d,f) mH =600GeV. (b,d,f) mH =600GeV. Higgses; no H → ZZ → 4µ decay Higgs Physics,Beyond Georg the Standard Weiglein, Model Nikhef (Higgs), Topical Georg Lectures, Weiglein, IMFP13,Amsterdam, Santander, 04 05/ 201 / 20136 –68 p. 36 101 101 27 27

200 400 600 800 1000 200 400 600 800 1000 (a)muud¯ d¯ [GeV] (b) muud¯ d¯ [GeV] 103 103 ⇒ ILC: Potential+ sensitivity¯ beyond the kinematic+ reach¯ of Higgs pair N e e− νν¯uud¯ d, √s = 1 TeV N e e− νν¯uud¯ d, √s = 1 TeV +→ +→ productionPol(e ,e−)=(0.3, 0.8) Pol(e ,e−)=(0.3, 0.8) Renormalisation of unstable particles in quantum− ﬁeld theories, Georg Weiglein, W. Zimmermann Memorial Symposium, MPI− Munich, 05 / 2017 42 2HDM, sβ α =0.98 2HDM, sβ α =0.98 − − Events mh = 125 GeV, mH = 400 GeV Events mh = 125 GeV, mH = 600 GeV

102 102

101 101

200 400 600 800 1000 200 400 600 800 1000 (c)muud¯ d¯ [GeV] (d) muud¯ d¯ [GeV] 103 103 + ¯ + ¯ N e e− νν¯uud¯ d, √s = 1 TeV N e e− νν¯uud¯ d, √s = 1 TeV +→ +→ Pol(e ,e−)=(0.3, 0.8) Pol(e ,e−)=(0.3, 0.8) − − 2HDM, sβ α =0.99 2HDM, sβ α =0.99 − − Events mh = 125 GeV, mH = 400 GeV Events mh = 125 GeV, mH = 600 GeV

102 102

101 101

200 400 600 800 1000 200 400 600 800 1000 (e)muud¯ d¯ [GeV] (f) muud¯ d¯ [GeV] Figure 15: Event rates for e+e− e+e−uud¯ d¯ for √s =1TeVand dt =500fb−1 after the → L cut pT,4j > 75 GeV as a function of the invariant mass of the 4 jets! muud¯ d¯ in the context of atypeII2HDM with tan β =1fordiﬀerent values of (a,b) sβ−α := sin(β α)=0.95; (c,d) − sβ−α =0.98 and (e,f) sβ−α =0.99 and the two mass scenarios (a,c,e) mH =400GeVand (b,d,f) mH =600GeV.

26 Higgs mixing: possible interference effects General case: inclusion of interference effects Total cross section:

σtot = σ(b¯bH)+σ(b¯bA) (incoherent sum) holds only in the CP-conserving case

But: in reality we don’t know whether CP in the Higgs sector is conserved or not

In the general case: Complex parameters ⇒ loop corrections induce CP-violation Two Higgs states, nearly mass degenerate, large mixing ⇒ Large (destructive) interference possible MSSM Higgs at the LHC: Interpretation of limits and search reach, Georg Weiglein, CMS Higgs Meeting, DESY, Hamburg, 11 / 2011 – p.13 Renormalisation of unstable particles in quantum ﬁeld theories, Georg Weiglein, W. Zimmermann Memorial Symposium, MPI Munich, 05 / 2017 43 Higgs production via gluon fusion in the MSSM with CP-violation: extension of the SusHi code [S. Liebler, S. Patel, G. W. ’16] gg → h2h3, dependence on phase ɸAt: 10 896.6 mod+ gg H2/H3 (13 TeV) ! mh tan = 40 [fb] m = 900 GeV 8 H± 896.4 [GeV]

6 m 896.2

4 896.0

LO H 2 3 H3 895.8 H2 H2 w/o t,˜ ˜b 0 0 ⇡ 2⇡ 1.24 ⇡/2 3⇡/2 LO 1.22 At

1.20

/ 1.18 1.16 Only production process shown here 0 ⇡/2 ⇡ 3⇡/2 2⇡ At ⇒ Full result for σ x BR needs to incorporate interference contribution Renormalisation of unstable particles in quantum field theories, Georg Weiglein, W. Zimmermann Memorial Symposium, MPI Munich, 05 / 2017 44 ratios from the existing literature including available higher-order corrections. Regarding the determination of the relative interference contributions, we restrict ourselves to the computation of the relevant 2 2partonicprocessesatleadingorder(LO).Thismeans ! that process (24), which involves b¯bha-associated production, is treated at tree-level, while the leading contributions to gluon fusion entering process (25)consistofone-loopdiagrams with quarks q and squarksq ˜, in particular those of the third generation. Besides, we take propagator-type corrections into account by using Higgs masses, total widths and Zˆ- factors from FeynHiggs including the full one-loop and dominant two-loop corrections. The procedure to evaluate the relative interference contributions on the basis of leading- order diagrams is motivated by the fact that the vertex corrections to the production and decay factorise (apart from non-factorisable initial-to-final state radiation). It furthermore avoids double-counting of higher-order contributions that are incorporated in the separate production and decay processes. We calculate the LO contributions to the amplitude and the cross section of the full process of production and decay using FeynArts [70–74] with a model file containing Zˆ-factors for the Higgs vertices (in this way the Higgs propagators Incorporationare expressed in terms ofofZˆ -factorsinterference and lowest-order contributions Breit–Wigner propagators as given in (20)), FormCalc [75–79] and, as mentioned above, FeynHiggs for quantities from the Higgs sector.

2 2 3 b h ⌧ 3 g g g a h ⌧ h ⌧ h ⌧ , q a + q˜ a + q˜ a • • • • • • • • g ⌧ + g ⌧ + g ⌧ + a=1 ¯b ⌧ + a=1

P P + Figure 2: Higgs boson production at LO via b¯b (left) and gg (right) with decay into ⌧ ⌧ . +The higher-order couplings of the masscontributions eigenstates ha = h1,h2,h3 (blue, dashed) to the initial and ﬁnal state contain a combination of Zˆ-factors (denoted by blue circles). for bb associated production and gluon fusion process with subsequent decay into �+�- Quantifying the interference In order to determine the interference term in each of the two processes (24, 25), we distinguish between the coherent sum of the 2 2 ! amplitudes with h1,h2,h3-exchange including the interference and their incoherent sum without the interference,

3 2 3 2 2 = , 2 = . (26) |A|coh Aha |A|incoh Aha a=1 a=1 Renormalisation of unstable particles in quantum X ﬁeld theories, Georg Weiglein, W. ZimmermannX Memorial Symposium, MPI Munich, 05 / 2017 45 The interference term is then obtained from the di↵erence,

2 2 2 = = 2Re h ⇤ . (27) |A|int |A|coh |A|incoh A a Ahb a

11 Cross sections with and without interference contributions vs. experimental limits [E. Fuchs, G. W. ’17] 1000 1000 CMS, 8 TeV, 24.6 fb-1 CMS, 8 TeV, 24.6 fb-1 500 bb expected gg expected observed observed mod+ mod+ h2+h3, Mh h2+h3, Mh

] ] 100 = 1TeV, At = /4 = 1TeV, At = /4 fb fb tan = 29 tan = 29

)[ 100 )[ 50

( ( 10 BR BR × × ) ) ( 10 ( bb gg 1 5 2×2 At =0 2×2 At =0 3×3 incoherent 3×3 incoherent 3×3 coherent 3×3 coherent 0.1 1 200 400 600 800 1000 200 400 600 800 1000

M [GeV] M [GeV] (a) (b) ⇒ CP-violating mixing induces resonance-type enhancement + large destructive interference contributions

Renormalisation of unstable particles in quantum ﬁeld theories, Georg Weiglein, W. Zimmermann Memorial Symposium, MPI Munich, 05 / 2017 46

(c) (d)

+ Figure 5: Comparison of predicted Higgs cross sections times branching ratio into ⌧ ⌧ with and without the interference and with 2 2 and 3 3 mixing, for a ﬁxed value of tan ,to ⇥ ⇥ experimental exclusion bounds. Left column (a, c): production via b¯b. Right column (b, d): production via gg. Upper row (a, b): strongest interference e↵ect at tan = 29. Lower row: (c) tan = 25, (d) tan = 19. Each plot shows the CMS observed (black, solid) and expected 1 (black, dotted) exclusion bounds at 95% CL at 8 TeV with = 24.6fb from Ref. [25,80,81], L as well as the theory prediction in the CM mod+ scenario for the combined cross section of h h R 2 and h as the incoherent sum restricted to -conserving mixing for = 0 (turquoise, dotted, 3 CP At labelled as “2 2”), the incoherent sum with -violating mixing for = ⇡/4 (turquoise, ⇥ CP At dashed, labelled as “3 3 incoherent”) and the coherent sum (i. e. including the interference ⇥ term) with -violating mixing for = ⇡/4 (turquoise, solid, labelled as “3 3 coherent”). CP At ⇥

16 complex M mod+ scenario with µ = 1000 GeV, interference included via rescaling by⌘ ˜ . • h a I plotted the excluded and allowed points in Fig. 10.6 in my thesis (=Fig. 1(a) here). mod+ As a cross check, I could directly reproduce the results for the real Mh scenario with µ = 200 GeV and µ = 1000 GeV. However, in the complex scenario (even neglecting the interference) I obtained a stronger limit than Oscar due to the Zˆ-factor enhanced individual cross sections of h2 and h3 with the FeynHiggs flag higgsmix=3.OnlyifIsethiggsmix=2 (not appropriate in the complex case), I can reproduce the bounds from Oscar’s HiggsBounds run, also the intermediate results CS bb hj ratio and CS gg hj ratio.SoIsent him a comparison of our results, a description of what I did and asked him in some emails if he had adjusted higgsmix. Because I did not get an answer (I fully understand that he is busy in his new job, but I should finally finish this comparison...), I read his file that he used for his HB scan. There were flags and comments to switch on/o↵ the ⌘ contributions and to set At = 0 or ⇡/4. On the other hand, higgsmix=2 and there was no comment to change it. Conclusion: higgsmix=2 instead of 3 seems to explain the di↵erence between Oscar’s and my results. I continue with my results where I set higgsmix=3 in the complex scenario. The conclusion also implies that Fig. 10.6 of my thesis (see Fig.1(a)) is not completely “kosher”. The qualitative e↵ect remains the same, but the bounds shift noticeably. Confirmation: When I just finished writing this paragraph, an email from Oscar arrived! He confirmed that it is very likely that higgsmix=2 was used in his scan file.

bb gg 2.2 Implementation of ⌘a and ⌘a The interference terms in the b¯b- and the gg-initiated processes are taken into account in the following way:

MSSM(b¯b h ) MSSM(b¯b h ) ! a ! a (1 + ⌘bb), (9) SM(b¯b h ) ! SM(b¯b h ) · a ! a ! a MSSM(gg h ) MSSM(gg h ) ! a ! a (1 + ⌘gg). (10) SM(gg h ) ! SM(gg h ) · a Search! a for heavy! Higgsa bosons at the LHC: impact Fig.1(a) shows the previous result where the interference is included only in b¯b and higgsmix=2 is used despite the complex scenario. In contrast,of interference Fig. 1(b) is based on higgsmix=3effectsand shows the exclusion bounds if no interference is considered (blue), interference in only gg (gray), bb (black) and a combination of both (red)[E. Fuchs, G. W. ’17] according to Eqs. (9,10). Exclusion limits from neutral Higgs searches in the MSSM with and without interference eﬀects:

CP-violating case, ɸAt = π / 4

H, A are nearly mass degenerate: large mixing possible in CP- violating case! mod+ mh scenario, Incoherent sum is μ = 1000 GeV not suﬃcient!

bb bb gg (a) Old: only ⌘⇒a , higgsmix=2Large .CP-violating interference(b) New: ⌘a and eﬀ⌘aects, higgsmix=3 possible. Renormalisation of unstable particles in quantum ﬁeld theories, Georg Weiglein, W. Zimmermann Memorial Symposium, MPI Munich, 05 / 2017 47

Figure 1: Parameter regions excluded by HiggsBounds for µ =1000GeV, At = ⇡/4 without the interference term (blue) and including the interference term (red) by modifying the input data for HiggsBounds with ⌘ (see text).

2.3 Next steps Discuss this new result, include it in Blois proceedings. Then replace FH production prediction by complex SusHi.

2 Field renormalisationCP-violating case: for SUSY unstable with complex particles parameters + complex parameters: RenormalisationMSSM with complex for unstable parameters particles Occurrence of imaginary parts:

From complex parameters

From absorptive parts of loop integrals ↔ unstable particles

⇒ MSSM with complex parameters: absorptive parts of loop integrals can contribute to real part of 1-loop quantities ⇒ Consistent renormalisation procedure needed for complex parameters and unstable particles [A. Bharucha, A. Fowler, G. Moortgat-Pick, G. W. ’12] [A. Fowler, G. W. ’09] Collider Physics (Hamburg), Georg Weiglein, Theory Jamboree, 02 / 2013 – p.18 Renormalisation of unstable particles in quantum ﬁeld theories, Georg Weiglein, W. Zimmermann Memorial Symposium, MPI Munich, 05 / 2017 48 RenormalisationRenormalisation of the in chargino the chargino / neutralino / neutralino sector with sectorcomplex of theparameters MSSM with complex parameters Allow ﬁeld renormalisations to be different for in- and outgoing fermions:

1 L 1 ¯L ωLχ˜i− (1 + 2 δZ )ijωLχ˜j−, χ˜i−ωR χ˜i−(1 + 2 δZ )ijωR, → − → − 1 R 1 ¯R ωRχ˜i− (1 + 2 δZ )ijωRχ˜j−, χ˜i−ωL χ˜i−(1 + 2 δZ )ijωL, → − → − In -conserving case: can choose a scheme where hermiticityCP relation holds (up to purely imaginary terms that do not contribute to squared matrix elements at 1-loop)

¯ δZij = δZij† Decomposition of fermion self-energies:

2 L 2 R 2 SL 2 SR 2 Σij(p )=p ωLΣij(p )+p ωRΣij(p )+ωLΣij (p )+ωRΣij (p ) Renormalisation of unstable̸ particles in quantumSUSY ﬁeld̸ Higgstheories, Production Georg and Weiglein, Decays at W. the LHC,Zimmermann Georg Weiglein, MemorialHiggs Days Symposium, at Santander MPI 2010, Munich, Santander, 05 10 / 201 / 20107 – p.3649 Field renormalisationField renormalisation conditions conditions (1-loop) (1-loop)

Vanishing of off-diagonal contributions and unit residues:

(2) (2) ˆ 2 2 ˆ 2 2 Γij χ˜i(p) p =m ˜ =0, χ˜i(p)Γij p =m ˜ =0 | χj | χi 1 1 lim Γˆ(2)χ˜ (p)=iχ˜ , lim χ˜ (p)Γˆ(2) = iχ˜ 2 2 ii i i 2 2 i ii i p m ˜ p mχ˜ p m ˜ p mχ˜ → χi ̸ − i → χi ̸ − i Additional conditions needed for the general case with violation: CP Loop-corrected propagator should have the same Lorentz structure in the on-shell as at tree level ( vanishing of γ5 contributions) → ˆ SL 2 ˆ SR 2 Σii (mχ˜i )=Σii (mχ˜i ) R ¯R ¯L L Exploit additional freedom: δZii δZii = δZii δZii Renormalisation of unstable particles in quantum fieldSUSY theories, Higgs Production Georg Weiglein, and− Decays W. at theZimmermann LHC, Georg Weiglein,MemorialHiggs− Symposium, Days at Santander MPI Munich, 2010, Santander, 05 / 201 107 / 201050 – p.37 CharginoChargino / neutralino / neutralino field field renormalisation renormalisation in the in the MSSM withMSSM complex with parameters complex parameters

In general δZ¯ = δZ† ij ̸ ij Hermiticity relation does not hold, but it can be shown ⇒ that the CPT theorem is fulﬁlled

If absorptive parts of the loop integrals are discarded from the ﬁeld renormalisation constants Hermiticity relation is restored, but correct on-shell ⇒ properties are spoiled Additional mixing contributions needed to obtain the ⇒ correct on-shell properties

SUSY Higgs Production and Decays at the LHC, Georg Weiglein, Higgs Days at Santander 2010, Santander, 10 / 2010 – p.40 Renormalisation of unstable particles in quantum ﬁeld theories, Georg Weiglein, W. Zimmermann Memorial Symposium, MPI Munich, 05 / 2017 51 Conclusions

The ground-breaking work of Wolfhart Zimmermann continues to have a big impact on our present-day applications of quantum ﬁeld theory for describing the fundamental laws of nature and confronting this description with the latest experimental results ⇒ Window for probing possible eﬀects of new physics

Treatment of unstable particles in spontaneously broken gauge theories: both challenging and phenomenologically very rich

Many issues related to gauge invariance, unitarity, etc. have to be solved in order to reach the level of precision that will be needed for the comparison with experimental results in the coming years

⇒ Vibrant ﬁeld with many new results

Renormalisation of unstable particles in quantum ﬁeld theories, Georg Weiglein, W. Zimmermann Memorial Symposium, MPI Munich, 05 / 2017 52