Narrow-width approximation accuracy Nikolas Kauer in collaboration - - PowerPoint PPT Presentation

SLIDE 1

Narrow-width approximation accuracy

Nikolas Kauer

in collaboration with

• D. Berdine, D. Rainwater and C.F

. Uhlemann

Particle Theory Seminar Paul Scherrer Institut May 29, 2008

1 / 29 funded by

SLIDE 2

Outline

• Introduction
• Accuracy
• Limitations
• Improvements
• Conclusions

2 / 29

SLIDE 3

Search for the Higgs boson and BSM physics

◮ Unitarity-violating processes, e.g. W − L W + L → W − L W + L ◮ Consistent mass generation via SSB → Higgs mechanism ◮ SUSY, composite Higgs, extra dimensions, ...

LHC at CERN: a powerful Terascale collider

◮ pp collisions at ECMS = 14 TeV ◮ Luminosity L = 10–100 fb−1/year

Introduction 3 / 29

SLIDE 4

BSM physics via supersymmetry?

SUSY: invariance under boson ↔ fermion transformation Particle spectrum of the Minimal Supersymmetric Standard Model (MSSM): gauge bosons S = 1 gluon, W ±, Z, γ fermions S = 1

2

`uL

dL

´ , `νeL

eL

´ uR, dR, eR Higgs S = 0 ` H0

d

H−

d

´ , `H+

u

H0

u

´ gauginos S = 1

2

gluino, f W ±, e Z, e γ sfermions S = 0 `e

uL e dL

´ , `e

νeL e eL

´ e uR, e dR, e eR Higgsinos S = 1

2

` e

H0

d

e H−

d

´ , ` e

H+

u

e H0

u

´ MX = M e

X ⇒ SUSY broken Introduction 4 / 29

SLIDE 5

Unstable particles and QFT

“Ordinary” QFT: unstable particle fields → asymptotic states (t = ±∞) Eliminate these states → Hilbert space of stable particle states Theory still unitary, causal and renormalizable Veltman (1963) [No discussion of gauge invariance!]

Introduction 5 / 29

SLIDE 6

Unstable particles and perturbation theory

LO propagators for massive gauge bosons and fermions: −i p2 − M 2 “ gµν − pµpν M 2 ” , i(p / + M) p2 − M 2 Fixed-order propagator: real pole at p2 = M 2

= + + + : : : t t t t t t t W b W b W b t

i p / − Mt

∞

X

n=0

„ (−iΠt) i p / − Mt «n = i p / − Mt − Πt

[analytic continuation to resonant region]

Im Πt = − 1 64π g2 M2

W

1 − M2

W

p2 !2 (2M2

W +p2) p

/PL

p2 ≈ M 2

t : Im Πt ≈ −MtΓt

Breit-Wigner propagator: i(p / + Mt) p2 − M 2

t + iMtΓt

Dyson-resummed propagator: complex pole at p2 = M 2 − iMΓ

[gauge invariance?]

◮ Physical fixed-order amplitudes with unstable particles

include contributions from all orders in perturbation theory

Introduction 6 / 29

SLIDE 7

Generic resonant enhancement

g g g b e

• e
• +
• b

W + W

• t
• t

t (a) g g g b e

• e
• +
• b

W + W

• t
• t

t (b) g g g b e

• e
• +
• b

W + W

• t

b b (d) g g g b e

• e
• +
• b

W + W

• b
• ;

Z (e) g g g b e

• e
• +
• b

W + W

• b

H (f ) g g g b e

• e
• +
• b

W + W

• t

b b (c) g g g b e

• e
• +
• b

W + W

• t
• t

t (a) g g g b e

• e
• +
• b

W + W

• t
• t

t (b) g g g b e

• e
• +
• b

W + W

• t

b b (d) g g g b e

• e
• +
• b

W + W

• b
• ;

Z (e) g g g b e

• e
• +
• b

W + W

• b

H (f ) g g g b e

• e
• +
• b

W + W

• t

b b (c)

pa pb p5 p1 p2 p3 p4 pR

• pR

2

Σ

Generic resonant enhancement: M 2 Z ∞

−∞

dp2 (p2 − M 2)2 + (MΓ)2 ∼ M Γ

Γ/M 1% → nonresonant contribution typically negligible

Theoretically consistent method to extract dominant resonant contribution?

Introduction 7 / 29

SLIDE 8

Resonant production and decay factorization

◮ Off-shell MC generators with complete |M|2 at tree level

use WHIZARD, MADEVENT, SHERPA, ... (but: often requires too much CPU time)

◮ Precise LHC/ILC predictions: need NnLO calculations, n ≥ 1 ◮ (S)particle decay chains → many-particle final states ◮ Already 4,5,6-leg one-/two-loop calculations tech. very demanding

Narrow-width approximation (NWA) to the rescue Sub- and nonresonant/nonfactorizable contributions can be neglected in a theoretically consistent way (gauge inv., ...)

Γ → 0 : D(p2) ≡ 1 (p2 − M 2)2 + (MΓ)2 ∼ π MΓ · δ(p2 − M 2)

• n-shell

Scales occurring in D(p2) → error estimate O(Γ/M) Branching ratio measurement implies NWA (and spin averaging):

BRX ≡ ΓX Γ = σNWA σp ≈ σX P

X σX Introduction 8 / 29

SLIDE 9

NWA accuracy: what to compare to?

Tree-level finite-width schemes [safe → agreement up to O((Γ/M)2)] Running-width scheme (Dyson-resummed propagator) 1/(p2 − M 2) → 1/(p2 − M 2 + ip2Γ/M) not gauge invariant, typically unsafe Fixed-width scheme (FWS) 1/(p2 − M 2) → 1/(p2 − M 2 + iMΓ) not gauge invariant, typically safe Overall-factor scheme MΓ=0 · (p2 − M 2)/(p2 − M 2 + iMΓ) gauge invariant, unsafe for complex resonance structure

Baur, Vermaseren, Zeppenfeld (1992)

Complex-mass scheme M → √ M 2 − iMΓ ⇒ complex cos θW , yt gauge invariant, no known practical problems

Lopez Castro, Lucio, Pestieau (1991); Denner, Dittmaier, Roth, Wackeroth (1999) Accuracy 9 / 29

SLIDE 10

Complex-mass scheme at one-loop level

Denner, Dittmaier, Roth, Wieders (2005) For each unstable particle: real bare mass → complex renormalized mass | {z }

→ free propagator (resummed)

+ complex counterterm | {z }

→ CT vertex (not resummed) ◮ Gauge invariance relations fulfilled order by order ◮ No double counting ◮ 1-loop integrals with complex internal masses required ◮ Potential unitarity violations are of higher order

Applications: radiative corrections to

◮ e+e−(→ W +W −) → 4 fermions Denner, Dittmaier, Roth, Wieders (2005) ◮ H → W +W −/ZZ → 4 fermions Bredenstein, Denner, Dittmaier, M. Weber (2006-7) ◮ (q/¯

q)2 → (q/¯ q)2H Ciccolini, Denner, Dittmaier (2008)

Accuracy 10 / 29

SLIDE 11

Unstable-particle effective field theory

Chapovsky, Vy. Khoze, Signer, Stirling (2002) Beneke, Chapovsky, Signer, Zanderighi (2004) Γ ≪ M → hierarchy of scales → Expand cross section in powers of α and δ ≡ (p2 − M 2)/M 2 ∼ Γ/M

• 1. Integrate out hard momenta k ∼ M in underlying theory:

underlying theory → effective theory with short-distance matching coefficients

• 2. Identify remaining dynamical modes (corresponding to momentum configurations near mass

shell) → field operators → effective Lagrangian

• 3. Match coefficients (up to a certain order in α and δ) to underlying theory by calculating and

comparing on-shell n-point functions ◮ Gauge invariance, resummation of self-energy insertions ◮ Systematic extension to higher orders , no double counting ◮ Well suited for inclusive calculations close to threshold ◮ Not well suited for fully differential calculations ◮ Underlying theory results with “Γ → 0” required for matching

Corrections to e+e− → u ¯ dµ−¯ νµ at WW threshold Beneke, Falgari, Schwinn, Signer, Zanderighi (2008)

Accuracy 11 / 29

SLIDE 12

Limitations of the O(Γ/M) uncertainty estimate

Relative deviation R ≡ σFWS/σNWA − 1 = R(1) + O(Γ2) for scalar process:

mp md md M, Γ mp M M

R(1) = (

M(s−m2 d) π(m2 d−M2)(s−M2)

+[πM(m2

d−M2)(s−M2)(s+m2 p)(s−M2+m2 p)] −1

"

m2

d s(s−M2+m2 p) 2 ln s m2 d

+(s+m2

p)

„

m2

d

“

s

`

s+m2

p

´

−2M2s+M4

”

−M4m2

p

«

ln

M2−m2 d s−M2

+M4m2

p(s−m2 d+m2 p) ln s−m2 d+m2 p m2 p

3 5

) · Γ Note: additional scales (√s, particle masses in production or decay, . . . ) Not { . . . } ≈ 1/M when

√s → M (production threshold ) or md → M

Limitations 12 / 29

SLIDE 13

Scalar process

md M, Γ mp

√s M

0.01 0.1 1 1.01 1.1 2 10

1 − md/M Contour lines (dashed, solid, dot-dashed) for R/(Γ/M) ∈ {−10, −3, −1, 0, 1, 3, 10} with Γ/M = 1% and mp = 0.02 M Dash length increases with magnitude

Limitations 13 / 29

SLIDE 14

Scalar process with distributed

√ ˆ s

Constant PDFs

• σ[NWA](
• ˆ

smax) ∝ 1 s ˆ

smax

dˆ s σ[NWA](ˆ s) · ln s ˆ s and

• R ≡
• σ
• σNWA

− 1

• R

Γ M

0.5 1 1.5 2 2.5 3 2.5 5 7.5 10 12.5 15 17.5 20

• ˆ

smax/M

Cross sections σNWA (dotted) and σ (dashed) and relative deviation

• R/(Γ/M) for md = 0.1 M (solid) [and md = 0.001 M (dot-dashed)],

√s = 3 M, Γ = 0.02 M, mp = 0.01 M

Limitations 14 / 29

SLIDE 15

• R

Γ M

1 1.5 2 2.5 3 10 5 5 10 15 20

• ˆ

smax/M

As before, but with md = 0.9 M.

Limitations 15 / 29

SLIDE 16

Nonscalar process

md M, Γ mp

√s M

0.01 0.1 1 1.01 1.1 2 10

1 − md/M Contour lines (dashed, solid, dot-dashed) for R/(Γ/M) ∈ {−10, −3, −1, 0, 1, 3, 10, 30, 100, 300} with Γ/M = 1% and mp = 0.02 M Dash length increases with magnitude

Limitations 16 / 29

SLIDE 17

Nonscalar process with distributed

√ ˆ s

Constant PDFs

• R

Γ M

1 1.5 2 3 5 7 10 20 20 40 60 80 100

• ˆ

smax/M

• R/(Γ/M) for md = 0.95 M (solid) and md = 0.9 M (dashed),

√s = 10 M, Γ = 0.02 M, mp = 0.01 M

Limitations 17 / 29

SLIDE 18

Nonscalar process with distributed

√ ˆ s

Proton-inspired PDFs e σ′

[NWA](

p ˆ smax) ∝ 1 s Z ˆ

s max

dˆ s σ[NWA](ˆ s) Z 1

ˆ s/s

dx x f(x)f( ˆ s xs), f(x) ∝ (1 − x)9/√x and e R′ ≡ e σ′/e σ′

NWA − 1

• R′

Γ M

1 1.5 2 3 5 7 10 20 10 10 20

• ˆ

smax/M

• R′/(Γ/M) for md = 0.95 M (solid) and md = 0.9 M (dashed),

√s = 10 M, Γ = 0.02 M, mp = 0.01 M

Limitations 18 / 29

SLIDE 19

MSSM example at the LHC

u ˜ g ¯ d ˜ dL ¯ s

• sL
• χ+

1

¯ s

• sL

˜ g

• uL

u

• χ+

1

¯ d

Γ( g)/M( g) ≈ 6 GeV/600 GeV = 1%, SPS1a

Limitations 19 / 29

SLIDE 20

Resonant 1 → 3 decays in the MSSM

Systematic on-/off-shell deviation analysis for 48 generic processes TI(PI, MI) → T1(p1, m1), T(q, M, Γ) and T(q, M, Γ) → T2(p2, m2), T3(p3, m3)

TI, T1, T, T2, T3 ∈ {scalar (S), fermion (F), vector boson (V)}

Example: S → SS followed by S → SV m3 MI

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 R>50 R>25 R>10 R> 5 ... R< 5

M/MI

R in units of Γ/M = 1%, m1 = m2 = 0 NK, Uhlemann (2008)

Limitations 20 / 29

SLIDE 21

Large deviations at SPS points

process SPS R/(Γ/M) Γ/M [%] ˜ g → d ˜ d∗

L → d ¯

d˜ χ0

1

1a 9.54 0.935 ˜ g → d ˜ d∗

L → d ¯

d˜ χ0

1

5 11.4 0.956 ˜ g → u˜ u∗

L → u¯

u˜ χ0

1

1a 5.98 0.976 ˜ g → u˜ u∗

L → u¯

u˜ χ0

1

5 9.46 0.975 ˜ χ+

1 → ˜

χ0

1W + → ˜

χ0

1u ¯

d 1a 5.21 2.49 ˜ χ+

1 → ˜

χ0

1W + → ˜

χ0

1e+νe

1a 5.21 2.49 ˜ g → ¯ b˜ b2 → ¯ bb˜ χ0

1

4 6.43 1.11 ˜ g → ¯ u˜ uL → ¯ ud˜ χ+

1

9 114 1.19 ˜ g → d ˜ d∗

L → d¯

u˜ χ+

1

9 209 1.19 and many 1 → 3 decay chain segments with |R|/(Γ/M) > 5 → consider all amplitude contributions to i → f

Limitations 21 / 29

SLIDE 22

˜ χ+

1 → ˜

χ0

1u ¯

d at SPS 1a

BR = 1.3% (dominant decay modes: ˜ χ+

1 → ˜

τ +

1 ντ, ˜

ντ1τ +, ˜ νµLµ+, ˜ νeLe+)

˜ χ0

1

¯ d ¯ d u u u ¯ d ˜ χ+

1

˜ χ+

1

˜ χ+

1

˜ χ0

1

˜ χ0

1

W + ˜ uL ˜ d∗

L

˜ χ+

1 → ˜

χ0

1 (W + → u ¯

d): resonant intermediate state → dominant contribution

• 1. decay stage: m1 + M

MI = 0.975 → R/(Γ/M) = 5.21 with Γ/M = 2.49% No QCD corrections for 1. decay stage Also small, nonresonant contributions from ˜

uL and ˜ dL decay channels Total on-/off-shell decay rate deviation RW +˜

uL+ ˜ dL = 11%

Limitations 22 / 29

SLIDE 23

Origin of amplified deviations

Off-shell vs. NWA cross section: σ = 1 2s dp2 2π D(p2)

• dφp(p2)
• dφd(p2) |Mr(p2)|2

σNWA = 1 2sKNWA

• dφp(M 2)
• dφd(M 2) |Mr(M 2)|2

with KNWA =

1 2MΓ =

∞

−∞ dp2 2π D(p2)

Deforming factors in phase space element and |Md|2: dφd dΩ∗ = 1 32π2

• 1 − (m2 + m3)2

p2 1 − (m2 − m3)2 p2

• =

1 32π2 p2 − m2 p2 for m ≡ m2, m3 = 0

md M, Γ

Strategy for improvements: absorb amplifying factors into K

Improvements 23 / 29

SLIDE 24

Breit-Wigner shape deformation by threshold factors

• R ≡ KINWA/KNWA =

Z p2

max

m2

dp2 2π 1 (p2 − M2)2 + (M Γ)2 (p2 − m2)2/p2 (M2 − m2)2/M2 ! , „Z ∞

−∞

dp2 2π 1 (p2 − M2)2 + (M Γ)2 «

0.98 0.99 1 1.01 1.02

p p2/M

Integrand of numerator (solid) and denominator (dashed) for m = M − 2Γ and Γ/M = 1%

Improvements 24 / 29

SLIDE 25

Explicitly

e R (≡ R′

2) = 1

π » tan−1 β2 γ + tan−1 λ γ – + γ π "„ 2 β2 − 1 « ln λ β2 + „ 1 β2 − 1 «2 ln p2

max

m2 #

with γ ≡ Γ/M, β = p 1 − m2/M 2 and λ ≡ p2

max/M 2 − 1

(R′

2 − 1)/(Γ/M)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 10 10 20 30

m/M p p2

max/M ∈ {1.05, 1.1, 2, 10}, dash length decreases with increasing p2 max, Γ/M = 1% Improvements 25 / 29

SLIDE 26

MSSM example at the LHC

u ˜ g ¯ d ˜ dL ¯ s

• sL
• χ+

1

¯ s

• sL

˜ g

• uL

u

• χ+

1

¯ d

Γ( g)/M( g) ≈ 6 GeV/600 GeV = 1%, SPS1a

Improvements 26 / 29

SLIDE 27

Application to MSSM cascade decay at the LHC

pp → ˜ g uL followed by the cascade decay ˜ g → ( sL → χ−

1 c)¯

s

(σoff-shell/σ[I]NWA − 1)/(Γ/M)

1.05 1.1 1.15 1.2 10 20 30 40 50 60 70

M(e sL)/M(e χ−

1 )

(σoff-shell/σ[I]NWA − 1)/(Γ/M)

0.92 0.94 0.96 0.98 10 20 30 40 50 60 70

M(e sL)/M(˜ g)

Variable strange squark mass approaching the chargino (left), gluino mass (right) Deviations for the standard NWA (diamonds) and an improved NWA (boxes) SPS1a

Improvements 27 / 29

SLIDE 28

Pole Approximation

Stuart (1991); Aeppli, v. Oldenborgh, Wyler (1994)

1PI 1PI 1PI

Laurent series expansion of exact amplitude about exact complex pole M 2: A(s) = Vir(s)Vrf(s) s − m2 − Πrr(s) + A′(s) , M 2 − m2 − Πrr(M 2) = 0 , A(s) = R(M 2) s − M 2 + N(s) M 2, R and N are gauge invariant → perturbative expansion

Off-shell phase space → pole approximation better than NWA But: amplifying factors in residual amplitude not mitigated → Unexpectedly large deviations can also occur in pole approximation

Improvements 28 / 29

SLIDE 29

Conclusions

◮ Unstable particles → formal and practical issues in QFT ◮ Consistent perturbative methods for complete amplitudes exist,

but such calculations are either very involved or not yet feasible

◮ For resonant cross sections the nonresonant contribution is suppressed ◮ Consistent methods to extract dominant resonant contribution exist

For configurations with kinematical bounds in an extended “threshold”- vicinity of resonances the conventional approximation uncertainty esti- mate O(Γ/M) is unreliable, because the p2-dependence of the phase space elements and residual matrix elements causes a significant distor- tion of the Breit-Wigner peak and tail Case study: MSSM benchmark scenarios feature decay chains with sim- ilar intermediate masses leading to approximation errors that are of the

• rder of QCD corrections when on-shell intermediate states are used

A suggestive approach exists to improve the NWA in such cases

Conclusions 29 / 29