A Higgs

“A Higgs- analysis

Vade Mecum”

ABRIDGED VERSION

Find a “Higgs” Candidate Particle

σ [Prod] × BR(s) ∼ STANDARD

Very L-dt demanding What is it’s J, P, C ???

What is it’s ^not J, P, C ???

Is it ^{Pure J ?}

^PC

^{Mixed ? Compo} _{- site ?}

Is it *the Higgs*

or an Impostor ?

2

Duhrssen, ATL- PHYS-2003-030

√ s = 14 TeV

WARNING

Not a WORD of

what I am saying is

TRULY NOVEL

Motivation ?

•

A good fraction of the theoretical considerations

(e.g. regarding C and CP properties of the observed object and the use of Bose statistics. or of the helicity formalism are not right)

_

Some fraction of the incorrect answers have made it to the currently-available data-analysis “protocols”

_

+ Some room for improvement :-)

5

The published protocols (that we have found) are not demonstrably the best one could use optimal

_

Surely people are planning to do much betteroptimal

+ ¡¡It pays TO DO IT RIGHT ... NOW!!

The contest:

• Team 1 is doing things “right”

Right from Starters

Teams may or may not

belong to the same experiment

• Team 2 is applying “currently

published” analysis methods

Discovery-time Plots

L = 10 ³³ cm ^{− 2} s ^{− 1}

√ s = 10 TeV

Snowmass factor of 3 (LHC start) Background: Currently estimated

7

Poisson(S,B) ^Team 1 Team 2

∼ 10 fb

⁻¹

Nominal BackGround ~ 6 times more BG

∼ 30 fb

⁻¹

9

While gathering significance faster, Team 1 is checking

agreement with _J ^P _{= 0} ⁺

With a few extra lines of code, Team 1 is checking

whether the “object”

is an impostor , e.g.:

Team 1

∼ 40 fb

⁻¹

Back to

this all detail in

5σ

We ARE

“translating”

to #-standard deviations,

but NOT assuming that anything is

Gaussian

12

Whaaat am I

talking about ?

H or HI → ZZ or ZZ ^∗

Z, Z ^{( ∗ )} → µ ⁺ µ ⁻

e.g.

13

(e ⁺ e ⁻

exercise)

“Pessimise”:

•

To OPTIMIZE

J, P, C

properties of the object and its couplings with NO extra assumptions

14

η p

_T

•

WHATEVER the object happens to be:

•

Not to use relative BRs (more than one H !)

•

Assume and distributions ~ Standard (actually, roughly expected)

d|

|

-10 -8 -6 -4 -2 0 2 4 6 8 10

[GeV/c] Tp

0 50 100 150 200 250 300 350 400 450 500

d|

|

-10 -8 -6 -4 -2 0 2 4 6 8 10

[GeV/c] Tp

0 50 100 150 200 250 300 350 400 450 500

1

0 ⁺

-

M=145 GeV

M=145 GeV

15

6 “handles” !!!

6 dimensions !!??

η

Z-pair decays Z-pair

production

1

θ

₂

µ

⁻

µ

⁺

z y

e

⁻

e

⁺

q

π − Θ Z

₂

θ

Z

₁

ϕ

₁

ϕ

₂

g,

H (HI) Rest System (undo and boosts)

p

_T

1

�

ω ≡ { cos θ

₁

, cos θ

₂

, ϕ ≡ ϕ

₂

− ϕ

₁

} Ω � ≡ { cos Θ, Φ ≡ ϕ

₂

}

M

^∗

in H (HI ) → ZZ

^∗

Exp. method is not new

• (Para)-Positronium

•

Helicity can be measured: Thomson scattering

•

J.A. Wheeler (Ann .N.Y. Acad.Sci. 48 (1946) 219) got the theory wrong

•

Sneyder, Pasternak & Hornbostel (P.R. 73 [1948] 440) and Pryce and Ward (Nat. 160 [1947] 435) got it right

•

Bleuler & Bradtʼs experiment (P.R. 73 [1948] 1398) was not conclusive and Hannaʼs (Nature 162 [1948] 332) was wrong.

•

C.S. Wu & Shaknov (P.R. 77 [1950] 136) did it right !!!

Hopefully history will not fully repeat

[J ^P = 0⁻] → (γ_Rγ_L − γ_Lγ_R)

Why is this so deeeeeply

interesting?

� = ^{P (cos θ}

¹

^{, cos ¯ θ}

²

⁾

P ˆ (cos θ

₁

)

Spooky Action at a distance

θ

₂

µ

⁻

e

⁻

q

Z

₂

θ

₁

Z

₁

P (cos θ ₁ , cos θ ₂ )

(γ

₁

) (γ

₂

)

What used to be paradoxes (EPR) first became the

QM paradigms and are now

DISCOVERY PROCEDURES

Spin

H (k )

Z ^µ (p ₁ ) Z ^ν (p ₂ )

X g

_µν ^Standard

=

L − inv

Pseudoscalar

+(P + i Q)�

_µναβ

p

^α₁

p

^β₂

M

_Z²

Derivative

− (Y + i Z ) k

_µ

k

_ν

M

_Z²

21

Some

Inteferences

XP is T-odd XQ is C-odd

XY is C,CP-even

=

L − inv

Vector X (g

^ρµ

p

^α₁

+ g

^ρα

p

^µ₂

)

+(P + i Q) �

^ρµα

(p

₁

− p

₂

) Axial

Spin 1 Z ^µ (p ₁ )

H

^ρ

(k ) Z ^α (p ₂ )

L = 1

Λ² (∂^µ H^α + ∂^α H^µ) (A₁ W� _µ^λ · W� _αλ + A₂ B_µ^λB_αλ) + 1

Λ² �^µναρ[A₃(W� _µ^λD_αW� _νλ)H_ρ + A₄(B_µ^λ∂_αB_νλ)H_ρ]

Some

interferences XP in T-odd

XQ is C-odd

�

ω ≡ { cos θ

₁

, cos θ

₂

, ϕ ≡ ϕ

₂

− ϕ

₁

}

angles: Kept^Lepton-

Ω � ≡ { cos Θ, Φ ≡ ϕ

₂

}

(Z-pair)-angles: Averaged

1

θ

₂

µ

⁻

µ

⁺

z y

e

⁻

e

⁺

q

π − Θ Z

₂

θ

Z

₁

ϕ

₁

ϕ

₂

g,

In the Literature

1

Deletes polarization-memory for spin > 0 Good only for uniform 4 acceptance π

23

Literature

H → Z Z

^∗

Median significance curves

Disfavouring Axial vs Vector

c ≡ cos Θ , s ≡ sin Θ c₁ ≡ cos θ₁ , s₁ ≡ sin θ₁

c₂ ≡ sin θ₂ , s₂ ≡ sin θ₂ m²_d ≡ m²₁ − m²₂ ,

M₁² ≡ M ² − 3m²₁ − m²₂ , M₂² ≡ M ² − m²₁ − 3m²₂ , M₃² ≡ M ² − 2(m²₁ + m²₂) , M₄² ≡ M ² − (m²₁ + m²₂)

P ²g₁�

M²s²s²₁s²₂ �

M₂⁴m²₁ cos 2(φ + ϕ) + M₁⁴m²₂ cos 2φ� +8m²₁m²₂m⁴_ds² �

(c²₁ + c²₂ + s²₁s²₂ sin ϕ²) + 2¯η²c₁c₂� +(1 + c²)M²�

2M₁⁴m²₂s²₁ + 2M₂⁴m²₁s²₂ − (M₂⁴m²₁ + M₁⁴m²₂)s²₁s²₂�

−8M m²_dm₁m₂c s �

M₂²m₁s₂ �

c₂s²₁ sin (φ + ϕ) cos ϕ + c₁(c₁c₂ + ¯η²) sin φ�

−M₁²m₂s₁ �

c₁s²₂ sin φ cos ϕ + c₂(c₁c₂ + ¯η²) sin (φ + ϕ)��

+2M²M₁²M₂²m₁m₂s₁s₂�

(1 + c²)(c₁c₂ − η¯²) cos ϕ − s²(c₁c₂ + ¯η²) cos ϕ + 2φ��

dPDF[Pure 1⁺]

dm^∗ d cos Θ dΦ d cos θ₁ d cos θ₂ dϕ ∝ P S(M, m₁, m₂ = m^∗)× g₁ ≡ �

c²_v + c²_a�2 �

g_v² + g_a²�

¯

η ≡ 2 c_v c_a

c²_v + c²_a

Z → l

⁺

l

⁻

g_v,a : qq¯ → HI

25

?

DIS-Favouring VECTOR interpretation of SCALAR signal Full

Full Full

Averaged Z-pair angles

P (x ₁ , ..., x _n )

P (x ₁ ) = �

P dx ₂ ...dx _n

P (x

₁

, ..., x

_n

) → P (x

₁

)...P (x

_n

)

P (x _n ) = �

P dx ₁ ...dx _{n − 1}

..

To handle multi-dimensional PDFs and their likelihoods, occasiona"y su#ested to:

27

5σ

P (x₁)...P (x_n)

�1.0

�0.5

0.0 0.5

1.0

�1.0

�0.50.00.51.0

4 6 8 10

�1.0 �0.5 0.0 0.5 1.0 4

5 6 7 8 9 10

1/4 1/2

3/4 cos(θ

1) = 1 0

cos(θ

2) ^{0 1 2 3 4 5 6}

6 7 8 9

ϕ cos(θ₁) = 0

1/4

1/2 3/4

1

0 ⁺

M=200 GeV

cos(θ₁) cos(θ₁)

cos(θ₂) ϕ

0

2

4

6

�1.0 �0.5 0.0 0.5 1.0

6 7 8 9

29

�1.0

�0.5 0.5 0.0

1.0

�1.0

�0.5 0.5 0.0

1.0

4 6 8 10

cos(θ₁)

cos(θ₂)

P[cos(θ

₁

), cos(θ

₂

)] P[cos(θ

₁

)] ∗ P[cos(θ

₂

)]

cos(θ₁)

cos(θ₂)

�0.5 �1.0 0.0

0.5 1.0

�1.0�0.5 0.0 0.5 1.0 6 8 10

Spin 0

H (k )

Z ^µ (p ₁ ) Z ^ν (p ₂ )

X g

_µν ^Standard

=

L − inv

Pseudoscalar

+(P + i Q)�

_µναβ

p

^α₁

p

^β₂

M

_Z²

Derivative (Composite)

− (Y + i Z ) k

_µ

k

_ν

M

_Z²

31

Some

Inteferences

XP is T-odd XQ is C-odd

XY is C,CP-even

r

Polarisation = v

c � 100%

µ ⁺

e ⁺

p

σ

Demonstrated maximal parity violation in a purely leptonic decay

O ^P ≡ �σ · p � ^P-odd

O

^P

→ − O

^P

{ � x → − � x }

EXP: J. Duclos, J. Heintze,V. Soergel, ADR; Phys. Lett. 9, 62 (1964).^.

�O

^P

� � = 0 ^(Γ ∝ A + B O

^P

)

e ⁺ ↔ e ⁺ _R

H => ZZ statistics will be In-Sufficient: You HAVE

O

^T

≡ p �

_e⁺

· p �

_µ⁺

× � p

_µ⁻

∝ sin θ

₁

sin θ

₂

sin ϕ

O

^T

→ −O

^T

{ t → − t } T-odd (& P-odd)

�O

^T

� � = 0

At the LHC

You HAVE NOT proven that T ( CP) is violated

Reminder ?

S → S

_{f i}

_| S |

²

= 1 S ≡ 1 + i T

T − T

^†

= i T T

^†

≡ i A

T

_{f i}

− T

_if^∗

= i A

^{f i}

≡ �

Γ

T

_{f Γ}

T

_iΓ^∗

�

Γ

|Γ��Γ| = 1

}

O ^T

Assume T (or CP) -inv

| T

_{f i}

|

²

− | T

_f˜˜i

|

²

= O (T

³

) + O (T

⁴

)

| T

_{f i}

|

²

= | T

_if

|

²

+ 2 Im( A T

^∗

) − |A|

²

H ^W

W

W ^Z Z H

T T

T

∼ : { � p → − � p, �σ → − �σ }

S → S

_{f i}

_| S |

²

= 1 S ≡ 1 + i T

T − T

^†

= i T T

^†

≡ i A

T

_{f i}

− T

_if^∗

= i A

^{f i}

≡ �

Γ

T

_{f Γ}

T

_iΓ^∗

�

Γ

|Γ��Γ| = 1

}

O ^T

Assume T (or CP) -inv

| T

_{f i}

|

²

− | T

_f˜˜i

|

²

= O (T

³

) + O (T

⁴

)

| T

_{f i}

|

²

= | T

_if

|

²

+ 2 Im( A T

^∗

) − |A|

²

∼ : { � p → − � p, �σ → − �σ }

� = 0 CP

2] [GeV/c mZZ

200 210 220 230 240 250 260 270 280 290 300

Events

0 10 20 30 40 50 60 70 80

2] [GeV/c mZZ

200 210 220 230 240 250 260 270 280 290 300

Events

0 10 20 30 40 50 60 70

80 Signal+Background

Background

Resonance already there !!

Do not Fourier- Transform!

c1

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Events

0 2 4 6 8 10 12 14 16 18

MC Truth sPlot

PDF Projection

s-Plot

MC Truth

c1

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Events

0 2 4 6 8 10 12 14 16 18

MC Truth sPlot

PDF Projection

cos θ SIGNAL

cos θ = cos θ

₁

or

cos θ

₂

c2

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Events

20 30 40 50 60 70 80 90

MC Truth sPlot

PDF Projection

cos θ BACKGROUND

38

s P lots

s W eights and

F.R. Le Diberder and M. Pivk, BABAR collab.

P. E. Condon & P. L. Cowell, Phys. Rev. D 9 (1974) 2558

STATISTICALLY OPTIMAL FOR A NARROW M(ZZ) and, WITH A BIT MORE EFFORT, FOR A WIDE ONE

s-Separates signal and background

In our case with use of Maximum-Likelihood fit to the M(ZZ) or M(ZZ*) distribution => Ns, Nb

To result in what we need:

a signal-only 5-d (or 6-d) “potato”:

The distribution of events in

all 5 angles (+ M* in H=>ZZ*).

Ideal for testing what the signal is... and is not

Neyman-Pearson test

40

Text

Keep fixed

Λ ≡ log

₁₀

[ L ( H

₁

)/ L ( H

₀

)]

H ⁰ = 0 ⁺ H ¹ = 0 ⁻

‘Simple’ hypotheses (no free parameters).

Test statistically optimal

All a-priori statistical information

H⁰ is “true”. Generate events L(H₀) of events in true theory

L(H₁) of events in alternative theory

Bin events in Construct P(Λ|H0)

Λ H¹ is “true”. Generate events of events in true theory L(H₁)

of events in alternative theory L(H₀)

P(Λ|H1)

Bin events in Construct Λ

Easy to rule out Scalar, if it is Vector

Much tougher to

tell Scalar from Axial Vector

Number of Observed Events

0 20 40 60 80 100 120 140

Significance

0 1 2

3 4 5

Number of Observed Events

0 20 40 60 80 100 120 140

Significance

0 1 2

3 4 5

band 1 m

band 2 m

= 200 GeV/c2

mH

= 0+

H1

= 2+

H0

5 m

3 m

Number of Observed Events

0 20 40 60 80 100 120 140

Significance

0 2 4 6 8 10

Number of Observed Events

0 20 40 60 80 100 120 140

Significance

0 2 4 6 8 10

band 1 m

band 2 m

= 350 GeV/c2

mH

= 0+

H1

= 2+

H0

5 m

3 m

Higgs vs Kaluza-Klein (Heavy Graviton)

46

“Mixed” hypotheses

e.g.: Standard coupling + Opposite-parity coupling

cos ξ L[0

⁺

, ZZ ] + sin ξ L[0

⁻

, ZZ ]

Conventional likelihood approach, as in

N

_S

S [M (ZZ )] + N

_B

B [M (ZZ )]

Not ce!ifiably optimal, but "

alternatives are not significantly $fferent

WITH ONE EXCEPTION !!!

XP: CP-odd, P-even XQ: P-odd, CP-even

ξ ≡ arctan Q X

X g

_µν

^{L −}

^Standard

^inv = ^{Spin 0}

H (k )

Z ^µ ( p ₁ )

Z ^ν (p ₂ )

Pseudoscalar

+(P + i Q)�

_µναβ

p

^α₁

p

^β₂

M

_Z²

XP: CP-odd, P-even XQ: P-odd, CP-even

ξ ≡ arctan Q X

X g

_µν

^{L −}

^Standard

^inv = ^{Spin 0}

H (k )

Z ^µ ( p ₁ )

Z ^ν (p ₂ )

Pseudoscalar

+(P + i Q)�

_µναβ

p

^α₁

p

^β₂

M

_Z²

XZ: CP-odd, P-even XY: P-even, CP-even

ξ ≡ arctan Y X

X g

_µν

^{L −}

^Standard

^inv = ^{Spin 0}

H (k )

Z ^µ ( p ₁ )

Z ^ν (p ₂ )

Derivative (Composite)

− (Y + i Z ) k

_µ

k

_ν

M

_Z²

R(H) ∼ M ⁻¹ (↔ π) e.g.: observable for

some(large) values of

ξ

?

49

A few examples of ...

Dozens of questions and their answers [fair]

[careful]

Some examples of ...

NEXT-TO-LEADING

questions and their answers

e.g.: The Higgs-impostor has zero spin, but is

non-standard. What is its best description ???

51

e.g.: A “composite Higgs” of M ~ 145 GeV

Assumed value of the mixing angle

1 σ

^Measured

?

SHAPE ?

NNLQ (Next to Next to Leading Questions)

Once you know (to some c.l.) what you have found, you can (to some c.l.) measure its parameters

cos(ξ ) g

_µν

− sin(ξ ) k

_µ

k

_ν

/M

_Z²

Two terms with the same Quantum Numbers

Feldman-Cousins belt construction

|M( )|

|M( )| 0

2 2

ξ

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 0.1

1 10

ξ

Very destructive interferences

Analogous angular distributions

Proved by way of example

For given

Detector performance Integrated Luminosity No extra SF

Arguably true as well for many other physics items, which... ...

It pays to have ab-initio (!) an ANALYSIS combining DISCOVERY & SCRUTINY

Readily come to mind ...