Electroweak Correction for the Study of Higgs Potential in LC

Electroweak Correction for the Study of Higgs Potential in LC

LAPTH-Minamitateya Collaboration LCWS05

2005.3.19, Stanford U.

presented by K.Kato(Kogakuin U.)

Introduction

• LC ： high-precision experiments

Æ Requires the same level theoretical prediction = Needs higher order

calculation

• Higgs study : Central target

Æ Interesting channels = multi-body final states

• 1-loop correction to e⁺e^- Æ 3, 4-bodies : Great Progress since Sept. 2002

Higgs@LC:tree cross sections

2Æ3,4 processes important

Main channels

WWH vs ZZH

Higgs Potential Yukawa coupl.

number of diagrams

⁽GRACE:NLG model)

tree 1-loop

4(1) 341(119)

RADCOR / Loops & Legs Sept. 2002

12(2) 1350(249) 12(6) 2327(758) 27(6) 5417(1597) 42(2) 4470(510)

−

→

+

e e

ν H

ν

ZHH ZH

H t t

H e

e^{+ −}

A.Denner, J.Kublbeck, R.Mertig, M.Bohm, Z.Phys.C56(1992) 261;

B.A.Kniehl,

Z.Phys.C55(1992) 605.

with(without) e-scalar couplings 10 years

required to develop 2Æ3

tools

Automated Systems

Full 1-loop RC available

H t t e

e

^{+ −}

→

ZHH e

e

^{+ −}

→

H e

e

^{+ −}

→ ν ν

GRACE, PLB559(2003)252

Denner et al., NPB660(2003)289 GRACE, PLB571(2003)163

You et al., PLB571(2003)85

Denner et al., PLB575(2003)290 GRACE, PLB576(2003)152

Zhang et al., PLB578(2004)349

H e

e e

e

^{+ −}

→

^{+ −}

γ ν ν

−

→

+

e

e

ν =ν_μ,ν_e

c s d u e

e^{+ −} →

ν

τ

τ

⁺

μ

⁻

ν

μ , Denner etal., hep- ph/0502063

GRACE, PLB600(2004)65

GRACE, NIM A534(2004)334

HH e

e

^{+ −}

→ ν ν

GRACE, Talk by Y.Yasui at Durham(Sep.2004)

User input

Theory

model

Diagram generator file

amplitude generator

Library

CHANEL, loop Kinematics

library

kinematics

code generated code

diagram description

convergence information

Make file etc.

symbolic code

REDUCE, Form etc.

PS file

Drawer

BASES(MC integral)

SPRING (EG manager)

parameter file

Diagrams (figure)

Events Cross sections

distributions

TREE LOOP

FORTRAN code .fin .mdl .rin

Higgs Potential

H H

H

H H

H H

Something required to provide mass

Key for the structure of standard model

Window to New Physics

Double (triple) Higgs production Process

e

⁺

e

^-

Æ ννＨＨ

400 600 800 1000

0 1 [ab/GeV]

e⁺e⁻−−>hhνν− (without Z)

m_h =120 GeV RS = 1000 GeV (Λ+δΛ)/Λ

0.00.5 1.01.5 2.0

Invariant mass of hh [GeV]

tree

old parameter set HHH coupling

dependence SM: δΛ＝０

ννＨＨ ｖｓ ＺＨＨ

ZHH/nunuHH

0 100 200 300 400 500 600

0 500 1000 1500 2000 2500

W

σ(ab)

GeV

= 120 MH

ν HH

ν

Tree

ZHH

ZHH e

e

^{+ −}

→

HH e

e

^{+ −}

→ ν ν

dominant mechanism (approximation) Low W

High W

W +

W −

H H

FULL calculation

e

⁺

e

^-

Æ ννＨＨ

W

⁺

W

^-

Æ HH

W + _W ⁻

H

Annihilation B^oson

Exchange

C^ontact

Interaction

H

H _{W , χ}

W

⁺

W

^-

Æ HH

• tree … 6 diagrams

Annihilation, Boson exch., Contact int.

B ÅÆ A, C … opposite sign

• 1-loop … 827 diagrams Æ 13 groups

A : vertex corr .HHH, WWH, AWW, ZWW B : vertex corr. WWH, WχH

C : Box

Check : Cuv independence

Cuv=0 Cuv=100 Cuv=10000

LG

NLG Cuv=0 Cuv=100 Cuv=0

Linear Gauge vs Non-Linear Gauge

leading m

_t

formulas

2 2

4

3 2 _{W H}

t C

w

H M M

N m C s

π

− α

=

2 2 2

2

2 12

3 2

8 _w

W W

t C

w

W s

s M

N m

C s +

−

= π

α

2 2 2

2 4 2

6

3 8

8 _w

W W

t C

w s

s M

N m

C s +

−

= π

α A: C_H+C_W

B: 2C_W C: C₄

M_W=80.4163 M_Z=91.1876 M_H=120 m_t=180

α=1/137.0359895

-0.11779616

-0.07033663 -0.02535196

) 5 , 4 , 3 , 2 , 1 (

GeV 10

180× ^{( 1)} =

= ⁻ n

m_t ⁿ

proof of the formulas and

the performance of GRACE system real world : NOT m_t >> W, M_H, M_Z,…

OK

WWHH(tree)

0 1 2 3 4 5 6

0 500 1000 1500 2000

W

σ(pb) tree+leading mt correction

full EW correction ^{(w/o QED)}

δ(EW) & δ(mt)

-5%

0%

5%

10%

15%

20%

0 500 1000 1500 2000

W

δ(%)

W

⁺

W

^-

Æ HH

WWHH 1-loop corrections

-100%

-80%

-60%

-40%

-20%

0%

20%

40%

0 500 1000 1500 2000

W

δ(%)

δ^V

λ=10^-12 Gev δ^V_QED δ_W

δ_W(Gm)

e

⁺

e

^-

Æ ννＨＨ

• number of diagrams

ν_e ... tree= 81 loop=19638 (prod. set = 12 x 3416)

50M lines of Fortran code ν_mu ... tree= 27 loop=8292

(prod. set = 6 x 1754)

e

⁺

e

^-

Æ ννＨＨ

ZHH e

e

^{+ −}

→

hep-ph/030910

Phys.Lett.B 576(2003)152.

“s-channel”

e

⁺

e

^-

Æ ννＨＨ

nunuHH

0 5 10 15 20 25

0 500 1000 1500 2000 2500

W

δ（％）

) EW

δ (

) G ( _μ δ

nunuHH(tree)

0 100 200 300 400 500 600

0 500 1000 1500 2000 2500

W

σ(ab)

Full

calculation M_H=120GeV

e

⁺

e

^-

Æ ννＨＨ

0 1 2

0 1

GeV

=120 MH

W=0.5TeV W=2.5TeV W=2.0TeV W=1.5TeV W=1.0TeV

M(HH) (TeV) Tree

Invariant mass of HH

• Higgs Potential channel eeÆννＨＨ cross section and its EW correction is calculated.

σ＝Ｏ（１００ ａｂ）and greater than eeÆZHH in high-E LC region ( ≧1TeV). δ(Gmu) is

~10% for >700GeV. Large diviation from eeÆZHH dominated by s-channel.

• WWÆHH is studied to check the t-channel structure of eeÆννＨＨ and has shown

validity and limitation of the leading m_t formulas.

Summary

s s

e

⁺

e

^-

Æ ννＨＨ

H e

e

^{+ −}

→ ν ν

)

W (s δ

hep-ph/0212261

Phys.Lett.B 559(2003) 252.

s,t channels show different behavior

genuine weak correction in G-scheme is -2～-4%

total )

(α O

G

δ

W

δ

W

)

W(t

δ

16 1.5%

5

2 2

2

HWW = − = −

W W

t

M s

M π δ α

H t t e

e

^{+ −}

→

hep-ph/0307029

Phys.Lett.B 571(2003) 163.

system components

• Diagram generation for input process

• Amplitude/Matrix element generation

• Kinematics and Integration (efficiency)

• Event generation (efficiency & weight)

• Peripheral tools: rule generator,

diagram selection, QED radiation, PDF, loop integral library, multi-process, color flow and interface for hadronization, etc.

5-point functions

∑ ∫

=

4 3 2 1 0

5 G d D D D D D

I

σ ν

μ σ

μν l l Ll

L l

j j

j r X

D

X D

+

⋅ +

=

+

=

l l

l

2 2

0 2

0

F D

E

N =

∑

+

= 4

) (

α α l α

rank M

BOX rank M-1

0 0

2 = D − X

l

∑

=

+

= ⁴

0

)]

( [

1

α aα bα_j lr_j Dα

scalar 5-pnt

BOX

ν μ

μν

j ij i

j i

ij r r g r A r

A = ⋅ = ⁻¹

] [

)

( ¹₂ ¹ _{0 0}

1

j j

ij i

j ij

i A r r A D D X X

r = − + −

= ^{μ − ν μ −}

μ l

l

N

δ (QED), δ (EW)

) 1

0

( δ

QED

δ

W

σ

σ = + +

δ

W

hard QED soft

QED V

QED

QED

δ δ δ

δ = + +

f

LL

( ) ∫ ( )

∫

^{+ ⊗ + − ⊗}

= LL hard LL

v QED

QED dσ ₀δ dσ~₀ f dσ₁ dσ~₀ f

δ _γ

=radiator

phase space subtraction

non-QED virtural corrections

Non-linear gauge fixing terms

( )

²

( )

²

2 1 2

1

1 _{Z A}

Z W

GF F F F F

L ^{= −} ξ ⁻ ξ ⁻ ξ

− +

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛ + ±

+

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛∂

=

±

±

±

±

±

χ χ κ χ

δ χ

ξ

β

α ^{μ μ} _μ

μ

3

~ 2

~ 2

~ ~

W W

W W

W W

s i e s H

M e

W s Z

i ec A

ie

F m m

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛ +

+

∂

= ₃ ~ ₃

2 ε χ

χ

μ ξ

μ H

c s M e

Z F

W W W

Z Z

μ μ

A F ^A = ∂

Samples of NLG Feynman rules

)]

)(

~ / 1

(

) )(

~ / 1

(

) (

[

1 2

3 3

2 1

μρ ν

νρ μ

νρ μ μρ

ν ρ μν

ξ α

ξ α

g p g

p

g p g

p p p

g e

A W

W

W W

− +

+

− +

+

−

−

−

α μν

χ

g ieM

A W

W (1− ~)

−

− m

αgμν

e

W A

c

c ^A

~

− 2

−

−

− ^±

m

W W

A

ie s

H c

c

ξ δ χ

~ 2

2 1 m

m − − ^± −

ghost-ghost- vector-vector / ghost-ghost-scalar-scalar modified

• Numerator structure is the same as Feynman gauge Æ Loop integral library

• Vertices modified

• general valuesÆ #diagrams

Non-linear gauge

κ ε

δ β

α ~ , ~ , ~

~ ,

~ ,

) 1 for

(

ξ =

g

μν

χ α~ = 1 ⇒ no AW

Check gauge invariance

ÆIndependence on gauge parameters

“old” usage

Æreduce #diagram