EW Physics in High Energy Collisions

EW Physics in High Energy Collisions

LNF: ILC conference October 22, 2006

Denis Comelli

INFN of Ferrara

Work done in collaboration with Paolo Ciafaloni and Marcello Ciafaloni

Perturbation Theory→large log corrections at high energy

∆σ

σ = α



 

 

Log

Q

M

+ Log Q

M

| {z }

LHC+ILC

+ 1 + o( m

Q

| {z }

LEP

)



 

 

IN QCD and QED only single logs (α Log

) for sufficient inclusive observables! Typical size of one loop logs:

αW

4π Log²(1 T eV )²

M_W² = 6.7%, αW

4π Log(1 T eV )²

M_W² = 1.4%

α_W

α_S ∼ 0.36, α_S

4π Log(1 T eV )²

M_W² = 3.6%

Different kinds of Logs

• (Single Ultraviolet Logs) Running (renormalization) of the EW parameters from MW to Q.

• ( Double + Single Infrared Logs) Mass Singularities (Collinear and Infrared)

• Large αWLog^{2 Q2}

M 2W

corrections from virtual and real W^′s for cross sections with Q ≫ MW → Big Impact on LHC and ILC Physics.

• Eventually we need an improved perturbative expansion → all order resummation

• Two classes of Observables:

– Exclusive observables: (σ

) Virtual EW Corrections (EW Sudakov Form factor)

– Inclusive observables: (σe

e

→ q ¯ q + X

+ W

s) Virtual plus Real EW corrections

• Theoretical control of the large logs:

– KLN Theorem, Block-Norsieck Theorem for QED and for QCD

– Violation of Block-Norsieck Theorem for Spontaneously broken theories

• Phenomenology at ILC: e⁺e⁻ → qq¯+ XQCD + SoftW^′s and at LHC: pp → (t¯b, b¯t, t¯t)

| {z }

Q−Scale

+ XQCD + SoftW^′s

| {z }

QCD+EW“Background′′

QCD versus EW

• QCD, QED: massless gauge bosons ⇒ real radiation has to be added

• QCD: Confinement ⇒ average over colour charges⇒ compensation of leading double logs and factorization of collinear singularities

• EW: massive gauge bosons ⇒ real radiation not necessarily required

Free non abelian charges ⇒ average over weak charges not required

Not complete cancellation of double logs in inclusive observables, no

simple factorization

Pictorial idea of the transition of a particle (e

) from “effective” abelian charge (clouds of photons ) for low energies (Q ≤ M

) to “effective” non abelian charge (clouds of W ’s) at high energies (Q ≫ M

).

σ

W W

W γ γ

γ γ

γ γ

Unbroken Phase

Q >> M Q < M

Broken Phase: Abelian charge

Non Abelian charge

γ γ

e

e

Corrections to σ(e

e

→ 2 Jets + X

+ sof t W

s )

100 400 700 1000

1 3 5

Weak unp.

Weak LL Strong

Q

%

∆σ

σ ≃ α

(Q)

π + α

4π Log

Q

M

Violation of BN Theorem for spontaneously broken U (1) gauge abelian theories

In an unbroken abelian U (1) theory (see QED) mass eigenstates coincide with the gauge eigenstates.

In a broken abelian U (1) theory (see U

(1)) mass eigenstates do not coincide necessary with the gauge eigenstates.

Asymptotic states that are a coherent superposition of different gauge eigenstates:

• transverse polarized fermions are a coherent superposition of left and right fermionic gauge charges.

• Z and the photon are a superposition of the B

and the W

gauge fields.

• In the scalar sector the higgs and the longitudinals goldstone bosons

are a superposition of the gauge doublets Φ and Φ

(having opposite

U(1) charges).

ILC with Transverse Polarized e

e

beams: σ(e

e

→ 2Jets) Asymptotic states: |f

i ∝ |f

i + |f

i

Cross section:

-1 -0.5 0.5 1 CosHΘL

-4%

-2%

2%

4%

6%

8%

100 DΣ

€€€€€€€

Σ

Φ= Π

€€€€€€€

2 Φ= Π

€€€€€€

3 Φ= Π

€€€€€€

4 , UnP Φ=0

Figure 1: BN Electroweak corrections to the differential cross section for fully transverse polarized beams at √

s = 1 TeV for various values

of φ (azimuthal angle).

EW BN corrections at LHC :

Heavy Quark Production at high energy

P P → Q ¯ Q + X where Q ¯ Q = t¯ t, t¯b, b¯ t, b¯b two kind of observables:

“Sudakov

: (P P → tagged final states + X) with W, Z / ∈ X

“BN

: (P P → tagged final states + X) with W, Z ∈ X

Cross sections Isospin Structure Tree Level (X = 0) BN corrections δ_ij α²_s α²_sα_w log²sˆ pp → Q_iQ¯_j + X

i 6= j α²_w α²_sα_w log²sˆ

u

d

t

b W

W u t

d b

W

u

u

g W

t b

b

Tree level cross section

Virtual EW corrections

Real EW corrections

α_W

α²W

α_S

Log² Q/M2 ²

α_W Log² Q²/M²

1000 1500 2000 2500 3000 3500 4000

!!!!s`

-17.5 -15 -12.5 -10 -7.5 -5 -2.5 DΣ

€€€€€€€€

Σ^H %

8pT>400 GeV<

DΣ_BN

€€€€€€€€€€€€

Σ^H DΣ_Sud

€€€€€€€€€€€€€€

Σ^H

Sudakov and BN electroweak corrections at Leading Log order for the process P P → t¯t (similar

results hold for P P → b¯b) in the massless limit. Effects of radiative corrections on dσ

dsˆ, integrated over θ for pt > 400 GeV, ˆs being the partonic c.m. energy (which is also the t¯t invariant mass)

1500 2000 2500 3000 3500 4000

!!!s 5

10 15 20 25 30

Σ_BN

€€€€€€€€€€€€€€

Σ_Sud 9 dΣ

€€€€€€€€€€

d s` =

PP®tb

PP®bt

√sˆ dependence of the ratio of BN and Sudakov cross sections for p⊥ ≥ 400 GeV. Dashed (red) line for b¯t, continuous (blue) line for t¯b.

Sudakov versus BN EW corrections:

• Sudakov EW corrections:

σ^Sud(s) = σH e^{−LW (s)}

P

i(ti(ti+1)+y2

i tan2 θW )

ti, yi : i-th leg isospin and hypercharge, LW(s) ≡ ^αW_4π Log^{2 s}

M 2W

• BN EW corrections: The hard overlap matrix is first decomposed in total t-channel isospin basis:

O^H =

X

t1,t2...tn

O^H

t1t2...tnP_t1t2...tn

O^H_t1t2...tn → O_t1t2...tn(s) = e^{− 12LW (s)}

P

i=1 ti(ti+1)

O_t1t2...tn^H inclusive (BN) and exclusive (Sudakov) case significantly different patterns of

radiative corrections:

• For Sud ti is the external leg isospin (e.g., ti = ¹₂ for a fermion) while for BN ti is obtained by composing two single-leg isospins (e.g., ¹₂ ⊗ ¹₂ =0 or 1 for a fermion).

• There is a factor 2 of difference in the argument of the exponentials.

• while Sudakov corrections always depress the tree level cross section,

Structure Function approach to EW Correction: Probability to find a parton inside another parton.

σΗ p₁ z₁

p₂ z₂

σ_ij(p1, p₂, M_W) =

Z

dz₂ z2

X

kl

F_i^k(z1, Q, M_W) σ_kl^H(z1p₁, z₂p₂) F_j^l(z2, Q, M_W)

Evolution eqs for structure functions:

QCD: f

, f

, f

EW :f

, f

, f

, f

, f

, f

, f

, f

, f

, f

, f

, f

Due to mixing effects also mixed structure functions: f

, f

f

d

d = +

d d

= +

b

b b b

f f f

b f

i i

i

i i i

α β α β α β

A B A B A B

Resummation of the tower of logs and structure of the evolution equations:

• QCD : LL α

Log

Q, NLL α

Log

Q,...

∂f (x, µ)

∂logµ = α

Z

x

dz

z f( x

z , µ) P [z] as t = 0 for EW

• EW : LL α

Log

Q, NLL α

Log

Q,...

∂f

(x, µ)

∂logµ = α

Z

x

dz

z f

( x

z , µ) P

[z]

∂f

(x, µ)

∂logµ = α

Z

x

dz

z f

( x

z , µ) (P

[z] + Log µ M

P

[z])

EW Evolution eqs for the system (v, e)

and (W

, W

):

e e

ν

W⁻ W⁺

e e

e

W

e e

ν ν

W W

e

e e

e

− 3

3

−

W³

5 coupled eqs



 

 

 



 

 

 



− 4παW ∂fν

∂t = fν ⊗ (3PV

f f + P R

f f) + 2fe ⊗ PR

f f + 2f+ ⊗ PR

gf + f3 ⊗ PR gf

− 4παW ∂fe

∂t = fe ⊗ (3PV

f f + P R

f f) + 2fν ⊗ PR

f f + 2f− ⊗ PR

gf + f3 ⊗ PR gf

− 2π αW

∂f+

∂t = f¯e +fν

2 ⊗ PR

f g + f3 ⊗ PR

gg + f+ ⊗ (PR

gg + 2PV gg )

− 2παW ∂f−

∂t = f¯ν +fe

2 ⊗ PR

f g + f3 ⊗ PR

gg + f− ⊗ (PR

gg + 2PV gg )

− 2παW ∂f3

∂t = f¯e +fe +fν +fν¯

4 ⊗ PR

f g + (f− + f+) ⊗ PR

gg + 2f3 ⊗ PV gg

“Diagonalization” of the system of eqs: change of variables

f L

(1) = Tr{t3 F f

} = fν − fe 2

, f g

(0) = f+ + f3 + f−

3

, f g

(1) = f+ − f−

2

, f g

(2) = f+ + f− − 2f3 6

5 factorized eqs.



 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 



 

 

 

 

− ∂

∂t f L

(0) = αW 2π



3 4 f

L

(0) ⊗ (PR

f f + P V

f f) + 3 4 f

g

(0) ⊗ PR gf



− ∂

∂t f g

(0) = αW 2π



2 f g

(0) ⊗ (PR

gg + PV

gg ) + 1 2(f

L

(0) + f L¯

(0)) ⊗ PR f g





 



 



− ∂

∂t f L

(1) = αW 2π



f L

(1) ⊗ PV f f −

14 f L

(1) ⊗ (PR

f f + P V

f f) + 12 f g

(1) ⊗ PR gf



− ∂

∂t f g

(1) = αW 2π



f g

(1) ⊗ PV gg + f

g

(1) ⊗ (PR

gg + PV

gg ) + 1 2(f

L

(1) + f L¯

(1)) ⊗ PR f g





− ∂

∂t f g

(2) = αW 2π



3 f g

(2) ⊗ PV gg − f

g

(2) ⊗ (PR

gg + PV gg )



EW Corrections at high energies:

• Are “strong” for Q ≫ M

!

• Show the full IR structure of non abelian theories!

• In general affect any cross section with at least two external SU (2) charged legs.

• At ILC leading also in inclusive QCD observables (σ(e

e

→ q¯ q + X))

• Very large impact on Heavy quark production at LHC !