QCD AT THE LHC

QCD AT THE LHC

Massimiliano Grazzini (INFN, Firenze)

Italo-Hellenic School, june 2005

•

Introduction:

•

Asymptotic freedom and the parton model

•

Infrared divergences

•

The factorization theorem

•

DGLAP equations

Outline

•

QCD ad hadron colliders: factorization theorem

•

Parton distributions

•

^{QCD tools:}

•

Fixed order calculations: LO, NLO, NNLO

•

Parton shower Montecarlos

•

Resummed calculations

•

Concluding remarks

Corcella’s lectures

Basics

QCD Quantum - Chromo - Dynamics Quantum theory of colour:

It is a non abelian gauge theory with gauge group

Completely specified given the number of colours

L = − 1

4 F

_µν^a

F

^µνa

+

N_f

!

f =1

ψ ¯

_fⁱ

[iγ

^µ

(D

_µ

)

_ij

− m

_f

δ

_ij

] ψ

_f^j

(D

_µ

)

_ij

= δ

_ij

∂

_µ

− igt

^a_ij

A

^a_µ

F

_µν^a

= ∂

_µ

A

^a_ν

− ∂

^ν

A

^a_µ

+ gf

_abc

A

^b_µ

A

^c_ν

Covariant derivative

Gluon field tensor N^c

i, j = 1...N^c SU(N^c)

Difference with QED: gluons are charged and interact among themselves

Feynman rules

α β

i j

p i(/p + m)αβ

p² − m² + i! δ^ij

µ ν

a b

p

i

p² + i! d^µν(p) δ^ab

a¹ µ¹

p¹

µ² a²

p²

a³ µ³

p³

µ¹ a¹ p¹ µ² a²

p²

a³ µ³

p³

µ⁴ a⁴

p⁴ −ig²

!

f^ba1a2f^ba3a4(g^µ1µ3g^µ2µ4 − g^µ

1µ⁴

g^µ2µ3)

+(2 ↔ 3) + (2 ↔ 4)!

−gf^a

1a2a3

!

g^µ1µ2(p₁ − p₂)^µ3

+g^µ2µ3(p² − p³)^µ1 +g^µ3µ1(p³ − p¹)^µ2!

α β

i j

g

µ

−ig(t^a)_ij(γ^µ)_αβ

Gluon spin pol. tensor:

gauge dependent

d^µν(p) =











−g^µν + (1 − α) p^µp^ν

p² + i" covariant gauges

−g^µν + p^µn^ν + p^νn^µ

p · n − n²

p^µ p^ν

(p · n)² axial gauges

d^µν(p) = !

λ

ε^µ_(λ)(p)ε^ν_(λ)(p)

The spin polarization tensor is

and depends on the gauge choice

p

a b

i

p² + i!δ^ab _{a c}

µ b

gf^abcp^µ In covariant gauges Lorentz invariance is manifest but ghosts

must be included to cancel effect of unphysical gluon polarizations

In physical gauges (e.g. , arbitrary direction) only two tranverse polarizations are present

more transparent physical picture: for lowest order or

approximated calculations physical gauges make life simpler

nµA^µ = 0

n

Colour algebra

The calculation of Feynman diagrams is similar to QED: just keep into account colour factors

(T ^a)_ij = t^a_ij (T ^a)^bc = if^abc

T r(t^at^b) = T^Rδ^ab T^R = 1/2

C_F = N_c²

− 1 2N^c

C_A = N^c Useful relations:

(t^at^a)^il = C^F δ_il

f^adcf^bdc = C^Aδ^ab

fundamental adjoint

Exercise: prove the above expressions for

(hint: form a basis for hermitian matrices) The explicit form of colour matrices is not important in practice

t^a, I

C_F , C_A

[T ^a, T ^b] = if^abcT^c

T r(T^a) = 0

N^c × N^c

Asymptotic freedom

QCD is a renormalizable gauge theory

Ultraviolet (UV) singularities appear in loop diagrams but they can be removed by the renormalization procedure

•

Regularization: allows to make sense of divergent loop integrals

•

Subtraction: redefine the coupling

Renormalization has two steps:

α^S = g²/(4π)

∼ α_B!

1 + α_Bβ0

" ^Λ2_cut

p²

d⁴k

(k²)² + O(α²_B)#

∼ α^B!

1 + α^Bβ0(log Λ²cut

µ² + log µ²

p² ) + O(α^B2)"

+

=

UV cut-off

= α(µ²)!

1 + β⁰α(µ²) log µ²

p² + O(α_B²)"

coefficient of UV behaviour

How does the coupling depend on the scale ?

The theory does not predict the absolute value of the coupling but its dependence on the scale can be predicted

α(µ²) ≡ α^B!

1 + β⁰α^B log Λ²cut

µ² + O(α²B)"

dα^S(µ²)

d log µ² = −β⁰α²_S(µ²) + O(α³S) ^{! !"#$}

"

"

%

%

&

' %

%

"_%^# "^#

if for a given scale we have ^αS(µ20) << 1

µ⁰

a perturbative solution of the

renormalization group equation can be given where we have defined

the renormalized coupling

it is one the renormalization group equations

Divergence desappeared but an arbitrary scale is now present

µ

The vacuum around a pointlike charge becomes polarized due to the emission of pairs and produces a screening effect

e⁺e⁻

In QED the dependence of the coupling on the scale has a simple physical interpretation

! !

"

"

"

"

"

"

"

"

"

"

"

"

"

" !

!

!

!

!

! !

!

!

!

!

!

!

! " " "

! !

: the effective coupling decreases with the distance

In QCD gluons are charged and provide a positive contribution to ^β0

n_F < 16

β⁰ = 11Nc − 2nF

12π > 0

for

β⁰ < 0

But:

Intuitively: gluons are

charged and spread colour charge over larger distances:

anti-screening effect

ASYMPTOTIC FREEDOM

In processes with large

momentum transfer we can use perturbation theory

even if we have not solved the full theory

Nobel prize in Physics 2004

D. Gross, H.D. Politzer, F. Wilczek

scale at which the coupling becomes strong

Λ^QCD ∼

Asymptotic freedom

at large tranferred momenta hadrons behave as collections of free (weakly- interacting) partons

at small scales the interaction becomes strong but if we are not interested in the details of hadronic processes (consider inclusive enough final states) we can use the parton picture

annihilation

e

⁺

e

⁻

produce a hard pair at scale (short time scale ) which travel far apart as free

qq¯

Q

_{τ = 1/Q}

Parton model

Deep inelastic scattering

Let us consider the process ep → eX

Q² = −q² = (k − k^!)² " Λ²_QCD

If the cross section is dominated by one-photon exchange

Q² < M_Z²

zp

! q

p

The photon acts as a probe of the proton structure

σ(p) ∼

!

dzf (z) ˆσ(zp) Parton interactions in the proton

characterized by a large time scale τ ∼ 1/Λ^QCD

e

e

γ ^q

Scattering is incoherent on the single partons

p

X

with respect to ^τhard = 1/Q

The question:

does the parton picture survive when

radiative corrections are included ?

annihilation

e

⁺

e

⁻

Consider the corrections to the total cross section

Real:

O(α^S)

Kinematics:

xⁱ = 2pⁱ · Q/Q² = 2Eⁱ/Q energy fractions p1 + p2 + p3 = Q x₁ + x₂ + x₃ = 2

(p¹ + p³)² = (Q − p²)² = (1 − x²)Q²

Soft limit: ^x3 → 0 Collinear limit: ^{p1 ! p3}

p2 ! p³ x₁ → 1 x2 → 1

Virtual:

Squaring the amplitudes real and virtual have similar structure but different kinematics

real

virtual Cut lines on shell

σ^R =

! ¹

0

dx¹dx²dx³ δ(2 − x¹ − x² − x³) |M^real(x¹, x², x³)|²

|M^real(x¹, x2, x3)|² = σ⁰ C_F αS

2π

x²₁ + x²2

(1 − x¹)(1 − x²)

Singular when

Real contribution:

Phase space Matrix element

x₁, x₂ → 1

The singularity cannot be integrated ! Let’s look at its structure:

Numerator:

Denominator:

x²₁ + x²2 = 1 + (1 − x³)² − 2(1 − x₁)(1 − x²)

Does not produce singularities 1

(1 − x¹)(1 − x²) = 1

2 − x¹ − x₂

! 1

1 − x¹ + 1 1 − x²

"

= 1 x₃

! 1

1 − x¹ + 1 1 − x²

"

soft sing. collinear sing.

The real contribution can be seen as a sum of

independent terms divergent both in the soft and in the two collinear regions plus a finite remainder

The divergences in the real and virtual contributions cancel out and a finite result is recovered

p¹

p²

p³ Q

In the region

|M^real|² ∼ !α_S 2π

" 1

1 − x¹ P

qg(x³)

Virtual contribution:

! ¹

0

dx₁dx₂δ(2 − x¹ − x₂)

! ^∞

0

M_virtdx₃

Taking the square the diagrams

have the same structure of the real but different kinematics

Differences in kinematics affect only finite parts

The sign is opposite by unitarity

p₁

p₂ Q

Universal splitting kernel

P_gq(x³) = C_F 1 + (1 − x³)² x₃

q → g

x¹ → 1 (p² " p³)

Infrared divergences

Originate in theories with massless particles and are of two kinds

•

Soft: the energy of a gluon vanishes

•

Collinear: two partons become parallel

Physically a hard parton plus a soft gluon or two very close

partons are indistinguishable They correspond to degenerate states

hard parton hard parton

+soft gluon 2 collinear partons

1

(p + k)² = 1

2EpE_k(1 − cos θ)

H q

q¯

g

Q

Q¯

H g

g

Q

Q¯

p

k

Infrared divergences are a manifestation of long-distance effects

Even in QED we cannot separate an electron from an electron + a soft photon, or an electron from an electron plus a collinear

photon

Kinoshita-Lee-Naumberg theorem:

if we limit ourself to considering quantities inclusive over initial and final states soft and collinear (degenerate)

configurations infrared divergences cancel out

The answer:

intuitive parton picture survives to the

computation of radiative corrections provided we consider inclusive enough processes

NOTE:

Phase space is flat

Matrix elements are enhanced in soft and collinear regions

The hadronic final state is typically formed by

jets of collinear particles plus soft particles

In order to cancel the divergences it is not necessary to integrate over the full phase space

According to the KLN theorem we can define a wide class of

Leading

Order (LO): σLO =

!

m

|M⁽⁰⁾({pⁱ})|²Fm({pⁱ})dΦ^m

Measurement function

Tree level matrix element

Phase space

Next-to-Leading

Order (LO): σ_{N LO} =

!

m+1

dσ^R +

!

m

dσ^V

Here one more

parton Here same number of partons but one-loop matrix element

F_m+1(...p_i, ...p_j...) ∼ F_m(....p_i + p_j....) if p_i " p_j or p_j → 0

To be sure that the presence of does not spoil the cancellation we should have:

F

number of partons at LO

(e.g. 3 jet )

m

m = 3

IR safe observables

What about hadronization ?

We have seen that matrix elements are enhanced in soft and collinear regions: soft radiation fulfills the property of angular ordering: the angle at which soft gluons are

emitted is continuosly reduced

radiation is confined in

collimated cones around the primary parton

Moreover: the colour flow in the jet evolution is such that colour singlets

are close in phase space

Hadronization is LOCAL: it does not

spoil parton picture

zp

! q

p

Hard processes with hadrons in the initial state

DIS: Partonic cross section

Parton density

Virtual: Real:

Consider corrections to the partonic cross section^O(αS)

In the calculation we find infrared divergences

Inclusive final state KLN cancellation of final state singularities However in the initial state we have only one parton and we cannot sum over degenerate states uncancelled collinear singularity

∼ α^S(Q²)

! ¹

0

dθ²

θ² ∼

! Q² 0

dk_T² k_T²

Actually the collinear divergence would be regularized by a physical cutoff of the order of the typical hadronic scale

Q⁰

The singularity implies the existence of long distance effects Multiple emission

α

ⁿS

log

ⁿ

Q

²

/Q

²0 to be resummed

Both problems are solved by the FACTORIZATION THEOREM IN SHORT: Collinear singularities can be

absorbed in bare parton densities ^f0(x) → f(x, Q²)

Physical parton densities thus become scale dependent

This is possible only if this redefinition is independent on the hard process

d³k

2k₀ ∼ k⁰dk₀d cos θ ∼ dθ² 1

(p − k)² ∼

1

p0k0(1 − cos θ) ∼ 1 θ² Phase space:

Propagator:

Vertex: /p/!(k)(/p − /k) ∼ θ in a physical gauge

(helicity conservation in the quark gluon vertex !)

dθ² 1 θ²

1

θ² θθ ∼ dθ² θ²

dθ² 1 θ² θ

In a physical gauge interferences can be neglected

not singular enough !

. . 2PI

2PI

f⁽⁰⁾ 2PI

K_T

K_T µ

F

< µ

F

>

2PI =

General strategy: decompose diagrams in 2PI blocks (such that they cannot be disjoint by cutting only two lines)

2PI blocks are free of collinear singularities in a physical gauge

Introduce an arbitrary separation scale ^µF

Process dependent but finite

Universal but divergent

Reabsorb the divergent part in the redefinition of ^f(x)

analogy with renormalization

σ(p, Q) = !

a

" ¹

0

dzf^a(z, µ²F )ˆσ^a(zp, α^S(Q²); µ²F )

Physical cross sections cannot depend on µ_F

µ_F

µ_F dependence must cancel between and

The choice of is arbitrary but if is too different from the hard scale reappear that spoil the

perturbative expansion

µ_F µ_F

Q ^{log Q2/µ2}_F

Introduction of an arbitrary scale

Parton densities become scale dependent

Such scale dependence is associated with the resummation of large collinear logs

The scale dependence is perturbatively computable

DGLAP equations

f

^a

σ ˆ

_a

. .

f⁽⁰⁾ P

P1 n Q²

Q₀

k_n~ x_n p+k_{T n}

k₁~ x₁ p+k_{T 1} z₁

z_n

DGLAP equations

f (µ²_F ) = f⁰ ⊗ E(µ²_F /Q²0)

Q² > k_{T n}² > ...k_{T 1}² > Q²₀

f (x, Q²) = f⁰(x) +

! Q² Q²₀

dk_{T n}² k_{T n}²

! ¹

x

dz_n

z_n P_n(α^S(k_{T n}² ), z_n)f (x/z_n, k_{T n}² )

The important region is:

where the maximum power of ^{log Q2/Q20} is generated iterative structure

Taking derivative with respect to

Q² ∂f (x, Q²)

∂Q² =

! ₁

x

dz

z P (α^S(Q²), z)f (x/z, Q²)

Q²

First order differential

equation: can be solved if f is known at a reference scale ^Q0

!

n

Q²

f Pab

x/z x

a

b b

f

Q² x

a a

Q² ∂

∂Q²

=

Probability to find parton in parton at scale

P^ab(α^S(Q²), z)

Pab(α^S, z) = αS

2π P_ab⁽⁰⁾(z) + !αS

2π

"2

P_ab⁽¹⁾(z) + ! αS

2π

"3

P_ab⁽²⁾(z) + ...

P^ab(0) and P^ab(1) are known (now also !)P^ab(2)

Solving DGLAP equation using allows us to resum Leading Logarithmic (LL) contributions

P^ab(0)

αSⁿ logⁿ Q²/Q²0

αSⁿ logⁿ⁻¹ Q²/Q²0

With we resum also Next-to-leading terms (NLL)_Pab⁽¹⁾

Probabilistic interpretation:

b a

Q²

Convolution conservation of longitudinal momentum

+ ! 11

6 C_A

− 2

3 T_R n_F

"

δ(1 − z)

P ⁽⁰⁾

qq (z) = CF

! 1 + z² (1 − z)+

+ 3

2δ(1 − z)

"

P ⁽⁰⁾

gg (z) = 2C_A! z

(1 − z)⁺ + 1 − z

z + z(1 − z)"

P⁽⁰⁾

gq (z) = C_F ! 1 + (1 − z)² z

"

P ⁽⁰⁾

qg (z) = T_R !z² + (1 − z)²"

! ¹

0

f(z)

(1 − z)⁺ ≡

! ¹

0

f(z) − f(1) 1 − z

+ distribution defined as

Exercise: compute Real:

Use Sudakov parametrization:

k^µ = zp^µ + k_T^µ + k² − k_T²

2zp · n n^µ

(p − k)^µ = (1 − z)p^µ − k

µ T −

k_T²

2(1 − z)p · nn^µ

Work in axial gauge

and check that ^Pqq^real(z) = CF

1 + z² 1 − z

Can be obtained from real by using fermion number conservation

! ¹

0

Pqq(x)dx = 0

Extract the leading contribution in _1/k²

P ^virt

qq = A δ(1 − z)

Virtual:

must be of the form

integral of valence quark density

cannot depend on ^Q2

gauge vector

Pqq⁽⁰⁾

Solving DGLAP equation

Consider for simplicity the case of non-singlet Q² ∂f (x, Q²)

∂Q² =

! ₁

x

dz

z P_qq(α^S(Q²), z)f (x/z, Q²)

f_N =

! ¹

0

f(x)x^{N −1}dx

(f ⊗ g)(x) =

! ₁

x

dz

z f(z)g(x z )

(f ⊗ g)_N =

! ¹

0

(f ⊗ g)(x) x^{N −1}dx

=

! ¹

0

dxx^N−1

! ¹

x

dz

z f(z)g "x z

#

=

!

z^{N −1} t^{N −1}dt dzf(z) g(t) = f_N g_N Define Mellin moments

In Mellin space DGLAP becomes

Q² ∂f_N (Q²)

∂Q² = γN,qq !α^S(Q²)" f_N (Q²)

DGLAP equation is diagonal

Anomalous dimension

•

Positive at small x

•

Slightly negative at large x

Scaling violations are:

Since:

γ^ab(N ) =

! ¹

0

P^ab(x)x^N−1

The small region corresponds to small , whereas selects large x → 1

x

γ_gg ∼ 2C^A N − 1

N N

N small as _{N → ∞}

Q

x

2

f(x,Q²)

Main effect of increase in is shift of partons from larger to smaller x

Q²

γ^aa → −2C^a log N

QCD at hadron colliders

In hadron collisions all phenomena are QCD related but we must distinguish between hard and soft

processes

x₁p₁ x₂p₂

h2

h1

Only hard scattering events can be controlled via the factorization theorem

Hard subprocess

Soft underlying event Production of low pt

hadrons: most common events

Same parton densities measured in DIS !

x1

x2

h2

h1

a

b

X

Proof of factorization:

similar to DIS: absorb initial state collinear divergences into a redefinition of parton distributions but....

The proof that soft gluons do not spoil factorization is very difficult: soft gluons (large wavelenght) can transfer colour information between the two initial state hadrons

Explicit calculations have shown that factorization breaking effects are present but are power suppressed in the high

energy limit

σ(p¹, p²; Q²) = !

a,b

" ¹

0

dx¹

" ¹

0

dx² f_h1,a(x¹, µ²_F ) fh₂,b(x², µ²_F)

׈σ^ab(x₁p₁, x₂p₂, α^S(Q²), µ²F ) Q²

10^-7 10^-6 10^-5 10^-4 10^-3 10^-2 10^-1 10⁰ 10⁰

10¹ 10² 10³ 10⁴ 10⁵ 10⁶ 10⁷ 10⁸ 10⁹

fixed target HERA

x_1,2 = (M/14 TeV) exp(±y) Q = M

LHC parton kinematics

M = 10 GeV

M = 100 GeV

M = 1 TeV

M = 10 TeV

6 6

y = 4 2 0 2 4

Q2 (GeV2 )

x

Varying Q and y

Sensitivity to different

x

₁

, x

₂

x1 x

Q

² 2

y

h₂

h1

x_1,2 = Q/√S e^±y y = 1

2 ln E + p^z E − p^z

Define rapidity

x₁x₂S = Q²

At LO we have:

Parton distributions:

Determined by global fits to different data sets

Partonic hard cross section: computed with different methods

Predictions at hadron colliders require good knowledge of:

We limit ourselves to considering hard scattering events:

QCD tools

the starting point is the factorization theorem

σ(p¹, p²; Q²) = !

a,b

" ¹

0

dx¹

" ¹

0

dx² f_h1,a(x¹, µ²_F ) fh₂,b(x², µ²_F )

׈σ^ab(x₁p₁, x₂p₂, α^S(Q²), µ²F )

Fixed order calculations: truncate at a given order in

Resummed calculations: resum (where possible) large

logarithmic contributions improving the reliability of fixed- order expansion: necessary when two different scales are

present

Parton shower simulations:

use colour coherence to approximate multiparton QCD matrix elements

LO, NLO, NNLO

quantum interferences approximated by

angular ordering constraint See G. Corcella lectures

α

_S

Parton distributions

Determined by global fits to different data sets

Parametrize at input scale

xf(x, Q²0) = Ax^α(1 − x)^β(1 + !√x + γx + ...) Q0 = 1 − 4 GeV

Impose momentum sum rule: ^!

a

" ¹

0

dxxf(x, Q²0) = 1

Then fit to data to obtain the parameters Standard procedure:

Evolve to desired and compute physical observables^Q2

Main groups: MRST, CTEQ

Now also: Alekhin, ZEUS, NNPDF..

DIS:

Drell-Yan

Fixed target: valence quark densities HERA:

f

_g

, f

_sea at small x

quark densities

Typical processes:

collisions: sensitive to antiquarks and sea densities

pp

p p ¯

collisions: sensitive to flavour

asymmetries of valence quarks q

¯ q

V

h1

h2

d(x)/¯¯ u(x) from FNAL E866/Nusea 800 GeV p + p and p + d → µ⁺µ⁻

σ^pp ∼ 4

9u(x₁)¯u(x₂) + 1

9d(x₁) ¯d(x₂) σ^pn ∼

4

9u(x₁) ¯d(x₂) + 1

9d(x₁)¯u(x₂)

Obtain neutron pdfs

from isospin symmetry:

σ^pd

2σ^pp ∼ 1 2

!

1 + d¯

2

u¯

2

"

x1 ! x₂

u ↔ d u ↔ ¯¯ d

Assuming ^{σpd ∼ σpp + σpn} and using _d(x) ! 4u(x)

σ^pd 2σ^pp

!

!

!x₁!x₂ ∼ 1 2

1 + ¹_{4 u}^d(x_(x¹⁾

1)

1 + ¹_{4 u}^d(x_(x^{1) ¯d(x2)}

1)¯u(x₂)

"

1 + d¯(x₂)

¯

u(x₂)

#

-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25

0 0.5 1 1.5 2

!Lepton Rapidity!

Charge Asymmetry

CDF 1992-1995 (110 pb^-1 e+!)

CTEQ-3M RESBOS

CTEQ-3M DYRAD

MRS-R2 (DYRAD) MRS-R2 (DYRAD)(d/u Modified) MRST (DYRAD)

u

d = d¯ u¯ = u

W asymmetry in collisions^pp¯ d

W ⁻

in the proton in the proton

If in the proton is faster than

(u(x) > d(x)) u

d

W ⁺

produced mainly in direction

W ⁺(W ⁻)

p (¯ p )

A(y) =

dσ(W⁺)

dy −

dσ(W⁻) dy dσ(W⁺)

dy + ^dσ(W_dy ⁻⁾

The W asymmetry is a measure of

u(x₁)d(x₂) − d(x₁)u(x₂) u(x₁)d(x₂) + d(x₁)u(x₂)

measure the charged lepton asymmetry

In practice W → lν

Jets at Tevatron

f

_{g at large x}

MRST 2002 and D0 jet data, !_S(M_Z)=0.1197 , "²= 85/82 pts

0 0.5 1

0 50 100 150 200 250 300 350 400 450

0.0 !"# # "#"! 0.5

0 0.5 1

0 50 100 150 200 250 300 350 400 450

0.5 !"# # #"! 1.0

0 0.5 1

0 50 100 150 200 250 300 350 400 450

1.0 !"# # #"! 1.5

(Data - Theory) / Theory

-0.5 0 0.5 1

0 50 100 150 200 250 300 350 400 450

1.5 !"# # #"! 2.0

E_T (GeV)

1 10 10 ² 10 ³ 10 ⁴ 10 ⁵ 10 ⁶ 10 ⁷

50 100 150 200 250 300 350 400 450 500

0.0 !!"""!# 0.5 0.5 !!"""!# 1.0 1.0 !!"""!# 1.5 1.5 !!"""!# 2.0 2.0 !!"""!# 3.0

Note:

Strong interplay between possible new physics effects at large

and extraction of the gluon

E_T

x

0 0.2 0.4 0.6 0.8 1 1.2

x f(x,Q)

10^-4 10^-3 10^-2 10^-1 .2 .3 .4 .5 .6 .7 Q = 5 GeV

.8 Overview of Parton Distribution Functions of the Proton

CTEQ5M ^{Gluon / 15}_dbar u_bar

s c u_v d_v

(d_bar-u_bar) * 5

Typical behaviour of parton densities in the proton

All densities vanish as _{x → 1} At _{x → 0}

- Valence quarks vanish

- Strong rise of the gluon, which becomes dominant - Also sea quarks increase

driven by the gluon through

g → q ¯ q

- propagation of errors on parameters (”hessian” method) - parameter scan (”lagrange-multiplier” method)

In recent years a consistent effort has been devoted to the study of pdf uncertainties

Experimental errors:

pdf with errors are now available

Overall picture is that quark densities are reasonably well constrained by DIS and DY data

The gluon is instead more uncertain

Also prompt photon production is sensitive to at large x but...

Theoretical problems and possible inconsistencies between data sets suggested not to use it

f_g

10² 10³

!s (GeV)

Luminosity function at LHC

Fractional Uncertainty ^0.1

0.1

0.1

-0.1 -0.1 -0.2 0.2

Q-Q --> W⁺

Q-Q --> W^- G-G 0

0

0

"

" "

"

#

#

!

50 200 500

^

10²

!s (GeV) Luminosity function at TeV RunII

Fractional Uncertainty ^0.1

0.1

0.1

-0.1 -0.1 -0.2 0.2

Q-Q --> W⁺ (W^-)

Q-Q --> "* (Z) G-G

0

0

0

^

#

# #

$ #

$ 0.3

0.4

200 400

50 20

±

Note that:

What about theoretical errors ? They are more difficult to quantify

Hints:

How to asses bias from initial parametrization ? Alternative approach: Neural network pdfs

S. Forte et al.

Conservative partons: MRST2003

Study stability with respect to cuts

From 2000 to 1200 data points

Select a data set where consistent

predictions are available: Alekhin 2003

DIS only at LO, NLO, NNLO

Bottom line: more work is needed in this direction...

QCD calculations at LO

Powerful techniques are available to compute multiparton matrix elements (ME):

These predictions are combined with automatic integration over multiparticle phase space

Cross sections for up to 8 jet production can be computed

Important because multijet final states can mimic signatures induced by physics beyond the standard model

helicity amplitudes / colour - subamplitude decomposition

Contrary to standard MC event generator they correctly include large angle radiation

Important to devise search strategies

See G. Corcella lectures for a more complete description

direct ME generation from interaction Lagrangian

When precision is an issue LO predictions are not sufficient

At LO jets are partons: it is only including NLO corrections that we can distinguish among jet algorithms

LO calculations are affected by large uncertainties Physical quantities should be

independent on

renormalization and

factorization scales but

truncation of perturbative

series at n-order induces a scale dependence of (n+1) order

At LO the scale at which to evaluate is not even defined !

0 0.5 1 1.5 2 2.5

!_R/E_T

0 0.2 0.4 0.6 0.8

d!/dET (nb/GeV)

E_T = 100 GeV 0.1 < |"| < 0.7 NLO LO

CDF Data

α

_S