2018-12-05 LeifLönnblad PartonDistributionsandtherunningcoupling

DIS Running coupling

Parton Distributions and the running coupling

Leif Lönnblad

Institutionen för Astronomi och teoretisk fysik Lunds Universitet

2018-12-05

DIS Running coupling

Kinematics Cross section ˇPartons

Deeply inelastic scattering

e⁻

p

e⁻ k^′

X q

k

P

P^′ γ^⋆

e⁻(k)p(P) → e⁻(k^′)X (P^′)

Q² = −q²= −(k − k^′)²= 2kk^′− 2m²e

≈ 2EE^′(1 − cos θ) = 4EE^′sin²θ/2 W² = (P + q)²= m_p²+ 2Pq − Q².

PDFs & α(Q²) 2 Leif Lönnblad Lund University

DIS Running coupling

Kinematics Cross section ˇPartons

W²is the invariant mass squared of X

◮ W = mp: Elastic scattering

◮ W > mp: Inelastic, the proton is excited

◮ W ≫ m^p: Deeply inelastic scattering

DIS Running coupling

Kinematics Cross section ˇPartons

Can we write down the matrix element without knowing X ? If it had been an elastic scattering we could have written

M ∼ e(¯k^′γ^µk)ǫ_µν

q²e( ¯P^′γ^νP)

Let’s try that, but introduce a form factor F(Q²). We expect that F(0) = 1. If F (Q²) < 1 it means that the proton is not point-like.

PDFs & α(Q²) 4 Leif Lönnblad Lund University

DIS Running coupling

ˆ Cross section Partons ˇParton Densities

Partons

Assume that the proton is a bound state of point-like constituents (a.k.a. partons) and look at

e⁻(k)q( ˆP) → e⁻(k^′)q( ˆP^′), where we have ˆP= xP.

Assuming that the constituents are (nearly) massless Pˆ^′2= (xP + q)²= x²P²+ 2xPq + q²≈ 2xPq − Q²≈ 0

⇒ x = _2Pq^Q2 ≡ xBj

DIS Running coupling

ˆ Cross section Partons ˇParton Densities

Mi ≈ e(¯k^′γ^µk)ǫ_µν

q²eQi(P¯ˆ^′γ^µPˆ) ≈ e²Qi

ˆs

−Q² where ˆs= (k − xP)²≈ x(k + P)²= xs

dσˆi =  Mi

2

√ˆs/2

32π²ˆs^3/2d ˆΩ = e⁴Q_i²ˆs² Q⁴

d ˆΩ 64π²ˆs

= (4π)²α²Q²_iˆs² Q⁴

d ˆΩ 64π²ˆs

= α²Q_i²ˆs 4Q⁴ d ˆΩ, In the collision rest frame.

PDFs & α(Q²) 6 Leif Lönnblad Lund University

DIS Running coupling

ˆ Cross section Partons ˇParton Densities

Q²= −(k − k^′)²≈ 2

√ˆs 2

√ˆs

2 (1 − cos ˆθ) so

dQ²= ˆs

2d(cos ˆθ) and

d ˆΩ = dφd(cos ˆθ) = 2dφdQ² integrating overφ gives the cross section

dσˆ

dQ² = πα² Q⁴ Q²_i.

DIS Running coupling

ˆ Cross section Partons ˇParton Densities

Compare this with standard Rutherford scattering using Q²= ˆs(1 − cos ˆθ²)/2 = ˆs sin²θ/2 = ˆˆ s sin²θ/2 We get

dσ

dΩ ∝ 1

sin⁴θ/2.

PDFs & α(Q²) 8 Leif Lönnblad Lund University

DIS Running coupling

ˆ Partons Parton Densities Evolution

Parton Densities

dσ_i

dQ²dx ≈ πα²

Q⁴ Q_i²f_i(x).

If we now define the structure function F2(x) =X

i

Q²_ixfi(x)

we can write the cross section dσ

dQ²dx = πα² Q⁴

F2(x)

x 2(1 + (1 − y)²)

(they-dependencecomes from proper spin treatment, with y = P · q/P · k)

DIS Running coupling

ˆ Partons Parton Densities Evolution

From this we can actually measure the density of quarks species. But not the individual ones.

F2(x) = 4

9(xfu(x) + xf¯u(x) + xfc(x) + xfc¯(x)) + 1

9 xfd(x) + xf¯d(x) + xfs(x) + xf¯s(x)

PDFs & α(Q²) 10 Leif Lönnblad Lund University

DIS Running coupling

ˆ Partons Parton Densities Evolution

We can also do “charged current” DIS, where the electron exchanges a W^±with the target: e⁻+ p → ν^e+ X . This is of course a bit tricky, since we cannot detect the neutrino. But we can infer its momentum by measuring all particles in the X system.

1

Q⁴ ⇒ 1

(m_W² − q²)² ≈ 1 m⁴_W We can then measure

X

i

Q_i²xfi(x) ⇒ xf^u(x) + xfd¯(x) + xfc(x) + xf¯s(x).

With e⁺+ p → ¯ν^e+ X we can also measure xfd(x) + xf¯u(x) + xfs(x) + xf¯c(x).

DIS Running coupling

ˆ Partons Parton Densities Evolution

Eight unknown PDFs, only three measurements. But:

◮ fs≈ f^¯s

◮ fc ≈ f^¯c

◮ f¯u≈ fd^¯

◮ assume valens is similar, fu− fu¯ ≈ 2(fd− fd¯)

◮ look for D-mesons in the final state

PDFs & α(Q²) 12 Leif Lönnblad Lund University

DIS Running coupling

ˆ Partons Parton Densities Evolution

Evolution of parton densities

In reality the parton densities also depends on Q².

◮ 1/Q²corresponds to resolution power.

◮ For very small Q²we only see the whole proton.

◮ when 1/Q . rpwe start to see quarks.

◮ For even larger Q²we can resolve fluctuations q → qg → q.

◮ The larger Q²the more parton fluctuations can be

resolved, the more partons we see, the smaller x they will have.

DIS Running coupling

ˆ Partons Parton Densities Evolution

PDFs & α(Q²) 14 Leif Lönnblad Lund University

DIS Running coupling

ˆ Partons Parton Densities Evolution

Small Q²

DIS Running coupling

ˆ Partons Parton Densities Evolution

Large Q²

PDFs & α(Q²) 14 Leif Lönnblad Lund University

DIS Running coupling

ˆ Partons Parton Densities Evolution

DGLAP evolution

δfi(x, Q²) δ ln Q² ∝ αs

2π



−Pⁱfi(x, Q²) +X

j

Z 1 x

dz

z fj(z, Q²)P_j→i(x z)





The Dokshitzer–Gribov–Lipatov–Altarelli–Parisi equation (enables indirect measurement of gluon density)

DIS Running coupling

The QED case The QCD case

Running couplings

What is a coupling constant and how do we measure it?

α_EM= e²/4π, but an electric charge is always surrounded by an EM field where there are virtual fluctuations that will screen the charge.

e⁻

−+

− +

+− +−

PDFs & α(Q²) 16 Leif Lönnblad Lund University

DIS Running coupling

The QED case The QCD case

So far we have only calculated “leading order” Feynman diagrams. Higher order typically can get messy:

M ∝ α

⇒

M ∝ α⁴ Every vertex comes with a factor√

α. But there can be many and there are loops.

DIS Running coupling

The QED case The QCD case

We can never measure the bare coupling. Any observable will depend on higher orders:

R= αⁿ

R0+ R1α + R2α²+ . . . and the sum is in general divergent.

The trick to renormalise the coupling, which makes it scale dependent, so that the sum in

R= αⁿ(Q²)

R0+ R₁^′(Q²)α + R₂^′α²(Q²) + . . . becomes convergent (with R0αⁿ(Q²) a good approximation).

PDFs & α(Q²) 18 Leif Lönnblad Lund University

DIS Running coupling

The QED case The QCD case

+ k kk^′′

q

+ p

+ · · ·

M = e¯u(k^′)γ^µu(k)ε_µ

−

Z d⁴p

(2π)⁴[e¯u(k^′)γ^µu(k)] 1 q² ×

×[e¯u(p)γ_µu(p − q)][e¯u(p − q)γ^λu(p)]

(p²− m²e)((p − q)²− me²) ε_λ + . . .

(minus sign due to fermion loop – a boson loop gives +)

DIS Running coupling

The QED case The QCD case

e¯u(k^′)γ^µu(k)ε_µ(1 − I(q²) + I²(q²) − I³(q²) + . . .) with

I(q²) =

Z d⁴p (2π)⁴

1 q²

[e¯u(p)γµu(p − q)][e¯u(p − q)γ^λu(p)]

(p²− m²e)((p − q)²− m²e)

= · · ·

= α

3π Z ∞

m²_e

dp² p² −2α

π Z 1

0 dx · x(1 − x) ln



1 − q²x(1 − x) m²_e



Where the first term is divergent.

PDFs & α(Q²) 20 Leif Lönnblad Lund University

DIS Running coupling

The QED case The QCD case

Introducing an ultra-violet cutoffΛ and noting that ln



1 −q²x(1 − x) m²_e



≈ lnQ²

m_e² for Q²= −q²≫ me²

we get

I(Q²) ≈ α 3πln Λ²

me²− α 3πlnQ²

m²e

= α 3π ln Λ²

Q².

DIS Running coupling

The QED case The QCD case

We can now define an observable couplingα_obs

M = αM⁰

"

1 − α 3πln Λ²

Q² + α 3πln Λ²

Q²

2

− α 3πln Λ²

Q²

3

+ . . .

#

= M⁰ α

1+ _3π^α ln_Q^Λ2₂ → M0αobs(Q²)

Note that we should take the limit Λ → ∞, in which αobs→ 0, but let’s ignore that for now.

PDFs & α(Q²) 22 Leif Lönnblad Lund University

DIS Running coupling

The QED case The QCD case

At some reference scaleµ²we measureα(µ²) and define α₀ through

α(µ²) = 1

1

α₀ +_3π¹ ln^Λ_µ²₂ ⇒ 1 α0

= 1

α(µ²) − 1 3πlnΛ²

µ² We can now calculateα(Q²) for any other scale:

α(Q²) = 1

1

α0 +_3π¹ ln_Q^Λ2₂ = 1

1

α(µ²)− _3π¹ ln^Λ_µ²₂ +_3π¹ ln_Q^Λ2₂

= 1

1

α(µ²)+_3π¹ ln_Q^µ2₂ = α(µ²) 1+^α(µ_3π²⁾ln_Q^µ2₂ which is finite forΛ → ∞!

DIS Running coupling

The QED case The QCD case

We can also have other fermions in the loop. However, the loop contribution if m²_f > Q²is small, and we get

α(Q²) = α(µ²) 1 − N^α(µ_3π²⁾ln^Q_µ2² with

N= nl+ 3 2 3

2

nu+ 3 1 3

2

nd

α_EM(m²_e) ≈ 1/137, αEM(m_Z²) ≈ 1/128.

PDFs & α(Q²) 24 Leif Lönnblad Lund University

DIS Running coupling

The QED case The QCD case

Note that we can have arbitrarily complex bubbles.

α(Q²) = α(µ²)

1 − N^α(µ_3π²⁾ln^Q_µ2² + β2α²(µ²) ln^{2 Q}_µ2² + · · ·

DIS Running coupling

The QED case The QCD case

Running coupling in QCD

The same procedure can be repeated for QCD, looking at a quark–gluon vertex and inserting quark-loops. We then get

αs(Q²) = αs(µ²) 1 − ⁿ2^f

αs(µ²) 3π ln^Q_µ2²

But we can also have gluon loops (with opposite sign) and in the end we get

αs(Q²) = αs(µ²)

1+ ^αs_12π^(µ2)(33 − 2nf) ln^Q_µ2² which, contrary to QED, decreases with Q².

PDFs & α(Q²) 26 Leif Lönnblad Lund University

DIS Running coupling

The QED case The QCD case

The Landau pole

Not that for small Q²there is actually a divergence when the denominator becomes zero. We define the scale where this happens to beΛ_QCD

12π

αs(µ²) = −(33 − 2n^f) lnΛ²_QCD µ² and we write

αs(Q²) = 12π (33 − 2n^f) ln_Λ^Q₂²

QCD

DIS Running coupling

The QED case The QCD case

PDFs & α(Q²) 28 Leif Lönnblad Lund University