## Full text

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DIS Running coupling

### Parton Distributionsand the running coupling

Institutionen för Astronomi och teoretisk fysik Lunds Universitet

2018-12-05

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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) = 4EEsin2θ/2 W2 = (P + q)2= mp2+ 2Pq − Q2.

PDFs & α(Q2) 2 Leif Lönnblad Lund University

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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 ≫ mp: Deeply inelastic scattering

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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(¯kγµk)ǫµν

q2e( ¯PγνP)

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

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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 = 2PqQ2 ≡ xBj

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DIS Running coupling

ˆ Cross section Partons ˇParton Densities

Mi ≈ e(¯kγµk)ǫµν

q2eQi(P¯ˆγµPˆ) ≈ e2Qi

ˆs

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

dσˆi = Mi

2

√ˆs/2

32π2ˆs3/2d ˆΩ = e4Qi2ˆs2 Q4

d ˆΩ 64π2ˆs

= (4π)2α2Q2iˆs2 Q4

d ˆΩ 64π2ˆs

= α2Qi2ˆs 4Q4 d ˆΩ, In the collision rest frame.

PDFs & α(Q2) 6 Leif Lönnblad Lund University

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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 Q2i.

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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

dΩ ∝ 1

sin4θ/2.

PDFs & α(Q2) 8 Leif Lönnblad Lund University

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DIS Running coupling

ˆ Partons Parton Densities Evolution

### Parton Densities

i

dQ2dx ≈ πα2

Q4 Qi2fi(x).

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

i

Q2ixfi(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)

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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

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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 → νe+ 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

(mW2 − q2)2 ≈ 1 m4W We can then measure

X

i

Qi2xfi(x) ⇒ xfu(x) + xfd¯(x) + xfc(x) + xf¯s(x).

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

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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

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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.

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DIS Running coupling

ˆ Partons Parton Densities Evolution

PDFs & α(Q2) 14 Leif Lönnblad Lund University

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DIS Running coupling

ˆ Partons Parton Densities Evolution

Small Q2

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DIS Running coupling

ˆ Partons Parton Densities Evolution

Large Q2

PDFs & α(Q2) 14 Leif Lönnblad Lund University

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DIS Running coupling

ˆ Partons Parton Densities Evolution

### DGLAP evolution

δfi(x, Q2) δ ln Q2 ∝ αs

−Pifi(x, Q2) +X

j

Z 1 x

dz

z fj(z, Q2)Pj→i(x z)

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

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DIS Running coupling

The QED case The QCD case

### Running couplings

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

αEM= 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

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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.

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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 + R2α2(Q2) + . . . becomes convergent (with R0αn(Q2) a good approximation).

PDFs & α(Q2) 18 Leif Lönnblad Lund University

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DIS Running coupling

The QED case The QCD case

+ k kk

q

+ p

+ · · ·

M = e¯u(kµu(k)εµ

Z d4p

(2π)4[e¯u(kµu(k)] 1 q2 ×

×[e¯u(p)γµu(p − q)][e¯u(p − q)γλu(p)]

(p2− m2e)((p − q)2− me2) ελ + . . .

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

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DIS Running coupling

The QED case The QCD case

e¯u(kµu(k)εµ(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)γλu(p)]

(p2− m2e)((p − q)2− m2e)

= · · ·

= α

3π Z

m2e

dp2 p2 −2α

π Z 1

0 dx · x(1 − x) ln



1 − q2x(1 − x) m2e



Where the first term is divergent.

PDFs & α(Q2) 20 Leif Lönnblad Lund University

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DIS Running coupling

The QED case The QCD case

Introducing an ultra-violet cutoffΛ and noting that ln



1 −q2x(1 − x) m2e



≈ lnQ2

me2 for Q2= −q2≫ me2

we get

I(Q2) ≈ α 3πln Λ2

me2− α 3πlnQ2

m2e

= α 3π ln Λ2

Q2.

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DIS Running coupling

The QED case The QCD case

We can now define an observable couplingαobs

M = αM0

"

1 − α 3πln Λ2

Q2 + α 3πln Λ2

Q2

2

− α 3πln Λ2

Q2

3

+ . . .

#

= M0 α

1+ α lnQΛ22 → 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

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DIS Running coupling

The QED case The QCD case

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

α(µ2) = 1

1

α0 +1 lnΛµ22 ⇒ 1 α0

= 1

α(µ2) − 1 3πlnΛ2

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

α(Q2) = 1

1

α0 +1 lnQΛ22 = 1

1

α(µ2)1 lnΛµ22 +1 lnQΛ22

= 1

1

α(µ2)+1 lnQµ22 = α(µ2) 1+α(µ2)lnQµ22 which is finite forΛ → ∞!

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DIS Running coupling

The QED case The QCD case

We can also have other fermions in the loop. However, the loop contribution if m2f > Q2is small, and we get

α(Q2) = α(µ2) 1 − Nα(µ2)lnQµ22 with

N= nl+ 3 2 3

2

nu+ 3 1 3

2

nd

αEM(m2e) ≈ 1/137, αEM(mZ2) ≈ 1/128.

PDFs & α(Q2) 24 Leif Lönnblad Lund University

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DIS Running coupling

The QED case The QCD case

Note that we can have arbitrarily complex bubbles.

α(Q2) = α(µ2)

1 − Nα(µ2)lnQµ22 + β2α22) ln2 Qµ22 + · · ·

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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) = αs2) 1 − n2f

αs2) lnQµ22

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

αs(Q2) = αs2)

1+ αs12π2)(33 − 2nf) lnQµ22 which, contrary to QED, decreases with Q2.

PDFs & α(Q2) 26 Leif Lönnblad Lund University

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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 beΛQCD

12π

αs2) = −(33 − 2nf) lnΛ2QCD µ2 and we write

αs(Q2) = 12π (33 − 2nf) lnΛQ22

QCD

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DIS Running coupling

The QED case The QCD case

PDFs & α(Q2) 28 Leif Lönnblad Lund University

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