Basics of QCD Lecture 2: Higher Orders, Divergences

Basics of QCD Lecture 2: higher orders, divergences

Gavin Salam

CERN Theory Unit

ICTP–SAIFR school on QCD and LHC physics July 2015, S˜aoPaulo, Brazil Neither lattice QCD nor perturbative QCD can oﬀer a full solution to using QCD at colliders

What the community has settled on is 1) factorisation of initial state non-perturbative problem from 2) the “hard process,” calculated perturbatively supplemented with 3) non-perturbative modelling of ﬁnal-state hadronic-scale processes (“hadronisation”).

Gavin Salam (CERN) QCD basics 2 2 / 17 Factorization

Z H Cross section for some hard σ x process in hadron-hadron p 1 2 p x 1 2 collisions

p1 p2 Z Z 2 2 2 σ = dx1fq/p(x1, µ ) dx2fq¯/p(x2, µ )σ ˆ(x1p1, x2p2, µ ) , ˆs = x1x2s

2 I Total X-section is factorized into a ‘hard part’σ ˆ(x1p1, x2p2, µ ) Calculated, e.g. with methods discussed in many of the other courses

2 I and parton distribution functions (PDFs): fq/p(x, µ ) is the probability of ﬁnding a quark q inside a proton p, and carrying a fraction x of its momentum. Determined experimentally, cf. later [For now, don’t worry about µ2 “factorisation scale” argument]

Gavin Salam (CERN) QCD basics 2 3 / 17 Factorization

Z H Cross section for some hard σ x process in hadron-hadron p 1 2 p x 1 2 collisions

p1 p2 Z Z 2 2 2 σ = dx1fq/p(x1, µ ) dx2fq¯/p(x2, µ )ˆσ(x1p1, x2p2, µ ) , ˆs = x1x2s

2 I Total X-section is factorized into a ‘hard part’σ ˆ(x1p1, x2p2, µ ) Calculated, e.g. with methods discussed in many of the other courses

Gavin Salam (CERN) QCD basics 2 3 / 17 Factorization

Z H Cross section for some hard σ x process in hadron-hadron p 1 2 p x 1 2 collisions

p1 p2 Z Z 2 2 2 σ = dx1fq/p(x1, µ ) dx2fq¯/p(x2, µ )ˆσ(x1p1, x2p2, µ ) , ˆs = x1x2s

2 I Total X-section is factorized into a ‘hard part’σ ˆ(x1p1, x2p2, µ ) Calculated, e.g. with methods discussed in many of the other courses

Gavin Salam (CERN) QCD basics 2 3 / 17 Factorisation is a term that has several related meanings in QCD.

Intimately connected with infrared divergences

We can start understanding those by studying a process that’s simpler than hadron collisions: e+e− collisions with hadronic ﬁnal states.

Gavin Salam (CERN) QCD basics 2 4 / 17 p1 p1 −ie γ −ie γ µ k ,ε µ k ,ε

p2 p2

[e+e− → qq¯] [Soft-collinear emission] Soft gluon amplitude Start with γ∗ → qq¯: p1

−ieγ µ Mqq¯ = −u¯(p1)ieqγµv(p2)

p2 Emit a gluon:

A i Mqqg¯ =u ¯(p1)igs /t ieqγµv(p2) p/1 + /k i A − u¯(p1)ieqγµ igs /t v(p2) p/2 + /k

Make gluon soft ≡ k p1,2; ignore terms suppressed by powers of k: A p1. p2. p/v(p) = 0, Mqqg¯ ' u¯(p1)ieqγµt v(p2) gs − p1.k p2.k p//k + /kp/ = 2p.k

Gavin Salam (CERN) QCD basics 2 5 / 17 [e+e− → qq¯] [Soft-collinear emission] Soft gluon amplitude Start with γ∗ → qq¯: p1

−ieγ µ Mqq¯ = −u¯(p1)ieqγµv(p2)

p2 Emit a gluon:

p p A i 1 1 M =u ¯(p )ig /t ie γ v(p ) −ie γ −ie γ qqg¯ 1 s p/ + /k q µ 2 µ k ,ε µ 1 k ,ε i A p p − u¯(p1)ieqγµ igs /t v(p2) 2 2 p/2 + /k

Make gluon soft ≡ k p1,2; ignore terms suppressed by powers of k: A p1. p2. p/v(p) = 0, Mqqg¯ ' u¯(p1)ieqγµt v(p2) gs − p1.k p2.k p//k + /kp/ = 2p.k

Gavin Salam (CERN) QCD basics 2 5 / 17 [e+e− → qq¯] [Soft-collinear emission] Soft gluon amplitude Start with γ∗ → qq¯: p1

−ieγ µ Mqq¯A= −iu¯(p1)ieqγµv(p2) p/1 + /k A u¯(p1)igs /t ieqγµv(p2) = −igs u¯(p1)/ 2 eqγµt v(p2) p/1 + /k (p1 + k) p Use /A/B = 2A.B − /B/A: 2 Emit a gluon:

1 A = −igs u¯(p1)[2.(p1 + k) − (p/ + /k)/] eqγpµt v(p2) p A i 1 2 1 1 M =u ¯(p )ig /t ie γ v(p ) (p1−ie+ γ k) −ie γ qqg¯ 1 s p/ + /k q µ 2 µ k ,ε µ 1 k ,ε Useu ¯(p1)p/1 i= 0 and k p1 (p1, k massless) − u¯(p )ie γ ig /t Av(p ) p2 p2 1 q µ p/ + /k s 2 2 1 A ' −igs u¯(p1)[2.p1] 2 eqγµt v(p2) Make gluon soft ≡ k p1,2; ignore(p terms1 + k) suppressed by powers of k: p1. A = −igs u¯(p1)eqγµt v(p2) p .k p1. p2. p/v(p) = 0, M ' u¯(p )ie γ t1Av(p|) g {z − } qqg¯ 1 q µ pure2 QEDs spinor structure p1.k p2.k p//k + /kp/ = 2p.k

Gavin Salam (CERN) QCD basics 2 5 / 17 [e+e− → qq¯] [Soft-collinear emission] Soft gluon amplitude Start with γ∗ → qq¯: p1

−ieγ µ Mqq¯ = −u¯(p1)ieqγµv(p2)

p2 Emit a gluon:

p p A i 1 1 M =u ¯(p )ig /t ie γ v(p ) −ie γ −ie γ qqg¯ 1 s p/ + /k q µ 2 µ k ,ε µ 1 k ,ε i A p p − u¯(p1)ieqγµ igs /t v(p2) 2 2 p/2 + /k

Make gluon soft ≡ k p1,2; ignore terms suppressed by powers of k: A p1. p2. p/v(p) = 0, Mqqg¯ ' u¯(p1)ieqγµt v(p2) gs − p1.k p2.k p//k + /kp/ = 2p.k

Gavin Salam (CERN) QCD basics 2 5 / 17 [e+e− → qq¯] [Soft-collinear emission] Squared amplitude

3~ 2 2 d k 2 2p1.p2 dΦqqg¯ |Mqqg¯ | ' (dΦqq¯|Mqq¯|) 3 CF gs 2E(2π) (p1.k)(p2.k) | {z } dS Note property of factorisation into hard qq¯ piece and soft-gluon emission piece, dS. dφ 2α C 2p .p θ ≡ θ dS = EdE dcos θ · s F 1 2 p1k 2π π (2p1.k)(2p2.k) φ = azimuth

Gavin Salam (CERN) QCD basics 2 6 / 17 [e+e− → qq¯] [Soft-collinear emission] Squared amplitude

Gavin Salam (CERN) QCD basics 2 6 / 17 [e+e− → qq¯] [Soft-collinear emission] Squared amplitude

Gavin Salam (CERN) QCD basics 2 6 / 17 [e+e− → qq¯] [Soft-collinear emission] Squared amplitude

Gavin Salam (CERN) QCD basics 2 6 / 17 [e+e− → qq¯] [Soft-collinear emission] Squared amplitude

Gavin Salam (CERN) QCD basics 2 6 / 17 [e+e− → qq¯] [Soft-collinear emission] Squared amplitude

Gavin Salam (CERN) QCD basics 2 6 / 17 [e+e− → qq¯] [Soft-collinear emission] Soft & collinear gluon emission

Take squared matrix element and rewrite in terms of E, θ,

2p1.p2 1 = 2 2 (2p1.k)(2p2.k) E (1 − cos θ) So ﬁnal expression for soft gluon emission is 2α C dE dθ dφ dS = s F π E sin θ 2π NB:

I It diverges for E → 0 — infrared (or soft) divergence

I It diverges for θ → 0 and θ → π — collinear divergence

Soft, collinear divergences derived here in speciﬁc context of e+e− → qq¯ But they are a very general property of QCD

Gavin Salam (CERN) QCD basics 2 7 / 17 [e+e− → qq¯] [Soft-collinear emission] Soft & collinear gluon emission

Take squared matrix element and rewrite in terms of E, θ,

2p1.p2 1 = 2 2 (2p1.k)(2p2.k) E (1 − cos θ) So ﬁnal expression for soft gluon emission is 2α C dE dθ dφ dS = s F π E sin θ 2π NB:

I It diverges for E → 0 — infrared (or soft) divergence

I It diverges for θ → 0 and θ → π — collinear divergence

Soft, collinear divergences derived here in speciﬁc context of e+e− → qq¯ But they are a very general property of QCD

Gavin Salam (CERN) QCD basics 2 7 / 17 [e+e− → qq¯] [Soft-collinear emission] Soft & collinear gluon emission

Take squared matrix element and rewrite in terms of E, θ,

2p1.p2 1 = 2 2 (2p1.k)(2p2.k) E (1 − cos θ) So ﬁnal expression for soft gluon emission is 2α C dE dθ dφ dS = s F π E sin θ 2π NB:

I It diverges for E → 0 — infrared (or soft) divergence

I It diverges for θ → 0 and θ → π — collinear divergence

Soft, collinear divergences derived here in speciﬁc context of e+e− → qq¯ But they are a very general property of QCD

Gavin Salam (CERN) QCD basics 2 7 / 17 [e+e− → qq¯] [Soft-collinear emission] Soft & collinear gluon emission

Take squared matrix element and rewrite in terms of E, θ,

2p1.p2 1 = 2 2 (2p1.k)(2p2.k) E (1 − cos θ) So ﬁnal expression for soft gluon emission is 2α C dE dθ dφ dS = s F π E sin θ 2π NB:

I It diverges for E → 0 — infrared (or soft) divergence

I It diverges for θ → 0 and θ → π — collinear divergence

Soft, collinear divergences derived here in speciﬁc context of e+e− → qq¯ But they are a very general property of QCD

Gavin Salam (CERN) QCD basics 2 7 / 17 If probability of gluon emission diverges, then how can you calculate anything beyond leading order?

Kinoshita-Lee-Nauenberg theorem tells as that if you sum over allowed states, then result must be ﬁnite.

Gavin Salam (CERN) QCD basics 2 8 / 17 2α C Z dE Z dθ σ = σ 1 + s F R(E/Q, θ) tot qq¯ π E sin θ 2α C Z dE Z dθ − s F V (E/Q, θ) π E sin θ

[Total cross sections] [Real-virtual cancellation] Real-virtual cancellations: total X-sctn

Total cross section: sum of all real and virtual diagrams

2 p1 −ie γ −ie γ ie γ µ k ,ε +µ x µ

Total cross section must be ﬁnite. If real part has divergent integration, so must virtual part. (Unitarity, conservation of probability)

I R(E/Q, θ) parametrises real matrix element for hard emissions, E ∼ Q. I V (E/Q, θ) parametrises virtual corrections for all momenta (a “physical fudge” — exact way is to do calc. in dim. reg.)

Gavin Salam (CERN) QCD basics 2 9 / 17 2α C Z dE Z dθ − s F V (E/Q, θ) π E sin θ

[Total cross sections] [Real-virtual cancellation] Real-virtual cancellations: total X-sctn

Total cross section: sum of all real and virtual diagrams

2 p1 −ie γ −ie γ ie γ µ k ,ε +µ x µ

Total cross section must be ﬁnite. If real part has divergent integration, so must virtual part. (Unitarity, conservation of probability) 2α C Z dE Z dθ σ = σ 1 + s F R(E/Q, θ) tot qq¯ π E sin θ

Gavin Salam (CERN) QCD basics 2 9 / 17 [Total cross sections] [Real-virtual cancellation] Real-virtual cancellations: total X-sctn

Total cross section: sum of all real and virtual diagrams

2 p1 −ie γ −ie γ ie γ µ k ,ε +µ x µ

Gavin Salam (CERN) QCD basics 2 9 / 17 [Total cross sections] [Real-virtual cancellation] Total X-section (cont.)

2α C Z dE Z dθ σ = σ 1 + s F (R(E/Q, θ) − V (E/Q, θ)) tot qq¯ π E sin θ

I From calculation: limE→0 R(E/Q, θ) = 1. I For every divergence R(E/Q, θ) and V (E/Q, θ) should cancel:

lim (R − V ) = 0 , lim (R − V ) = 0 E→0 θ→0,π Result:

I corrections to σtot come from hard (E ∼ Q), large-angle gluons

I Soft gluons don’t matter: 2 I Physics reason: soft gluons emitted on long timescale ∼ 1/(Eθ ) relative to collision (1/Q) — cannot inﬂuence cross section. I Transition to hadrons also occurs on long time scale (∼ 1/Λ) — and can also be ignored.

I Correct renorm. scale for αs: µ ∼ Q — perturbation theory valid.

Gavin Salam (CERN) QCD basics 2 10 / 17 [Total cross sections] [Real-virtual cancellation] Total X-section (cont.)

2α C Z dE Z dθ σ = σ 1 + s F (R(E/Q, θ) − V (E/Q, θ)) tot qq¯ π E sin θ

I From calculation: limE→0 R(E/Q, θ) = 1. I For every divergence R(E/Q, θ) and V (E/Q, θ) should cancel:

lim (R − V ) = 0 , lim (R − V ) = 0 E→0 θ→0,π Result:

I corrections to σtot come from hard (E ∼ Q), large-angle gluons

I Correct renorm. scale for αs: µ ∼ Q — perturbation theory valid.

Gavin Salam (CERN) QCD basics 2 10 / 17 [Total cross sections] [Real-virtual cancellation] Total X-section (cont.)

2α C Z dE Z dθ σ = σ 1 + s F (R(E/Q, θ) − V (E/Q, θ)) tot qq¯ π E sin θ

I From calculation: limE→0 R(E/Q, θ) = 1. I For every divergence R(E/Q, θ) and V (E/Q, θ) should cancel:

lim (R − V ) = 0 , lim (R − V ) = 0 E→0 θ→0,π Result:

I corrections to σtot come from hard (E ∼ Q), large-angle gluons

I Correct renorm. scale for αs: µ ∼ Q — perturbation theory valid.

Gavin Salam (CERN) QCD basics 2 10 / 17 [Total cross sections] [Real-virtual cancellation] Total X-section (cont.)

2α C Z dE Z dθ σ = σ 1 + s F (R(E/Q, θ) − V (E/Q, θ)) tot qq¯ π E sin θ

I From calculation: limE→0 R(E/Q, θ) = 1. I For every divergence R(E/Q, θ) and V (E/Q, θ) should cancel:

lim (R − V ) = 0 , lim (R − V ) = 0 E→0 θ→0,π Result:

I corrections to σtot come from hard (E ∼ Q), large-angle gluons

I Correct renorm. scale for αs: µ ∼ Q — perturbation theory valid.

Gavin Salam (CERN) QCD basics 2 10 / 17 [Total cross sections] [Real-virtual cancellation] Total X-section (cont.)

2α C Z dE Z dθ σ = σ 1 + s F (R(E/Q, θ) − V (E/Q, θ)) tot qq¯ π E sin θ

I From calculation: limE→0 R(E/Q, θ) = 1. I For every divergence R(E/Q, θ) and V (E/Q, θ) should cancel:

lim (R − V ) = 0 , lim (R − V ) = 0 E→0 θ→0,π Result:

I corrections to σtot come from hard (E ∼ Q), large-angle gluons

I Correct renorm. scale for αs: µ ∼ Q — perturbation theory valid.

Gavin Salam (CERN) QCD basics 2 10 / 17 [Total cross sections] [Real-virtual cancellation] Total X-section (cont.)

2α C Z dE Z dθ σ = σ 1 + s F (R(E/Q, θ) − V (E/Q, θ)) tot qq¯ π E sin θ

I From calculation: limE→0 R(E/Q, θ) = 1. I For every divergence R(E/Q, θ) and V (E/Q, θ) should cancel:

lim (R − V ) = 0 , lim (R − V ) = 0 E→0 θ→0,π Result:

I corrections to σtot come from hard (E ∼ Q), large-angle gluons

I Correct renorm. scale for αs: µ ∼ Q — perturbation theory valid.

Gavin Salam (CERN) QCD basics 2 10 / 17 [Total cross sections] [Real-virtual cancellation] Total cross section at NLO

Our treatment so far was a bit rough: designed to emphasize physical nature of divergences.

In practice calculations will be done in 4 + dimensions and infrared divergences translate to powers of 1/.

Full ﬁnal answer for σtot at next-to-leading order (NLO) is, for massless quarks, 3 α C σ = σ 1 + s F + O α2 tot qq¯ 4 π s

Gavin Salam (CERN) QCD basics 2 11 / 17 [Total cross sections] [Beyond NLO] Beyond NLO: e.g. NNLO

x x x x x x x 0 loops (tree−level)

o o o o 1 loop

ø ø 2 loops

0 1 2 3 4 5 6

ij → Z + n partons