Particle Physics Phenomenology 2. Phase space and matrix...
Transcript of Particle Physics Phenomenology 2. Phase space and matrix...
Particle Physics Phenomenology2. Phase space and matrix elements
Torbjorn Sjostrand
Department of Astronomy and Theoretical PhysicsLund University
Solvegatan 14A, SE-223 62 Lund, Sweden
NBI, Copenhagen, 3 October 2011
Four-vectors
four−vector : p = (E ;p) = (E ; px , py , pz)
vector sum : p1 + p2 = (E1 + E2;p1 + p2)
vector product : p1p2 = E1E2 − p1p2
= E1E2 − px1px2 − py1py2 − pz1pz2
= E1E2 − |p1| |p2| cos θ12
square : p2 = E 2 − p2 = E 2 − p2x − p2
y − p2z = m2
transverse mom. : p⊥ =√
p2x + p2
y
transverse mass : m⊥ =√
m2 + p2x + p2
y =√
m2 + p2⊥
E 2 = m2 + p2 = m2 + p2⊥ + p2
z = m2⊥ + p2
z
Warning: No standard to distinguish p = (E ; px , py , pz) and
p = |p| =√
p2x + p2
y + p2z , but usually clear from context.
When we remember, we will try to use p = |p|, since p = p.
Torbjorn Sjostrand PPP 2: Phase sapce and matrix elements slide 2/48
Decay widths and cross sections
Decay width at rest, 1 → n:
dΓ =|M|2
2MdΦn
Integrated it gives exponential decay rate
dPdt
= Γe−Γt and 〈τ〉 = 1/Γ
Collision process cross section, 2 → n:
dσ =|M|2
4√
(p1p2)2 −m21m
22
dΦn
Integrated it gives collision rate
N = σ
∫L(t) dt with L ≈ f
n1n2
A
in a theorist’s approximation of the luminosity L for a collider.Torbjorn Sjostrand PPP 2: Phase sapce and matrix elements slide 3/48
Phase space
n-body phase space:
dΦn = (2π)4δ(4)(P −n∑
i=1
pi )n∏
i=1
d3pi
(2π)32Ei
Lorentz covariant:
d4pi δ(p2i −m2
i ) θ(Ei ) = d4pi δ(E2i − (p2
i + m2i )) θ(Ei )
=d3pi
2Ei
with Ei =√
p2i + m2
i and using
δ(f (x)) =∑
xj ,f (xj )=0
1
|f ′(xj)|δ(x − xj)
Application: Lorentz invariant production cross sections E dσ/d3pTorbjorn Sjostrand PPP 2: Phase sapce and matrix elements slide 4/48
Spherical symmetry
Spherical coordinates:
d3p
E=
dpx dpy dpz
E=
p2 dp dΩ
E=
p EdE dΩ
E= p dE dΩ
where Ω is the unit sphere,
dΩ = d(cos θ) dφ = sin θ dθ dϕ
px = p sin θ cos ϕ
py = p sin θ sin ϕ
pz = p cos θ
and E 2 = p2 + m2 ⇒ E dE = p dp.
Convenient for use e.g. in resonance decays,but not for standard QCD physics in pp collisions.Instead:
Torbjorn Sjostrand PPP 2: Phase sapce and matrix elements slide 5/48
Cylindrical symmetry and rapidity
Cylindrical coordinates:
d3p
E=
dpx dpy dpz
E=
d2p⊥ dpz
E= d2p⊥ dy
with rapidity y given by
y =1
2ln
E + pz
E − pz=
1
2ln
(E + pz)2
(E + pz)(E − pz)=
1
2ln
(E + pz)2
m2 + p2⊥
= lnE + pz
m⊥= ln
m⊥E − pz
The relation dy = dpz/E can be shown by
dy
dpz=
ddpz
(ln
E + pz
m⊥
)=
ddpz
(ln(√
m2⊥ + p2
z + pz)− lnm⊥
)
=
12
2p⊥√m2⊥+p2
z
+ 1√m2⊥ + p2
z + pz
=pz+E
E
E + pz=
1
E
Torbjorn Sjostrand PPP 2: Phase sapce and matrix elements slide 6/48
Lightcone kinematics and boosts
Introduce (lightcone) p+ = E + pz and p− = E − pz .Note that p+p− = E 2 − p2
z = m2⊥.
Consider boost along z axis with velocity β, and γ = 1/√
1− β2.
p′x ,y = px ,y
p′z = γ(pz + β E )
E ′ = γ(E + β pz)
p′+ = γ(1 + β)p+ =
√1 + β
1− βp+ = k p+
p′− = γ(1− β)p+ =
√1− β
1 + βp− =
p−
k
y ′ =1
2ln
p′+
p′−=
1
2ln
k p+
p′−/k= y + ln k
y ′2 − y ′1 = (y2 + ln k)− (y1 + ln k) = y2 − y1
Torbjorn Sjostrand PPP 2: Phase sapce and matrix elements slide 7/48
Pseudorapidity
If experimentalists cannot measure m they may assume m = 0.Instead of rapidity y they then measure pseudorapidity η:
y =1
2ln
√m2 + p2 + pz√m2 + p2 − pz
⇒ η =1
2ln|p|+ pz
|p| − pz= ln
|p|+ pz
p⊥
or
η =1
2ln
p + p cos θ
p − p cos θ=
1
2ln
1 + cos θ
1− cos θ
=1
2ln
2 cos2 θ/2
2 sin2 θ/2= ln
cos θ/2
sin θ/2= − ln tan
θ
2
which thus only depends on polar angle.η is not simple under boosts: η′2 − η′1 6= η2 − η1.You may even flip sign!Assume m = mπ for all charged ⇒ yπ; intermediate to y and η.
Torbjorn Sjostrand PPP 2: Phase sapce and matrix elements slide 8/48
The pseudorapidity dip
By analogy with dy/dpz = 1/E it follows that dη/dpz = 1/p.
Thus
dη
dy=
dη/dpz
dy/dpz=
E
p> 1
with limits
dη
dy→ m⊥
p⊥for pz → 0
dη
dy→ 1 for pz → ±∞
so if dn/dy is flat for y ≈ 0then dn/dη has a dip there.
η−y = lnp + pz
p⊥−ln
E + pz
m⊥= ln
p + pz
E + pz
m⊥p⊥
→ lnm⊥p⊥
when pz m⊥
Torbjorn Sjostrand PPP 2: Phase sapce and matrix elements slide 9/48
Two-body phase space
Evaluate in rest frame, i.e. P = (Ecm, 0).
dΦ2 = (2π)4δ(4)(P − p1 − p2)d3p1
(2π)32E1
d3p2
(2π)32E2
=1
16π2δ(Ecm − E1 − E2)
d3p1
E1E2
=1
16π2δ(√
m21 + p2 +
√m2
2 + p2 − Ecm)p2 dp dΩ
E1E2
=1
16π2
δ(p − p∗
| p
E1+
p
E2|p2 dp dΩ
E1E2
=1
16π2
E1E2
E1 + E2
p dΩ
E1E2
=p dΩ
16π2 Ecm
Torbjorn Sjostrand PPP 2: Phase sapce and matrix elements slide 10/48
The Kallen function – 1
√m2
1 + p2 +√
m22 + p2 = Ecm
gives solution
E1 =E 2
cm + m21 −m2
2
2Ecm
E2 =E 2
cm + m22 −m2
1
2Ecm
p =1
2Ecm
√(E 2
cm −m21 −m2
2)2 − 4m2
1m22 =
1
2Ecm
√λ(E 2
cm,m21,m
22)
where Kallen λ function is
λ(a2, b2, c2) = (a2 − b2 − c2)2 − 4b2c2
= a4 + b4 + c4 − 2a2b2 − 2a2c2 − 2b2c2
= (a2 − (b + c)2)(a2 − (b − c)2)
= (a + b + c)(a− b − c)(a− b + c)(a + b − c)
Torbjorn Sjostrand PPP 2: Phase sapce and matrix elements slide 11/48
The Kallen function – 2
Hides everywhere in kinematics, e.g.
dσ =|M|2
4√
(p1p2)2 −m21m
22
dΦn
has
4((p1p2)2 −m2
1m22) = (p2
1 + 2p1p2 + p22 −m2
1 −m22)
2 − 4m21m
22
= ((p1 + p2)2 −m2
1 −m22)
2 − 4m21m
22
= λ(E 2cm,m2
1,m22)
so
dσ =|M|2
2√
λ(E 2cm,m2
1,m22)
dΦn
Torbjorn Sjostrand PPP 2: Phase sapce and matrix elements slide 12/48
Mandelstam variables
For process 1 + 2 → 3 + 4
s = (p1 + p2)2 = (p3 + p4)
2
t = (p1 − p3)2 = (p2 − p4)
2
u = (p1 − p4)2 = (p2 − p3)
2
In rest frame, massless limit: m1 = m2 = m3 = m4 = 0,
p1,2 =Ecm
2(1; 0, 0,±)
p3,4 =Ecm
2(1;± sin θ, 0,± cos θ)
s = E 2cm
t = −2p1p3 = − s
2(1− cos θ)
u = −2p2p4 = − s
2(1 + cos θ) s + t + u = 0
Torbjorn Sjostrand PPP 2: Phase sapce and matrix elements slide 13/48
Mandelstam variables with masses
β34 =
√λ(s,m2
3,m24)
s
p3,4 =
√s
2
(1± m2
3 −m24
s;±β34 sin θ, 0,±β34 cos θ
)t = m2
1 + m23 −
s
2
(1 +
m21 −m2
2
s
)(1 +
m23 −m2
4
s
)+
s
2β12 β34 cos θ
dσ =|M|2
2√
λ(s,m21,m
22)
p34√s
d cos θ dϕ
16π2=|M|2
2sβ12
β34
2
d cos θ
8π
assuming no polarization ⇒ no ϕ dependence
dσ
dt=
dσ
dcos θ
dcos θ
dt=
|M|2
16πs2β212
Torbjorn Sjostrand PPP 2: Phase sapce and matrix elements slide 14/48
Mandelstam variables with final-state masses
Usually m1,2 ≈ 0, while often m3,4 non-negligible
t, u = −1
2
[s −m2
3 −m24 ∓ sβ34 cos θ
]dσ
dt=
|M|2
16πs2
s + t + u = m23 + m2
4
tu =1
4
[(s −m2
3 −m24)
2 − s2β234 cos2 θ
]=
1
4
[s2β2
34 + 4m23m
24 − s2β2
34 cos2 θ]
=1
4s2β2
34 sin2 θ + m23m
24 = sp2
⊥ + m23m
24
p2⊥ =
tu −m23m
24
s
Torbjorn Sjostrand PPP 2: Phase sapce and matrix elements slide 15/48
s-, t- and u-channel processes
Classify 2 → 2 diagrams by character of propagator, e.g.
Singularities reflect channel character, e.g. pure t-channel:
dσ(qq′ → qq′)dt
=π
s2
4
9α2
s
s2 + u2
t2
peaked at t → 0 ⇒ u ≈ −s, so
dσ(qq′ → qq′)dt
≈ 8πα2s
9t2=
32πα2s
9s2(1− cos θ)2=
8πα2s
9s2 sin4 θ/2≈ 8πα2
s
9p4⊥
i.e. Rutherford scattering!Torbjorn Sjostrand PPP 2: Phase sapce and matrix elements slide 16/48
Order-of-magnitude cross sections
With masses neglected:
s−channel :dσ
dt∼ π
s2
t−channel, spin 1 :dσ
dt∼ π
t2
t−channel, spin1
2:
dσ
dt∼ π
−stu−channel : same with t → u
Add couplings at vertices:
qqg : CFαs
ggg : Ncαs
f fγ : e2f αem
f f ′W : |Vff′ |2αem
4 sin2θW
f f ′Z : (v2f + a2
f )αem
16 sin2θW cos2θWTorbjorn Sjostrand PPP 2: Phase sapce and matrix elements slide 17/48
Closeup: qg → qg
Consider q(1) g(2) → q(3) g(4):
t : pg∗ = p1 − p3 ⇒ m2g∗ = (p1 − p3)
2 = t ⇒ dσ/dt ∼ 1/t2
u : pq∗ = p1 − p4 ⇒ m2q∗ = (p1 − p4)
2 = u ⇒ dσ/dt ∼ −1/su
s : pq∗ = p1 + p2 ⇒ m2q∗ = (p1 + p2)
2 = s ⇒ dσ/dt ∼ 1/s2
Contribution of each sub-graph is gauge-dependent,only sum is well-defined:
dσ
dt=
πα2s
s2
[s2 + u2
t2+
4
9
s
(−u)+
4
9
(−u)
s
]Torbjorn Sjostrand PPP 2: Phase sapce and matrix elements slide 18/48
Scale choice
What Q2 scale to use for αs = αs(Q2)?
Should be characteristic virtuality scale of process!But e.g. for q g → q g: both s-, t- and u-channel + interference.At small t the t-channel graph dominates ⇒ Q2 ∼ |t|,at small u the u-channel graph dominates ⇒ Q2 ∼ |u|,in between all graphs comparably important ⇒ Q2 ∼ s ∼ |t| ∼ |u|.Suitable interpolation:
→ −t for t → 0
Q2 = p2⊥ =
tu
s→ −u for u → 0
→ s
4for t = u = − s
2
but could equally well be multiple of p2⊥, or more complicated
⇒ one limitation of LO calculations.
Torbjorn Sjostrand PPP 2: Phase sapce and matrix elements slide 19/48
Resonances
Resonance shape given by Breit-Wigner
1 7→ ρ(s) =1
π
mΓ
(s −m2)2 + m2Γ2
7→ 1
π
sΓ(m)/m
(s −m2)2 + s2Γ2(m)/m2
where m 7→√
s in phase space and Γ(s) 7→ Γ(m)√
s/mfor gauge bosons, neglecting thresholds.Latter shape suppressed below and enhanced above peak; tilted.For s → 0 ρ(s) goes to constant or like s.PDF’s tend to be peaked at small x : convolution enhances small s.Can give secondary mass-spectrum “peak” in s → 0 region.But note that
|M|2 = |Msignal +Mbackground|2
so in many cases Breit-Wigner cannot be trusted except in theneighbourhood of the peak, where signal should dominate.
Torbjorn Sjostrand PPP 2: Phase sapce and matrix elements slide 20/48
Three-body phase space
Three-body final states has 3 · 3− 4 degrees of freedom.In massless case straightforward to show that, in CM frame,
dΦ3 = (2π)4δ(4)(P − p1 − p2 − p3)d3p1
(2π)32E1
d3p2
(2π)32E2
d3p3
(2π)32E3
=1
8(2π)5dE1 dE2 d cos θ1 dϕ1 dϕ21
with θ1, ϕ1 polar coordinates of 1 andϕ21 azimuthal angle of 2 around 1 axis (Euler angles).Phase space limits 0 ≤ E1,2 ≤ Ecm/2 andE1 + E2 = Ecm − E3 > Ecm/2.
Same simple phase space expression holds in massive case,but phase space limits much more complicated!
Higher multiplicities increasingly difficult to understand.One solution: recursion!
Torbjorn Sjostrand PPP 2: Phase sapce and matrix elements slide 21/48
Factorized three-body phase space
Drop factors of 2π, and don’t write implicit integral signs.Introduce intermediate “particle” 12 = 1 + 2.
dΦ3(P; p1, p2, p3)
∼ δ(4)(P − p1 − p2 − p3)d3p1
2E1
d3p2
2E2
d3p3
2E3δ(4)(p12 − p1 − p2) d4p12
= δ(4)(P − p12 − p3) d4p12d3p3
2E3
[δ(4)(p12 − p1 − p2)
d3p1
2E1
d3p2
2E2
]= δ(4)(P − p12 − p3) d4p12 δ(p2
12 −m212) dm2
12
d3p3
2E3dΦ2(p12; p1, p2)
= dm212
[δ(4)(P − p12 − p3)
d3p12
2E12
d3p3
2E3
]dΦ2(p12; p1, p2)
= dm212 dΦ2(P; p12, p3) dΦ2(p12; p1, p2)
Note: here 4 angles + 1 mass2; last slide 3 angles + 2 energies.
Torbjorn Sjostrand PPP 2: Phase sapce and matrix elements slide 22/48
Recursive phase space
Generalizes to
dΦn(P; p1, . . . , pn) = dm212...(n−1) dΦ2(P; p12...(n−1), pn)
× dΦn−1(P; p1, . . . , p(n−1))
Can be viewed as a sequentialdecay chain, with undeterminedintermediate masses.
Recall dΦ2(P; p1, p2) ∝
√λ(M2,m2
1,m22)
M2dΩ12
where dΩ12 is the unit sphere in the 1+2 rest frame.Now can write down e.g. 4-body phase space:
Torbjorn Sjostrand PPP 2: Phase sapce and matrix elements slide 23/48
The M-generator
dΦ4(P; p1, p2, p3, p4) ∝
√λ(M2;m2
4,m2123)
M2m123 dm123 dΩ1234
×
√λ(m2
123;m23,m
212)
m2123
m12 dm12 dΩ123
√λ(m2
12;m21,m
22)
m212
dΩ12
Mass limits coupled, but can be decoupled: pick two randomnumbers 0 < R1,2 < 1 and order them R1 < R2. Then
∆ = M − (m1 + m2 + m3 + m4)
m12 = m1 + m2 + R1∆
m123 = m1 + m2 + m3 + R2∆
uniformly covers dm12 dm123 space with weight√λ(M2;m2
4,m2123)
M
√λ(m2
123;m23,m
212)
m123
√λ(m2
12;m21,m
22)
m12
Torbjorn Sjostrand PPP 2: Phase sapce and matrix elements slide 24/48
RAMBO
For massless case a smart solution is RAMBO (RAndom Momentaand BOosts), which is 100% efficient:
RAMBO
1 Pick n massless 4-vectors pi according to
Eie−Ei dΩi
2 boost all of them by a common boost vector that brings themto their overall rest frame
3 rescale them by a common factor that brings them to thedesired mass M
Can be modified for massive cases, but then no longer 100%efficiency; gets worse the bigger
∑mi/M is.
MAMBO: workaround for high multiplicities
Torbjorn Sjostrand PPP 2: Phase sapce and matrix elements slide 25/48
Efficiency troubles
Even if you can pick phase space points uniformly, |M|2 is not!A n-body process receives contributions from a large number ofFeynman graphs, plus interferences.Can lead to extremely low Monte Carlo efficiency.Intermediate resonances ⇒ narrow spikes when (pi + pj)
2 ≈ M2res.
t-channel graphs ⇒ peaked at small p⊥.
Multichannel techniques:
|M|2 =|∑
i Mi |2∑i |Mi |2
∑i
|Mi |2
so pick optimized for either |Mi |2 according to their relativeintegral, and use ratio as weight.Still major challenge in real life!
Torbjorn Sjostrand PPP 2: Phase sapce and matrix elements slide 26/48
Composite beams
In reality all beamsare composite:p : q, g, q, . . .e− : e−, γ, e+, . . .γ : e±, q, q, g
Factorization
σAB =∑i ,j
∫∫dx1 dx2 f
(A)i (x1,Q
2) f(B)j (x2,Q
2) σij
x : momentum fraction, e.g. pi = x1pA; pj = x2pB
Q2: factorization scale, “typical momentum transfer scale”
Factorization only proven for a few cases, like γ∗/Z0 prodution,and strictly speaking not correct e.g. for jet production,
but good first approximation and unsurpassed physics insight .
Torbjorn Sjostrand PPP 2: Phase sapce and matrix elements slide 27/48
Subprocess kinematics
If pA + pB = (Ecm; 0), A,B along ±z axis, and 1, 2 collinear withA,B then convinently put them massless:
p1 = (Ecm/2)(1; 0, 0, 1)
p2 = (Ecm/2)(1; 0, 0,−1)
such that s =(p1 + p2)2 = x1 x2 s = τ s. Velocity of subsystem is
βz =pz
E=
x1 − x2
x1 + x2
and its rapidity
y =1
2ln
E + pz
E − pz=
1
2ln
x1
x2
dx1 dx2 = dτ dy convenient for Monte Carlo.Historically xF = 2pz/Ecm = x1 − x2.Subprocess 2 → 2 kinematics for σ: s, t, u..
Torbjorn Sjostrand PPP 2: Phase sapce and matrix elements slide 28/48
Matrix Elements and Their Usage
L ⇒ Feynman rules ⇒ Matrix Elements ⇒ Cross Sections+ Kinematics ⇒ Processes ⇒ . . .⇒
(Higgs simulation in CMS)
Torbjorn Sjostrand PPP 2: Phase sapce and matrix elements slide 29/48
Loops and legs – 1 (Peter Skands)
Torbjorn Sjostrand PPP 2: Phase sapce and matrix elements slide 30/48
Loops and legs – 2 (Peter Skands)
Torbjorn Sjostrand PPP 2: Phase sapce and matrix elements slide 31/48
Loops and legs – 3 (Peter Skands)
Torbjorn Sjostrand PPP 2: Phase sapce and matrix elements slide 32/48
Loops and legs – 4 (Peter Skands)
Torbjorn Sjostrand PPP 2: Phase sapce and matrix elements slide 33/48
Loops and legs – 5 (Peter Skands)
Torbjorn Sjostrand PPP 2: Phase sapce and matrix elements slide 34/48
Loops and legs – 6 (Peter Skands)
Torbjorn Sjostrand PPP 2: Phase sapce and matrix elements slide 35/48
Born level calculations – 1 (Frank Krauss)
Torbjorn Sjostrand PPP 2: Phase sapce and matrix elements slide 36/48
Born level calculations – 2 (Frank Krauss)
Remember: to be squared for number of squared MEs.
Torbjorn Sjostrand PPP 2: Phase sapce and matrix elements slide 37/48
Born level calculations – 3 (Frank Krauss)
Torbjorn Sjostrand PPP 2: Phase sapce and matrix elements slide 38/48
Born level calculations – 4 (Frank Krauss)
Torbjorn Sjostrand PPP 2: Phase sapce and matrix elements slide 39/48
Born level calculations – 5 (Frank Krauss)
Torbjorn Sjostrand PPP 2: Phase sapce and matrix elements slide 40/48
Born level calculations – 6 (Frank Krauss)
Torbjorn Sjostrand PPP 2: Phase sapce and matrix elements slide 41/48
Next-to-leading order (NLO) graphs
Torbjorn Sjostrand PPP 2: Phase sapce and matrix elements slide 42/48
Next-to-leading order (NLO) graphs
Torbjorn Sjostrand PPP 2: Phase sapce and matrix elements slide 42/48
Next-to-leading order (NLO) graphs
Torbjorn Sjostrand PPP 2: Phase sapce and matrix elements slide 42/48
NLO calculations – 1
σNLO =
∫ndσLO +
∫n+1
dσReal +
∫ndσVirt
Simple one-dimensional example: x ∼ p⊥/p⊥max, 0 ≤ x ≤ 1Divergences regularized by d = 4− 2ε dimensions, ε < 0
σR+V =
∫ 1
0
dx
x1+εM(x) +
1
εM0
KLN cancellation theorem: M(0) = M0
Phase Space Slicing:Introduce arbitrary finite cutoff δ 1 (so δ |ε| )
σR+V =
∫ 1
δ
dx
x1+εM(x) +
∫ δ
0
dx
x1+εM(x) +
1
εM0
≈∫ 1
δ
dx
xM(x) +
∫ δ
0
dx
x1+εM0 +
1
εM0
=
∫ 1
δ
dx
xM(x) +
1
ε
(1− δ−ε
)M0 ≈
∫ 1
δ
dx
xM(x) + ln δ M0
Torbjorn Sjostrand PPP 2: Phase sapce and matrix elements slide 43/48
NLO calculations – 1
σNLO =
∫ndσLO +
∫n+1
dσReal +
∫ndσVirt
Simple one-dimensional example: x ∼ p⊥/p⊥max, 0 ≤ x ≤ 1Divergences regularized by d = 4− 2ε dimensions, ε < 0
σR+V =
∫ 1
0
dx
x1+εM(x) +
1
εM0
KLN cancellation theorem: M(0) = M0
Phase Space Slicing:Introduce arbitrary finite cutoff δ 1 (so δ |ε| )
σR+V =
∫ 1
δ
dx
x1+εM(x) +
∫ δ
0
dx
x1+εM(x) +
1
εM0
≈∫ 1
δ
dx
xM(x) +
∫ δ
0
dx
x1+εM0 +
1
εM0
=
∫ 1
δ
dx
xM(x) +
1
ε
(1− δ−ε
)M0 ≈
∫ 1
δ
dx
xM(x) + ln δ M0
Torbjorn Sjostrand PPP 2: Phase sapce and matrix elements slide 43/48
NLO calculations – 2
Alternatively Subtraction:
σR+V =
∫ 1
0
dx
x1+εM(x)−
∫ 1
0
dx
x1+εM0 +
∫ 1
0
dx
x1+εM0 +
1
εM0
=
∫ 1
0
M(x)−M0
x1+εdx +
(−1
ε+
1
ε
)M0
≈∫ 1
0
M(x)−M0
xdx +O(1)M0
NLO provides a more accurate answer for an integrated cross section:
Warning!Neither approach operateswith positive definite quantities.No obvious event-generatorimplementation.No trivial connection tophysical events
Torbjorn Sjostrand PPP 2: Phase sapce and matrix elements slide 44/48
NLO calculations – 2
Alternatively Subtraction:
σR+V =
∫ 1
0
dx
x1+εM(x)−
∫ 1
0
dx
x1+εM0 +
∫ 1
0
dx
x1+εM0 +
1
εM0
=
∫ 1
0
M(x)−M0
x1+εdx +
(−1
ε+
1
ε
)M0
≈∫ 1
0
M(x)−M0
xdx +O(1)M0
NLO provides a more accurate answer for an integrated cross section:
Warning!Neither approach operateswith positive definite quantities.No obvious event-generatorimplementation.No trivial connection tophysical events
Torbjorn Sjostrand PPP 2: Phase sapce and matrix elements slide 44/48
Scale choices
Cross section depends on factorization scale µF
and renormalization scale µR :
σAB =∑i ,j
∫∫dx1 dx2 f
(A)i (x1, µF ) f
(B)j (x2, µF ) σij(αs(µR), µF , µR)
Historically common to put Q = µF = µR but nowadays variedindependently to gauge undertainty of cross section prediction.
Typical variationfactor 2±1 around“natural value”,but beware
Torbjorn Sjostrand PPP 2: Phase sapce and matrix elements slide 45/48
Current status (N)(N)LO (Frank Krauss)
Torbjorn Sjostrand PPP 2: Phase sapce and matrix elements slide 46/48
Colour flow in hard processes – 1
One Feynman graph can correspond to several possible colourflows, e.g. for qg → qg:
while other qg → qg graphs only admit one colour flow:
Torbjorn Sjostrand PPP 2: Phase sapce and matrix elements slide 47/48
Colour flow in hard processes – 2
so nontrivial mix of kinematics variables (s, t)and colour flow topologies I, II:
|A(s, t)|2 = |AI(s, t) +AII(s, t)|2
= |AI(s, t)|2 + |AII(s, t)|2 + 2Re(AI(s, t)A∗II(s, t)
)with Re
(AI(s, t)A∗II(s, t)
)6= 0
⇒ indeterminate colour flow, while• showers should know it (coherence),• hadronization must know it (hadrons singlets).Normal solution:
interferencetotal
∝ 1
N2C − 1
so split I : II according to proportions in the NC →∞ limit, i.e.
|A(s, t)|2 = |AI(s, t)|2mod + |AII(s, t)|2mod
|AI(II)(s, t)|2mod = |AI(s, t) +AII(s, t)|2(
|AI(II)(s, t)|2
|AI(s, t)|2 + |AII(s, t)|2
)NC→∞
Torbjorn Sjostrand PPP 2: Phase sapce and matrix elements slide 48/48