Twistors and UnitarityNLO six-gluon Amplitude in QCD
Pierpaolo Mastrolia
Institute for Theoretical Physics, University of Zurich
IFAE 2006, April 19, 2006
in collaboration with: Ruth Britto and Bo Feng → hep-ph/0602187
P. Mastrolia, UniZh - Twistors & Unitarity, 1
Outline
• Trees
• Spinor Formalism
• MHV Amplitudes
• CSW diagrams
• BCFW Recurrence Relation
• Loops
• Need for NLO
• Unitarity & Cut-Constructibility
• General Algorithm for Cuts
• NLO six-gluon amplitude in QCD
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Spinortrix
Xu, Zhang, Chang
Berends, Kleiss, De Causmaeker
Gastmans, Wu
Gunion, Kunzst
• on-shell massless Spinors
|i±〉 ≡ |k±i 〉 ≡ u±(ki) = v∓(ki) , 〈i±| ≡ 〈k±
i | ≡ u±(ki) = v∓(ki) ,
• k2 = 0 : kaa ≡ kµσµaa = λk
a λka or k/ = |k±〉〈k±|
• Spinor Inner Products
〈i j〉 ≡ 〈i−|j+〉 = εab λai λ
bj =
q
|sij| eiΦij , [i j] ≡ 〈i+|j−〉 = εab λ
ai λ
bj = −〈i j〉∗ ,
with sij = (ki + kj)2 = 2ki · kj = 〈i j〉[j i] .
• Polarization Vector
ε+µ (k; q) =
〈q|γµ|k]√2〈q k〉
, ε−µ (k; q) =
[q|γµ|k〉√2[k q]
,
with ε2 = 0 , kµ · ε±µ (k; q) = 0 , ε+ · ε− = −1 .
Changes in ref. mom. q are equivalent to gauge trasformations.
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MHV Amplitudes
• n-gluon Amplitudes
An(1±, 2
+, . . . , n
+) = 0
AMHVn (1
−, 2
−, 3
+, . . . , n
+) =
〈1 2〉4
〈1 2〉〈2 3〉 · · · 〈n 1〉 Holomorphic in 〈ij〉-product !!!
guessed by Parke & Taylor; proven by Berends & Giele via Recurrence Relation
ANMHVn (1−, 2−, 3−, 4+, . . . , n+) = · · · , does contain both 〈ij〉 and [ij] Kosower (1990)
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Gluons dance TwistWitten, hep-th/0312155
• Twistor Space Penrose (1967): (Z1, Z2, Z3, Z4) = (λ1, λ2, µ1, µ2), µa = −i∂
∂λa
as a Fourier transform with respect to plus-helicity spinor.
• Witten’s observation
In Twistor Space, as a consequence of the holomorphy,
A(λi, µi) =
ZY
i
d2λi exp
“
iX
j
µaj λja
”
A(λi, λi)
external points lie on specific curves!
MHV amplitudes for gluon scattering are associated with collinear arrangements of points
Collinear Operator:
Fi,j,k = 〈ij〉 ∂
∂λk
+ 〈ki〉 ∂
∂λj
+ 〈jk〉 ∂
∂λi
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CSW conjecture
All the non-MHV n-gluon tree amplitudes are expressed as sum of tree graphs whose vertices
are MHV amplitudes continued off-shell, and connected by scalar propagators
Cachazo, Svrcek, Witten; hep-th/040347 .
• MHV rules for An
`q−, (n − q)+
´
1. Draw all the skeletons with q-1 nodes and q-2 links and assign helicity to the links,
+− −+
2. Keep only MHV-node configurations
6+
1−
2−
+
3−
4+
5+
−6+
1−
2−
−
3−
4+
5+
+
YES NO
YES =〈1 2〉4
〈1 2〉〈2 P 〉〈P 6〉〈6 1〉 · 1
P 2· 〈P 3〉4
〈P 3〉〈3 4〉〈4 5〉〈5 P 〉
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• CSW off-shell continuation
〈i P 〉 → 〈i|P/ |η]
where η is an arbitrary reference momentum (which MUST disappear from any physical
quantity).
• Massless Projection Bena, Bern, Kosower Pµ = P [µ + P2
2P ·η ηµ ⇒ 〈i P 〉 → 〈i P [〉
• (some) Applications:
- fermion-gluon coupling Georgiou & Khoze
- massive Higgs Dixon, Glover & Khoze
- Vector Boson Currents Bern, Forde, Kosower & PM
- massless QED Ozeren & Stirling
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BCFW Recurrence RelationBritto, Cachazo, Feng; hep-th/0412308
& Witten; hep-th/0501052
Consider A(1, 2, . . . , n), and pick up
any two special legs, say k and n.
k
21
n
• Analytic continuation, A → A(z):
pµk → p
µk(z) ≡ p
µk + z〈k|γµ|n]
pµn → p
µn(z) ≡ p
µn − z〈k|γµ|n]
If n ∈ {i, . . . , j} && k /∈ {i, . . . , j}
⇒ P2ij ≡ (pi + . . . pj)
2 → P2ij(z) = P
2ij − z〈k|Pij/ |n]
∴ the propagator develops a simple pole @ z = zij ≡ P2ij
〈k|Pij/ |n] .
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• Cauchy Residue Theorem
1
2πi
Idz
zA(z) = C∞ = A(0) +
X
α
Res|z=zα
A(z)
z
To get back the physical amplitude , A(0), use the residue theorem (the surface term does
vanish C∞ = 0)
A(0) =
k
21
n
= −X
i,j
i + 1
k
j − 1
∓±
j
i
n
1
AL AR
A(0) =X
i,j,h
AhL(zij)
1
P 2ij
A−hR (zij)
B holds in the massive case too
Badger, Glover, Khoze & Svrcek: hep-th/0504159
Forde & Kosower: hep-th/0507292
Ozeren & Stirling: hep-ph/0603071
B Loops (!) unreal poles in
complex momentum space from
splitting functions: [a b]〈a b〉
∴ non-trivial factorization
Bern, Dixon & Kosower; hep-ph/0505055
Bern, Bjerrum-Bohr, Dunbar & Ita; hep-ph/0507019
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Need for NLO
• Less sensitivity to unphysical input scale (→ renormalization & factorization scale)
• Precision measurments
• More accurate estimates of background for new physics
• More physics: initial state radiation, final state parton merging in jets, richer virtuality
• Matching with resummed calculation (→ MC@NLO, NLOwPS)
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Need for NLO
• Less sensitivity to unphysical input scale (→ renormalization & factorization scale)
• Precision measurments
• More accurate estimates of background for new physics
• More physics: initial state radiation, final state parton merging in jets, richer virtuality
• Matching with resummed calculation (→ MC@NLO, NLOwPS)
NLO Bricks
• Born level n-point
• Real contribution (n + 1)-point
• Virtual contribution n-point
• IR safety: R + V
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Need for NLO
• Less sensitivity to unphysical input scale (→ renormalization & factorization scale)
• Precision measurments
• More accurate estimates of background for new physics
• More physics: initial state radiation, final state parton merging in jets, richer virtuality
• Matching with resummed calculation (→ MC@NLO, NLOwPS)
NLO Bricks
• Born level n-point
• Real contribution (n + 1)-point
• Virtual contribution n-point
• IR safety: R + V
)( Virtual contribution: Tensor Reduction → Iµνρ...
=
Z
dD
``µ`ν`ρ . . .
D1D2 . . .
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Situation @ NLO
QCD
• 2 → 3Bern, Dixon, Kosower (1993)
Kunszt, Signer, Trocsanyi (1994)
• 1 → 4 Campbell, Glover, Miller (1997)
• 2 → 4 Bern, Dixon, Kosower (1997)
relevant for status
pp → 3 jets ×pp → V + 2 jets ×pp → tt + jet w.i.p
pp → H + 2 jets w.i.p
EW • 2 → 4Denner, Dittmaier, Roth, Wieders (2005)
Boudjema et al. (2005)
relevant for status
e+e− → 4 fermions ×e+e− → HHνν ×
)( 6-point amplituderelevant for status
pp → W +W− + 2 jets ?
pp → tt + bb ?
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Situation @ NLO
QCD
• 2 → 3Bern, Dixon, Kosower (1993)
Kunszt, Signer, Trocsanyi (1994)
• 1 → 4 Campbell, Glover, Miller (1997)
• 2 → 4 Bern, Dixon, Kosower (1997)
EW • 2 → 4Denner, Dittmaier, Roth, Wieders (2005)
Boudjema et al. (2005)
)( 6-point amplitude
(some) Algebraic-Semi-Numerical Approaches
• Ferroglia, Passera, Passarino, Uccirati
• Ellis, Giele, Glover, Zanderighi ⇒ gg → gggg [EGiZ,hep-ph/0602185]
• Binoth, Guillet, Heinrich, Pilon, Schubert
• GRACE group
• Nagy, Soper
• Anastasiou, Daleo
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Analytic Approach
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One Loop AmplitudesBritto, Buchbinder, Cachazo & Feng; hep-ph/0503132
Britto, Feng & PM; hep-ph/0602187
PV Tensor Reduction
Ag =X
i
c4,i +X
j
c3,j +X
k
c2,k + rational
Since the D-regularised scalar functions associated to boxes (I(4m)4 , I
(3m)4 , I
(2m,e)4 , I
(2m,h)4 , I
(1m)4 ),
triangles (I(3m)3 , I
(2m)3 , I
(1m)3 ) and bubbles (I2) are analytically known
Bern, Dixon & Kosower (1993)
• Ag is known, once the coefficients c4, c3, c2 and the rational term are known: they all are
rational functions of spinor products 〈i j〉, [i j]
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Supersymmetric decomposition Bern, Dixon, Dunbar & Kosower (1994)
Ag = (Ag + 4Af + 3As)| {z }
−4 (Af + As)| {z }
+ As|{z}
N = 4 N = 1 N = 0
• non-zero coefficients
N Box Triangle Bubble rational
4 ×1 × × ×0 × × × ×
Massless SuSy 1-Loop amplitudes constructible from 4-dim tree amplitudes, i.e. fully
determined by their discontinuities (or absorptive parts)
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Unitarity & Cut-Constructibility
• Discontinuity accross the Cut
Cut Integral in the P 2ij-channel
i
i + 1
j
l1
l2
j + 1
i − 1
Ci...j = ∆(A1-loopn ) =
Z
dµ Atree
(`1, i, . . . , j, `2)Atree
(−`2, j + 1, . . . , i − 1,−`1)
with dµ = d4`1 d4`2 δ(4)(`1 + `2 − Pij) δ(+)(`21) δ(+)(`2
2)
Ci...j = ∆(A1-loopn ) =
X
i
c4,i +X
j
c3,j +X
k
c2,k
• The Cut carries information about the coefficients.
• In 4-dim we lose any information about the rational term
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IR Divergence
In Dim-reg, D = 4 − 2ε, the divergent behaviour of Ag is:
A1−loopg |singular =
1
εAtree
g
• Divergence of scalar functions
Function I(4m)4 I
(3m,2m,1m)4 I
(3m)3 I
(2m,1m)3 I2
Divergence none (−P2)−ε
ε2none (−P2)−ε
ε21ε
• No need for 1m- & 2m-triangles
Their coefficients are related to the Box-coefficients, as required by the mutual cancelation of
the corresponding divergent pieces.
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Reduced Basis
Ci...j = ∆(A1-loopn ) =
X
i
c4,i +X
j
c3,j +X
k
c2,k
with boxes (I(4m)4 , I
(3m)4 , I
(2m,e)4 , I
(2m,h)4 , I
(1m)4 ), triangles (I
(3m)3 ) and bubbles (I2)
The global divergence of the amplitude is carried by the bubble coefficients:
X
k
c2,k = Atreeg
• coefficients show up entangled in a given cut: how do we disentangle them?
The polylogarithmic structure of boxes, 3m-triangles, and bubbles is different. Therefore their
double cuts have specific signature which enable us to distinguish unequivocally among them.
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Quadruple Cuts
Boxes
• Multiple Cuts Bern, Dixon, Dunbar, Kosower (1994)
j
i + 1 i
m + 1
m
k + 1
j + 1
k
Aj
Ak
Ai
Am
The discontinuity across the leading singularity, via quadruple cuts, is unique, and corresponds
to the coefficient of the master box Britto, Cachazo, Feng (2004)
c4,i ∝ Atreej A
treek A
treem A
treei
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Double Cuts
3m-Triangles & Bubbles
Ci...j = ∆(A1-loopn ) =
Z
dµ Atree
(`1, i, . . . , j, `2)Atree
(−`2, j + 1, . . . , i − 1,−`1)
with dµ = d4`1 d4`2 δ(4)(`1 + `2 − Pij) δ(+)(`21) δ(+)(`2
2)
• Twistor-motivated Integration Measure Cacahazo, Svrcek & Witten (2004)
Use the δ(4) integral to reduce just to a single loop momentum variable
ˆ= `µγµ
= |`〉[`| ≡ t |λ〉[λ| , ∀t ∈ RZ
d4` δ
(+)(`
2) f(|`〉, |`]) =
Z ∞
0
t dt
Z
|λ]=|λ〉∗〈λ dλ〉[λ dλ] t
αf(α)(|λ〉, |λ])
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Cutting a Bubble
∆I2 = =
Z
d4` δ
(+)(`
2) δ
(+)((`−K)
2) =
Z ∞
0
t dt
Z
〈λ dλ〉[λ dλ] δ(+)
((`−K)2)
Since
δ(+)
((` − K)2) = δ(+)(K2 − 2` · K)
= δ(+)(K2 − 〈`|K|`]) =1
〈λ|K|λ]δ
(+)
„
t − K2
〈λ|K|λ]
«
• after the t-integration
∆I2 =
Z
〈λ dλ〉[λ dλ]K2
〈λ|K|λ]2
• Derivative Form (via Integration-by-Parts)
[λ dλ]
〈λ|K|λ]2= [dλ ∂|λ]]
[η λ]
〈λ|K|η]〈λ|K|λ], ∀η : η
2= 0
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• Holomorphic Anomaly Cachazo, Svrcek, Witten (2004)
The last integration is carried out along the contour |λ] = |λ〉∗, and that give rise to a
delta-function like contribution, similar to
∂
∂z
1
z − a= 2πδ(z − a)
In this case, Cauchy Residue Theorem would imply the contribution at the pole
|λ〉 = K|η] (⇔ |λ] = K|η〉)
With the final result
∆I2 = lim|λ〉→K|η]
〈λ|K|η]
„K2[η λ]
〈λ|K|η]〈λ|K|λ]
«
= 1.
• The discontinuity of a bubble is rational !!!
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Cutting a 3m-Triangle
∆I3 =
K2
K1 K3
=
Z
d4` δ
(+)(`
2)
δ(+)((` − K1)2)
(` + K3)2
• after the t-integration
∆I3 =
Z
〈λ dλ〉[λ dλ]1
〈λ|K1|λ]〈λ|Q|λ]
with Q = (K23/K2
1)K1 + K3
• Feynman parameter
∆I3 =
Z 1
0
dz
Z
〈λ dλ〉[λ dλ]1
〈λ|(1 − z)K1 + zQ|λ]2=
Z 1
0
dz1
[(1 − z)K1 + zQ]2
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Finally
∆I3 =1√∆3m
ln
„2a + b − √
∆3m
2a + b +√
∆3m
«
with a = (Q − K1)2, b = 2((K1 · Q) − K2
1) and
∆3m = (K21)
2 − 2K21K
22 − 2K
21K
23 − (K
22 − K
23)
• The discontinuity a 3m-Triangle is a ln(irrational argument) !!!
• The double cut detect box-coefficient as well. One can show that the discontinuity of a
1m-,2m-,3m-box is a ln(rational argument). When it appears, it is flagged and forgotten
(later used as crosscheck), since we prefer to compute it from 4-ple cut.
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Cut-Constructible Ag
Ag =k
n1
2
=X
i
c4,i +X
j
c3,j +X
k
c2,k
c2,i =
"k
n1
2
#
∝ rational
c3,i =
"k
n1
2
#
∝ ln(irr.)
c4,i =k
n1
2
=
"k
n1
2
#
∝ ln(rat.)
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General Algorithm
Sew Trees Remove `1 or `2dµCSW =
tdt 〈λ dλ〉[λ dλ]
Canonical
Decomposition
G(|λ〉,|λ])
〈λ|P |a]α 〈λ|P |λ]βG(|λ〉,|λ])
〈λ|P Q|λ〉α 〈λ|P |λ]βF (|λ〉)
〈λ|P |a]α 〈λ|P |λ][λ a]F (|λ〉)
〈λ|P Q|λ〉α 〈λ|P |λ]〈λ|Q|λ]
Feynman Param.
Derivative Form
Higher Poles
Holom. AnomalyCauhchy Residue Th.
Feynman Int.
3m-Triangle coeff.Bubble coeff.
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6-gluon Amplitude in QCD
Amplitude N = 4 N = 1 N = 0|CC N = 0|rat(− − + + ++) BDDK’94 BDDK’94 BDDK’94 BDK’05, KF’05
(− + − + ++) BDDK’94 BDDK’94 BBST’04
(− + + − ++) BDDK’94 BDDK’94 BBST’04
(− − − + ++) BDDK’94 BBDD’04 BBDI’05, BFM
(− − + − ++) BDDK’94 BBCF’05, BBDP’05 BFM
(− + − + −+) BDDK’94 BBCF’05, BBDP’05 BFM
Quadruple CutsBidder, Bjerrum-Bohr,
Dunbar & Perkins (2005)
Double Cuts
→
→ &
P. Mastrolia, UniZh - Twistors & Unitarity, 29
Analytic Tools for One-Loop Amplitudes
Unitarity-based methods
Bern, Dixon, Dunbar & Kosower (Jurassical)
Brandhuber, Mc Namara, Spence, & Travaglini (2004,2005)
Quigley & Rozali (2004)
Britto, Buchbinder, Cachazo, Svrcek & Witten (2004, 2005)
Britto, Feng & PM (2006)
B terms with discontinuities ⇔ 4-dimension cuts
B rational terms ⇔ D-dimension cuts
Recurrence Relations
Bern, Dixon & Kosower (2005)
Bern, Bjerrum-Bohr, Dunbar & Ita (2005)
Forde & Kosower (2005) = ⊗
When possible, are more elegant!
B terms with discontinuities ⇔ within the same class of polylogarithm
B rational terms ⇔ Correction Terms driven by the polylog structure
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On-shell scattering amplitudes seem to have properties not evident in the expansion of
off-shell Green-functions in terms of Feynman diagrams (later taken on-shell)
Trees
• CSW as a formalism dealing with on-shell objects: MHV amplitudes as vertices, and scalar
propagators
• BCFW recurrence relation for amplitudes, from general properties of complex functions,
without any specific process-dependence
Loops
• Unitarity & Cut-Constructibility allow to transfer knowledge from tree-amplitudes
• BBCF+BFM developed a systematic technique for finite cuts in any theory:
• CSW integration measure
• Canonical Decomposition: algebraic spinor reduction to Box, Bubble and 3m-Triangle functions
• One-Feynman Parameter for the residual integration
• (purely) Algebraic treatment of higher poles (characteristic of non-SYM)
• gg → gggg: numerically known (EGiZ), and its CC-part analytically known
multileg NLO QCD: squeezing out of the bottleneck!
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