Structure of Amplitudes in Gravity
II
Unitarity cuts, Loops, Inherited properties from
Trees, Symmetries
Playing with Gravity - 24th Nordic Meeting
Gronningen 2009Niels Emil Jannik Bjerrum-Bohr
Niels Bohr International AcademyNiels Bohr Institute
Outline
Outline of lecture II• Summery of lecture I
• Tree amplitudes and Helicity formalism• How to compute and New Techniques
– In this lecture we will consider loop amplitudes in gravity• Traditional methods vs. Unitarity• Supersymmetry and matter amplitudes• Organisation of amplitudes• Twistor Space and amplitudes beyond one-
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Simplicity…SUSY N=4, N=1,QCD, Gravity..
Loops simple and symmetric
Unitarity
Cuts
Trees (Witten)Twistor
s
Trees simple and symmetric
Hidden Beauty!New simple analytic expressions
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l
One-loop amplitudes
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Loop amplitudes in field theory
1
nStandard way:
• Choose gauge• Expand Lagrangian
Features:• 3pt vertex: approx 100 terms• 4pt vertex much worse
• Propagator: 3 terms• Number of topologies grows as n!
Problems: off-shell formalism• Not directly usable with spinor-
helicity
Much worse thantree level – one have to do integrations
In sums of contributionsto loop amplitudes cancellations appear (but only after doinghorrible integrals…)
Unitarity cuts Unitarity methods are building on
the cut equation
Singlet Non-Singlet
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log(z) = lnjzj + iarg(z)
General 1-loop amplitudes
Vertices carry factors of loop momentum
n-pt amplitudep = 2n for gravityp=n for Yang-Mills Propagators
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Mn = ¹ 2²Z dD `
(2¼)D
Q 2nj (q(2n;j )
¹ j¹̀ j ) + Q 2n¡ 1
j (q(2n¡ 1;j )¹ j
¹̀ j ) + ¢¢¢+ K2̀1 ¢¢¢̀ 2n
(Passarino-Veltman) reduction
Collapse of a propagator
RdD ` 2(`¢k1)`2(`¡ k1)2 (¢¢¢) = RdD ` 1
(`¡ k1)2 (¢¢¢) ¡ RdD ` 1`2 (¢¢¢)
@p
(Maximal graph)
Passarino-Veltman
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Due to this generic loop amplitudes have the form:
Illustrative Passarino-Veltman
Unitarity cuts
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Generic one-loop amplitude (without R term):
ImK i ; : : : ; j >0A1-loop =X
aca ImK i ; : : : ; j >0(I a)
Relate kinematic discontinuity of the one loop amplitude. This imposes constraints on the coefficients
M 1¡ loop = P c4I 4 + P c3I 3 + P c2I 2
Early problems in 60ties with cutting techniques is related to not having a integral basis (dimensionally regularised).
Quadruple Cut
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c = 12
X
S
µAtree( 1̀; i1; : : : ; i2; 2̀) £ Atree( 2̀; i3; : : : ; i4; 3̀)
£ Atree( 3̀; i5; : : : ; i6; 4̀) £ Atree( 4̀; i7; : : : ; i8; 1̀)¶
In 4D an algebraic expression!
2̀1 = 02̀2 = 02̀3 = 02̀4 = 0
Boxes only!
(Britto, Cachazo and Feng)
Having complex momentumCrucial for mass-less corners
Triple Cut
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C3 =X
s
Zd4li ±(l21)±(l22)±(l23)M (ls1; im;¢¢¢i j ; ¡ ls2) £ M (ls2; i j +1;¢¢¢i l ; ¡ ls3)
£ M (ls3; i l+1;¢¢¢im¡ 1;¡ ls1)
2̀1 = 02̀2 = 02̀3 = 0
In 4D still one integral left!
Scalar Boxes and Scalar Triangles
Double Cut
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2̀1 = 02̀2 = 0
In 4D still two integrals left!
Scalar Boxes and Scalar Trianglesand Bubbles
Supersymmetry
Unitarity Cuts for different theories
• Have to sum over multiplet to compute supersymmetric amplitudes
• Hence we need tree amplitudes with matter lines..
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Sum over particles in multiplet (singlet)
Sum over particles in multiplet (non-singlet states)
N=8 Supergravity
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DeWit, Freedman; Cremmer, Julia, Scherk; Cremmer, Julia
28 = 256 massless states (helicity)1+1=2 graviton states (+2,-2)8+8=16 gravitino states (+3/2, -3/2)28+28 = 56 vector states (-1,1)56+56 = 112 fermion states (-1/2,1/2)70 scalars (0)
Maximal theory of supergravity
Features:
Need to sum over multiplet of all 256 states… in cuts
KLT and N=4 Yang-Mills
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24 = 16 massless states (helicity)1+1=2 vector states (+1,-1)4+4=8 fermion states (+1/2, -1/2)6 scalars (0)
Maximal theory of super Yang-Mills
Features:
Uses two things:• KLT writes N=8 amplitudes as products of N=4 amplitudes.• [Spectrum of N=8] = [Spectrum of N=4] x [Spectrum of N=4]
Supersymmetric Ward Identities
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Need a method to sum over states in cut• Possibilities:• Use CSW, BCFW, other recursive techniques to generate
amplitudes • Use SUSY ward identities to sum over terms in Cut.
• Very useful for MHV amplitudes• Helps for NkMHV amplitudes but much more work...
Sum over particles in multiplet (singlet)
Sum over particles in multiplet (non-singlet states)
SUSY Ward identities
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[Qa;g§ (p)] = ¨ ¡ § (p;q)¹g§a
[Qa; ¹g§ (p)] = ¨ ¡ ¨ (p;q)g§ ±ab ¨ i¡ § (p;q)s§ab²ab
[Qa;s§ab(p)] = § i¡ (p;q)²ab¹g§
b
Atree(( 1̀)¨ ; i1; : : : ; i2;( 2̀)§ ) = (x)§ 2hAtree(( 1̀)s; i1; : : : ; i2;( 2̀)s)h = 1=2 (fermions) and h = 1 (gluons)x = hl1iai=hl2iai (with ia being the negative helicity gluon leg.)
½= ¡ X + 2¡ 1X = ¡ (X ¡ 1)2
X½2 = X 2 ¡ 4X + 6¡ 4 1
X + 1X 2 = (X ¡ 1)4
X 2
½4 = X 4 ¡ 8X 3 + 28X 2 ¡ 56X + 70¡ 56X + 28
X 2 ¡ 8X 3 + 1
X 4
N = 1
N = 4
N = 8
MHV
N=4
Ward identities
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Needed to work out For N=8 6pt SUGRAamplitudes
NMHV
Recipe for computations in N=8 SUGRA
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1. Write down 1-loop amplitude
2. Write down all helicity configurations
3. Write down all possible cuts (consider various cut channels)
4. Write down cut trees (including all trees with internal SUSY particles)
5. Fix box coefficients from quadruple cuts
6. Fix triangles and bubbles from triple and double cuts
7. Finally check that amplitude does not have rational parts:1. If rational parts exist either compute using cuts in 2. Or use new recursive techniques (will be discussed in lecture III)
4D ¡ 2²
Examples of cuts
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Example of quadruple cut
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c = 12
X
S
µAtree( 1̀; i1; : : : ; i2; 2̀) £ Atree( 2̀; i3; : : : ; i4; 3̀)
£ Atree( 3̀; i5; : : : ; i6; 4̀) £ Atree( 4̀; i7; : : : ; i8; 1̀)¶
2̀1 = 02̀2 = 02̀3 = 02̀4 = 0
Have to solve…)If corners is massive we can just solve constraints
If one corner is massless we have to assume complexmomenta of say Thereby we can write
Where either
1̀
1̀ = ®̧ p¹̧ q¸p is propotional to both ¸K 1 and ¸`4
or ¹̧ p is propotional to both ¹̧ K 1 and ¹̧ `4
2̀
3̀
4̀
1̀
Examples of cuts
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Lets consider 5pt 1-loop amplitude in N=8 Supergravity (singlet cut)
We have M4(1¡ ;2¡ ;`+
1 ;`+2 ) » h12i 7 [12]
h1`1 i h1`2 i h2`1 i h2`2 i h̀ 1 `2 i 2
M5( ¡̀1 ; ¡̀
2 ;3+;4+;5+) » h̀ 1 `2 i 7 (h4`1 i h̀ 2 3i [34] [̀ 1 `2]¡ h34i h̀ 1 `2 i [4`1] [`2 3])h34i h35i h45i h̀ 1 3i h̀ 1 4i h̀ 1 5i h̀ 2 3i h̀ 2 4i h̀ 2 5i
Cut = RdLIPSM4(1¡ ;2¡ ;`+1 ;`+
2 )M5( ¡̀1 ; ¡̀
2 ;3+;4+;5+) =h12i 7 [12]
h1`1 i h1`2 i h2`1 i h2`2 i h̀ 1 `2 i 2 £h̀ 1 `2 i 7 (h4`1 i h̀ 2 3i [34] [̀ 1 `2]¡ h34i h̀ 1 `2 i [4`1] [̀ 2 3])
h34i h35i h45i h̀ 1 3i h̀ 1 4i h̀ 1 5i h̀ 2 3i h̀ 2 4i h̀ 2 5i
Examples of cuts
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Cut = RdLIPSM4(1¡ ;2¡ ;`+1 ;`+
2 )M5( ¡̀1 ; ¡̀
2 ;3+;4+;5+) =h12i7 [12]
h1 1̀i h1 2̀i h2 1̀i h2 2̀i h̀ 1 2̀i2£
h̀ 1 2̀i7 (h4 1̀i h̀ 2 3i [34] [̀ 1 2̀]¡ h34i h̀ 1 2̀i [4 1̀] [̀ 2 3])h34i h35i h45i h̀ 1 3i h̀ 1 4i h̀ 1 5i h̀ 2 3i h̀ 2 4i h̀ 2 5i
In this example we have 4 terms (after some algebra…)» s12 £ M5(1¡ ;2¡ ;3+;4+;5+) £ tr(1; l1; l2;2)
hl1 1i hl2 2i [l1 1] [l2 2]» s12 £ h12i6 [23] [45]
h14i h15i h23i h34i h35i h45i £ tr(3; l1; l2;1)hl1 3i hl2 1i [l1 3] [l2 1]
» s45 £ h12i7 [34] [12]h13i h15i h23i h25i h34i h45i2 £ tr(5; l1; l2;3)
hl1 5i hl2 3i [l1 5] [l2 3]
» s12 £ h12i6 [23] [45]h14i h15i h23i h34i h35i h45i £ tr(3; l2; l1;1)
hl2 3i hl1 1i [l2 3] [l1 1]
Examples of cuts
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tr(1; l2;2; l2) = ¡ h2jP j2](l1 ¡ k1)2 + h1jP j1](l2 + k2)2 ++(l1 ¡ k1)2(l2 + k2)2
tr(1; l2;3; l2) = h1jP j1]h3jP j3]¡ P 2s13¡ h3jP j3](l1 ¡ k1)2 + h1jP j1](l2 + k3)2
+(l1 ¡ k1)2(l2 + k3)2
tr(5; l2;3; l2) = h5jP j5]h3jP j3]¡ P 2s35¡ h3jP j3](l1 ¡ k5)2 + h5jP j5](l2 + k3)2
+(l1 ¡ k5)2(l2 + k3)2
tr(k1; l1; l2;k2) = ¡ tr(k1; l1;k2; l2) + sk1 l1 sk2 l2
Using that
We have
Supergravity boxes
(Bern, NEJBB, Dunbar)
KLT
N=4 YM results can be recycled into results for N=8 supergravity
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Supergravity amplitudes
(Bern, NEJBB, Dunbar)
Box Coefficients
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• A way to organise cuts is through use the scaling behaviour of shifts
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Supergravity amplitudes
M tree¡(¡ 1̀)¡ ; i;¢¢¢;j ;( 2̀)¡ ¢£ M tree
¡(¡ 2̀)+; j + 1;¢¢¢;i ¡ 1;( 1̀)+¢
=X
i2C0
ci( 1̀ ¡ K i ;4)2( 2̀ ¡ K i ;2)2 +
X
j 2 D 0
dj( 1̀ ¡ K j ;3)2 + ek0 + D( 1̀; 2̀)
Let us consider this equation under the shift of the two-cut legs¸`1 ¡ ! ¸`1 + z¸`2 ;~̧̀
2 ¡ ! ~̧̀2 ¡ z ~̧̀
1
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This shift does not change the coe±cients but it does enterthe propagator terms (and possibly the D( 1̀; 2̀)).
In the large-z limit the propagators will vanish as 1z.
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Supergravity amplitudes
This can serve as a way to organise the amplitude. Especially if the large-z limit is zero then bubbles will be vanishing Terms corresponding to box terms will go as While triangles goes as
We will discuss this in more details in Lecture III
» 1z2
» 1z
Factorisationof
amplitudes
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Singularity structure of amplitude
Tree amplitude has factorisations:
Loop amplitudes has the following generic factorisation structure: (Bern and Chalmers)
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M one¡ loopn ¡ !K 2! 0 X
¸ =§
"M one¡ loop
r +1;1 (ki ; : : : ;ki+r ¡ 1;K ¸ ) iK 2 M tree
n¡ r +1((¡ K )¡ ¸ ;ki+r ; : : : ;ki ¡ 1)
+ M treer+1(ki ; : : : ;ki+r ¡ 1;K ¸ ) i
K 2 M one¡ loopn¡ r +1;1 ((¡ K )¡ ¸ ;ki+r ; : : : ;ki ¡ 1)
+ M treer+1(ki ; : : : ;ki+r ¡ 1;K ¸ ) i
K 2 M treen¡ r +1((¡ K )¡ ¸ ;ki+r ; : : : ;ki ¡ 1) Factn(K 2;k1; : : : ;kn)
#
33
IR singularities of gravity
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Gravity amplitudes have IR singularities of the formM one¡ loop
²¡ 1 (1;2;: : :;n) »µ P
i<j si j ln(¡ si j )2²
¶£ M tree(1;2;: : :;n) :
IR singularities can arise from both box and triangle integral functions
I (abc)def j1=² » ¡ 2sdesef
hln(¡ sde)+ln(¡ sef )¡ ln(¡ K 2
a bc )²
i
I a(bc)(de)f j1=² » ¡ 2sa f K 2
a bc
hln(¡ sa f )+2ln(¡ K 2
a bc )¡ ln(¡ sbc )¡ ln(¡ sde )2²
i
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Singularity structure of amplitude
• Singularity structure can be used to check validity of amplitude expressions
• Looking at IR singularities can be used to determine if certain terms are in amplitude
• Complete control of singularity structure can be used to do recursive computations–Will discuss more in Lecture III…
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Twistor space
symmetry
Twistor space properties of gravity loop amplitudes
• Unitarity : loop behaviour from trees– Cuts of the MHV box
– Consider the cut C123, where the gravity tree amplitude is Mtree(l5, 1, 2, 3, l3).
– This tree is annihilated by F3(123)• Hence F3(123)cN=8(45)123 = 0• Similarly F3(145)cN=8(45)123 = F3(345)cN=8(45)123 = 0. • Remaining choices of Fijk : consider more generalised cuts,
e.g., C(4512) and hence F4(124)cN=8(45)123 = 0.
• Summarising:
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Twistor space properties of gravity loop amplitudes
• Inspecting the general n-point case, we can now predict
• Similarly we can deduce that (consistent with the YM picture),
Topology : As N=4 super-Yang-Mills ) Points lie on three intersecting lines (Bern, Dixon and Kosower)
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Multi-loopamplitudes
Multi-loop amplitude• Most of the cut techniques we have
discussed can be applied also at multi-loop level
• Difficulties: more difficult factorisations + no set basis of integral functions
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Conclusions
Conclusions• We have seen how it possible to deal
with loop amplitudes in new and efficient ways
• On-shell tree amplitudes can be used as input for cuts. – Calculating all cuts we can compute the
amplitude– Feature: Symmetries for tree amplitudes leads to symmetries for loop amplitudes
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Outline af III• In Lecture III – we will discuss how new techniques for
gravity amplitudes can be used learn new aspects of gravity amplitudes
• Among other things we will discuss– Additional symmetry for gravity– No-triangle Property of N=8
Supergravity• Possible Finiteness of N=8 Supergravity
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