Structure of Amplitudes in Gravity II Unitarity cuts, Loops, Inherited properties from Trees,...

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-

loop

Gronningen 3-5 Dec 2009

3Playing with Gravity

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



l

One-loop amplitudes

3-5 Dec 2009 Playing with Gravity 6

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



log(z) = lnjzj + iarg(z)

General 1-loop amplitudes

Vertices carry factors of loop momentum

n-pt amplitude

p = 2n for gravityp=n for Yang-Mills

Propagators



Mn = ¹ 2²Z

dD `(2¼)D

Q 2nj (q(2n;j )

¹ j¹̀ j ) +

Q 2n¡ 1j (q(2n¡ 1;j )

¹ j¹̀ j ) + ¢¢¢+ K

2̀1 ¢¢¢̀ 2

n

(Passarino-Veltman) reduction

Collapse of a propagator

RdD ` 2(`¢k1)

`2(`¡ k1)2 (¢¢¢) =R

dD ` 1(`¡ k1)2 (¢¢¢) ¡

RdD ` 1

`2 (¢¢¢)

@p

(Maximal graph)

Passarino-Veltman



Due to this generic loop amplitudes have the form:

Illustrative Passarino-Veltman

Unitarity cuts



Generic one-loop amplitude (without R term):

ImK i ; : : : ; j >0A1-loop =X

a

ca 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



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



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



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..



Sum over particles in multiplet (singlet)

Sum over particles in multiplet (non-singlet states)

N=8 Supergravity



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



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



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



[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

X

½2 = X 2 ¡ 4X + 6¡ 41X

+1

X 2 =(X ¡ 1)4

X 2

½4 = X 4 ¡ 8X 3 + 28X 2 ¡ 56X + 70¡56X

+28X 2 ¡

8X 3 +

1X 4

N = 1

N = 4

N = 8

MHV

N=4

Ward identities



Needed to work out For N=8 6pt SUGRAamplitudes

NMHV

Recipe for computations in N=8 SUGRA



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

Playing with Gravity 23

Example of quadruple cut

3-5 Dec 2009

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 ¹̧p is propotional to both ¹̧

K 1 and ¹̧`4

2̀

3̀

4̀

1̀

Examples of cuts



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 =R

dLIPSM4(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



Cut =R

dLIPSM4(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



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



Supergravity amplitudes

(Bern, NEJBB, Dunbar)

Box Coefficients




• A way to organise cuts is through use the scaling behaviour of shifts

3-5 Dec 2009


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


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 as1z.

3-5 Dec 2009


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


Singularity structure of amplitude

Tree amplitude has factorisations:

Loop amplitudes has the following generic factorisation structure: (Bern and Chalmers)

3-5 Dec 2009

M one¡ loopn ¡ !K 2! 0

X

¸ =§

"

M one¡ loopr +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 ¸ )

iK 2 M one¡ loop

n¡ r +1;1 ((¡ K )¡ ¸ ;ki+r ; : : : ;ki ¡ 1)

+ M treer +1(ki ; : : : ;ki+r ¡ 1;K ¸ )

iK 2 M tree

n¡ r +1((¡ K )¡ ¸ ;ki+r ; : : : ;ki ¡ 1) Factn(K 2;k1; : : : ;kn)

#

33

IR singularities of gravity

3-5 Dec 2009 Playing with Gravity

Gravity amplitudes have IR singularities of the form

M one¡ loop²¡ 1 (1;2;: : :;n) »

µ Pi<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

abc

hln(¡ sa f )+2ln(¡ K 2

a bc )¡ ln(¡ sbc )¡ ln(¡ sde )2²

i


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…

3-5 Dec 2009

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:



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)



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



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



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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Structure of Amplitudes in Gravity II Unitarity cuts, Loops, Inherited properties from Trees,...

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