entanglement, conformal invariance, and the double copy€¦ · scattering amplitudes have revealed...

Entanglement and Conformal Invariance in the Double Copy

Cliff CheungCaltech

CC, James Mangan, Chia-Hsien Shen: (20xx.xxxxx)CC, Grant Remmen: (2002.10470)

Zoomplitudes (5/12/20)

Scattering amplitudes have revealed wondrous new structures lurking within QFT, for example:

• double copy (KLT, BCJ)

• scattering equations (CHY)

• assorted -hedra and -topes

which apply to gravity, gauge theory, and certain effective field theories.

Bern, Carrasco, Chiodaroli, Johansson, Roiban (1909.01358)

5 A WEB OF DOUBLE-COPY-CONSTRUCTIBLE THEORIES

Gravity Gauge theories Refs. Variants and notes

N > 4supergravity

• N = 4 SYM theory• SYM theory (N = 1, 2, 4) [1, 2, 31, 291,

292]

N = 4supergravity withvector multiplets

• N = 4 SYM theory• YM-scalar theory from dim.

reduction[1, 2, 31, 293]

• N = 2 ◊ N = 2 constructionis also possible

pure N < 4supergravity

• (S)YM theory with matter• (S)YM theory with ghosts [188] • ghost fields in fundamental rep

Einstein gravity• YM theory with matter• YM theory with ghosts [188]

• ghost/matter fields infundamental rep

N = 2Maxwell-Einsteinsupergravities(generic family)

• N = 2 SYM theory• YM-scalar theory from dim.

reduction[120] • truncations to N = 1, 0

• only adjoint fields

N = 2Maxwell-Einsteinsupergravities(homogeneoustheories)

• N = 2 SYM theory with halfhypermultiplet

• YM-scalar theory from dim.reduction with matter fermions

[121, 294] • fields in pseudo-real reps• include Magical Supergravities

N = 2supergravities withhypermultiplets

• N = 2 SYM theory with halfhypermultiplet

• YM-scalar theory from dim.red. with extra matter scalars

[121, 240]• fields in matter representations• construction known in

particular cases

N = 2supergravitieswith vector/hypermultiplets

• N = 1 SYM theory with chiralmultiplets


[239, 241, 295] • construction known inparticular cases

N = 1supergravities withvector multiplets


• YM-scalar theory with fermions

[188, 239, 241,295]

• fields in matter reps• construction known in

particular cases

N = 1supergravities withchiral multiplets


• YM-scalar with extra matterscalars

[188, 239, 241,295]

• fields in matter reps• construction known in

particular cases

Einstein gravitywith matter

• YM theory with matter• YM theory with matter [1, 188] • construction known in

particular cases

63


R + „R2 + R

3

gravity• YM theory + F

3 + F4 + . . .

• YM theory + F3 + F

4 + . . .[296]

• extension to N Æ 4 replacingone of the factors by undeformedSYM theory

Conformal(super)gravity

• DF2 theory

• (S)YM theory [152, 153]• N Æ 4• involves specific gauge theory

with dimension-six operators

3D maximalsupergravity

• BLG theory• BLG theory [119, 243, 297] • 3D only

Table 4: Non-inclusive list of ungauged gravities and supergravities for which a double-copyconstruction is presently known. Theories are given in four dimensions unless otherwise stated.

gauged. While this program has not yet been completed, important progress has beenmade in formulating double-copy constructions for theories which include, among others,pure supergravities, homogeneous N = 2 Maxwell-Einstein supergravities, homogeneousN = 2 theories with hypermultiplets, large classes of YME or gauged theories, and conformalsupergravities. A list of ungauged and gauged theories for which a double-copy constructionis currently known can be found in Table 4 and Table 5, respectively. Gauge theorieswith fields in various matter (non-adjoint) representations of the gauge group are a rathercommon building block for this class of extended constructions. Useful tools for treatingmatter representations in a way that makes manifest color and numerator relations will beintroduced in Sec. 5.2. We will then discuss systematics of the process of identifying thegravity theory given, through double copy, by a pair of gauge theories and study severalexamples in Sec. 5.3.

Double-copy constructibility is a property that goes beyond gravitational theories. Vari-ous theories without a graviton, most prominently some variants of the DBI theory have alsobeen shown to possess this property (see Table 6). We shall briefly review their constructionin Sec. 5.3.11.

5.1 The rules of the gameTo capture as many gravities as possible, we need to consider gauge theories which are moregeneral than the ones discussed at length in previous sections. At the same time, havingin mind a double-copy construction which leads to a sensible gravity theory with desirablebasic properties, it makes sense to impose some requirements on the gauge theories underconsideration. Some additional requirements will also be imposed for simplicity reasons; inboth cases, one can contemplate generalizations in which some of the stated rules of thegame bent or broken.

First of all, for simplicity, we choose to focus on theories for which amplitudes can beorganized exclusively in terms of cubic graphs. This is a natural generalization of the gaugetheories from the previous sections, which possess this property, and is a natural choice for

64


Gravity Gauge theories Refs. Notes

YMEsupergravities

• SYM theory• YM + „

3 theory

[120, 125, 133,134, 140, 214,216, 257, 283,285, 289]

• trilinear scalar couplings• N = 0, 1, 2, 4 possible

Higgsedsupergravities

• SYM theory (Coulomb branch)• YM + „

3 theory with extramassive scalars

[122] • N = 0, 1, 2, 4 possible• massive fields in supergravity

U(1)R gaugedsupergravities

• SYM theory (Coulomb branch)• YM theory with SUSY broken

by fermion masses[123]

• 0 Æ N Æ 8 possible• SUSY is spontaneously broken• only theories with Minkowski

vacua

gaugedsupergravities(nonabelian)

• SYM theory (Coulomb branch)• YM + „

3 theory with massivefermions

[284]• SUSY is spontaneously broken• only theories with Minkowski

vacua

Table 5: Gauged/YME gravities and supergravities for which a double-copy construction ispresently known.

describing gravities that are entirely specified by their three-point interactions. Hence, werestrict the space of gauge theories under consideration according to the following rule:

Working Rule 1: Consider gauge theories with only cubic invariant tensors or,alternatively, theories for which amplitudes can be organized in terms of cubicgraphs.

Allowed invariant tensors will include, for example, structure constants, representation ma-trices and cubic Clebsch-Gordan coe�cients. It should be emphasized that the gauge the-ories under consideration can and will possess quartic vertices. Our requirement constrainshigher-point interaction vertices to be made of color building blocks which are cubic. If thisproperty is satisfied, amplitudes can be expressed in terms of cubic graphs by including asuitable number of inverse propagators in the numerator factors. While this rule is quitedesirable for the sake of simplicity, it can in principle be broken. A notable violation are thethe Bagger-Lambert-Gustavsson (BLG) and Aharony-Bergman-Je�eris-Maldacena (ABJM)theories, which are most naturally organized in terms of quartic graphs [119, 243, 297].

Within the class of cubic theories, however, we need to consider cases which are as generalas possible. This motivates the second rule:

Working Rule 2: The gauge theories will include matter fields transforming ingeneral (not necessarily irreducible) representations of the gauge group (which isnot necessarily semisimple). Only one adjoint representation will be allowed.

Considering general gauge groups and representations will allow us to capture very largefamilies of (super)gravities which would not otherwise be accessible through double-copymethods. The main observation is that there is nothing in the double-copy construction

65


Double copy Starting theories Refs. Variants and notes

DBItheory

• NLSM• (S)YM theory

[125, 126, 285,298–301]

• N Æ 4 possible• also obtained as –

Õ æ 0 limitof abelian Z-theory

Volkov-Akulovtheory

• NLSM• SYM theory (external fermions) [125, 302–308] • restriction to external fermions

from supersymmetric DBI

Special Galileontheory

• NLSM• NLSM

[125, 285, 301,306, 309]

• theory is also characterized byits soft limits

DBI + (S)YMtheory

• NLSM + „3

• (S)YM theory

[125, 126, 156,285, 298–300,306, 310]

• N Æ 4 possible• also obtained as –

Õ æ 0 limitof semi-abelianized Z-theory

DBI + NLSMtheory

• NLSM• YM + „

3 theory[125, 126, 156,285, 298–300]

Table 6: List of non-gravitational theories constructed as double copies.

that requires that representations be divided into irreducible blocks. At the same time, wewant to obtain theories with a single graviton. This forces us to combine all gauge-theorygluons in a single adjoint representation, even when the gauge group is the product of severalfactors each possessing its own adjoint representation. In case of more than one semi-simplefactor in the gauge group, we need to take all gauge coupling constants to be the same. Sinceall fields in the gauge theory have canonical couplings with gluons, our second rule can alsobe regarded as the double-copy incarnation of the Equivalence Principle.

Additionally, massive fields are typically assigned to non-adjoint representations suchthat all the fields in a given representation have the same mass. This will be accompaniedby mass-matching conditions of the spectrum of the two sides of the double copy.

Combining the first two rules, we obtain a generic amplitude structure that involves cu-bic graphs in which internal and external legs carry definite representations of the gaugegroup. Cubic vertices between three representations are allowed only when it is possibleto extract a gauge singlet in their tensor product (or, alternatively, there exist a nonvan-ishing invariant tensor with the three corresponding indices). Whenever a vertex involvestwo lines carrying the same representation, its symmetry or antisymmetry will be dictatedby the representations under consideration (real representations will imply antisymmetry,pseudo-real representation will imply symmetry). Additionally, color factors will obey three-term identities following from the Jacobi relations, the generators’ commutation relationsand additional algebraic relations which may also involve the Clebsch-Gordan coe�cients.Consequently, the duality between color and kinematics must to be imposed in the followingway:

66

Case in point, the double copy construction is weirdly ubiquitous among “nice” theories.

Who ordered all this and why?

But KLT, BCJ, CHY all employ a handful of basic building blocks with the multiplication table:

ϕ3

YM

NLSM

YM

gravity

DBI

NLSM

DBI

Galileon

ϕ3

YM

NLSM

YM NLSM ϕ3

The fewer the parameters, the more the theory is constrained. Single-coupling theories are thus maximally constrained without being eliminated.

Poincare +factorization gravity, YM

Poincare +factorization +soft theorems

NLSM, DBI, Galileon

But unique theories need not be related! So why is DBI + Galileon connected to YM, etc.?

In fact, when several unique theories appear in the same framework it motivates us to look for a single unifying principle.

For the remainder of my talk I will discuss how:


• conformal invariance is actually a common thread of many theories in the double copy



• conformal invariance is actually a common thread of many theories in the double copy

• unity of gravitons, gluons, pions, and Galileons can be understood via dimensional reduction as well as entanglement


1) “double copy theories are conformal”

1) “double copy theories are conformal”

( classically )

( scalar )

Conditions like soft theorems or gauge invariance require interference between Feynman diagrams.

+pn4 pn4 pn6

p−2

2n4 − 2 = n6

Such theories are labeled by a single derivative counting parameter, .ρ

L = ∑n

cn ( ∂ϕ )2 ( g ∂ρϕ )n

numerical coefficient

coupling constant

To mandate scale invariance, set ,so given , the critical dimension is

[ g ∂ρϕ ] = 0[ ϕ ] = d/2 − 1

d = 2(1 − ρ)

which for each double copy theory is

(d = 6)ϕ3

YM

gravity

DBI

Galileon

NLSM

(d = 4)

(d = 2)

(d = 2)

(d = 0)

(d = − 2)!!?

Scale invariance is trivial. Conformal invariance usually follows automatically but here we will find additional constraints in formal . d = 0, − 2

Specifically, we calculate in general dimension throughout, fixing at the very end. Gram determinants ill-defined so we do not incorporate them here.

d = 0, − 2

Of course, SYM and NLSM have nice properties like integrability, so one can speculate this might generalize to other double copies.

d = 4 d = 2

where

T = 0

Conformal symmetry requires that the stress tensor vanishes up to on-shell zeros and so-called improvement terms.

In particular, the stress tensor is ambiguous up to additionally conserved tensors

Tμν → Tμν + ΔTμν ∂μΔTμν = 0

ΔS = ∫ −g RμνρσOμνρσ(ϕ) = ∫ hμνΔTμν + ⋯

automaticallyconserved

where we include a general ansatz for the latter to generate all possible improvement terms.

Stress tensor is computed by coupling the field theory to a gravitational background. We include minimal and non-minimal couplings.

S = ∫ −g [ 12 ∇ϕ2 + ⋯]

whereS = ∫ −g L(X)

ΔS = ∫ −g [RA(X) ϕ2 + RμνB(X) ϕ2 ∇μϕ∇νϕ]

X = ∇μϕ∇μϕ

together with the general improvement terms

where , , and are priori arbitrary.L(X) A(X) B(X)

Example #1) Consider a Nambu-Goldstone boson described by the action

ρ = 1

0 = 2A + XB − L′

0 = BL′ + 4A′ L′ + 2XB′ L′ − 2AL′ ′ − 3BXL′ ′

0 = 2A′ − XB′

The unique solution is the DBI action!

L = − 1g 1 + gX + λ A = − gX + 2

8 1 + gXB = g

4 1 + gX

Set , eliminating all appearances of via equations of motion. The tensor structures in

yield scalar differential equations:

T = 0 □ ϕ

T = 0

Example #2) Consider a Nambu-Goldstone boson described by the schematic action

ρ = 2

S = ∫ −g ∑n

cn ∇ϕ2(∇2ϕ)n

Including the many allowed improvement terms and setting on-shell, we constrain the operator coefficients.

T = 0 cn

The unique solution up to 8pt is the action of the special Galileon!

which are fixed by factorization and conformal. On-shell recursion may be possible.

A corollary is that we can “conformal bootstrap” the amplitudes of DBI and the special Galileon,

whereD ⋅ An = Kl ⋅ An = 0 l ∈ 1,2,⋯, n

D = n2 (d − 2) − d + ∑

i,j≠i

sij∂sij

Kl = ∑i,j≠i,k≠i

(sikslj−12 sjksli) ∂sij

∂sik+ d − 2

2 ∑i,j≠i

sjl∂sij

Loebbert, Mojaza, Plefka (1802.05999)

11) “gluons, pions, Galileons are gravitons”

11) “gluons, pions, Galileons are gravitons”

( dimensionally reduced )

massless

transverse

helicity basis

To begin, we think of gluon tree amplitudes as abstract functions of kinematic invariants.

Crucially, we maintain the on-shell conditions.

A = eμ11 eμ2

2 ⋯ eμnn Aμ1μ2⋯μn

scalar function of= pipj, piej, eiej

pipi = piei = eiei = 0

CC, Shen, Wen (1705.03025)

Here we recast the Ward identity as a differential operator that annihilates the amplitude.

A physical on-shell amplitude satisfies several constraints. The first is the Ward identity.

Aei=pi

= Wi A = 0

Wi = ∑v=pj, ej

(piv)∂

∂ (vei)

The second constraint is typically trivial: total momentum conservation.

Pv A = 0

As before, we can define a differential operator for this property of the amplitudes.

Pv = ∑i

piv

If the operator satisfies the conditions,

then if is gauge invariant and momentum-conserving then so too is .

AT ⋅ A

[ Wi , T ] ∼ 0 [ Pv , T ] ∼ 0

Wi ⋅ ( T ⋅ A ) = 0 Pv ⋅ ( T ⋅ A ) = 0

Now let us construct an operator that acts on the amplitude to produce a new one .

TA T ⋅ A

Tij =∂

∂ (eiej)

Ti = ∑j

pipj∂

∂ (pjei)

Tijk =∂

∂ (piej)−

∂∂ (pkej)

gg or → ϕϕ ππ

g → ϕ

g → π

From these vanishing commutators we derive the “transmutation operators”.

We proved transmutation for all graviton, gluon, pion tree amplitudes + explicit checks up to 8pt.

graviton

gluon BI photon

pionscalar ϕ3 Galileon

higherspin

lowerspin

CC, Shen, Wen (1705.03025)

T12 ⋅ T34 ⋅ A(g1, g2, g3, g4) = [ ∂∂ (e1e2)

∂∂ (e3e4) ] A(g1, g2, g3, g4)

Example #1: YM to SQED

=p1p3

p1p2= A(ϕ1, ϕ2, ϕ3, ϕ4) =

Extracting the term is dimensional reduction to two new flavors of charged scalars.

(e1e2)(e3e4)

eμ1 = eμ

2 = (0,1,0) eμ3 = eμ

4 = (0,0,1)d + 1 + 1 d + 1 + 1

Example #2: YM to NLSM

T14 ⋅ T2 ⋅ T3 ⋅ A(g1, g2, g3, g4) = [ ∂∂ (e1e4)

⋯] A(g1, g2, g3, g4)

= p1p3 = A(π1, π2, π3, π4) =

We can reinterpret -dimensional pions as very oddly polarized -dimensional gluons.

d(2d + 1)

eμ1 = eμ

4 = (0,1,0) eμ2 = (pα

2 ,0,ipβ2 )

d + 1 + d d + 1 + d

eμ3 = (pα

3 ,0,ipβ3 )

d + 1 + d

So -dimensional NLSM is an exotic dimensional reduction of -dimensional YM.

d(2d + 1)

Aμ = ( Xα + Zα

2, Y,

Xβ − Zβ

i 2 )(pα,0,ipβ)

(0,1,0)

In terms of the fields, the pion amplitude isX, Y, Z

A(π1, π2, ⋯, πn) = A(Y1, Z2, ⋯, Zn−1, Yn)( Bose symmetry is not manifest! )

From YM we derive manifestly color-kinematic dual actions realizing (NLSM)2 = Galileon.

LYM = − 14 Tr(FμνFμν) + LGF

LNLSM = Tr (Xα □ Zα+ 12 Y □ Y + iXαβ[Zα, Zβ] + iZα[Y, ∂αY])

LGal = Xαα □ Zαα+ 12 Y □ Y + XαβαβZααZββ + ZααY∂α∂αY

CC, Remmen, Shen, Wen (1709.04932)

( transmute the action )

( literally square the action )

111) “gravitons are gluons”

111) “gravitons are gluons”

( entangled )

Double copy relates gauge theory and gravity amplitudes, but an explicit state mapping,

( gluonstate )

2 graviton state

were it to exist, would provide a weak-weak dual description of any gravity setup.

It is tempting to fantasize about applications to black hole evaporation or finite coupling, e.g. ( lattice QCD )2 = numerical relativity.

Question: is there an entangled superposition of gluons which scatters just like gravitons?

• KLT is a relation on color-stripped amplitudes which obscures perm invariance

• BCJ is a relation on color-dressed amplitudes entailing unphysical numerators

We examine a “twin gauge theory”, , where both factors have gauge group.

YM ⊗ YMSU(N)

The answer is nontrivial since states are perm invariant and color-dressed, while

We have made several nontrivial assumptions:

• the kernel is an polynomialK(σ, p) sn−3

• color is contracted across copies only

|p1, ⋯, pn⟩ ≡ ∑a1,⋯,an

∑σ∈Sn

|p1, a1⟩⋯ |pn, an⟩ ⊗ |p1, σ(a1)⟩⋯ |pn, σ(an)⟩ K(σ, p)

|p⟩ ≡ ∑a,σ

|p, a⟩ ⊗ |p, σ(a)⟩ K(σ, p)

Assume an ansatz for an entangled -gluon state:2n

( abridged notation )

• is perm invariant, like gravitons|p⟩

Permutation invariance of the external momenta imposes stringent constraints on the state,

|p⟩ = |τ(p)⟩ = ∑a,σ

|τ(p), a⟩ ⊗ |τ(p), σ(a)⟩ K(σ, τ(p))

= ∑a,σ

|p, a⟩ ⊗ |p, σ(a)⟩ K(τστ−1, τ(p))

so any entanglement kernel is obtained from that of a representative of its conjugacy class in :Sn

K(σ, p) = K(τστ−1, τ(p))K(σ, p) = K(σ, τ(p))

and s.t. ∀τ [σ, τ] = 0

The “graviton” amplitudes from these states are

M = ⟨0 |T |p⟩ = ∑a,σ

⟨0 |TYM |p, a⟩ ⟨0 |TYM |p, σ(a)⟩ K(σ, p)

∝ ∑a,σ

A(p, a) A(p, σ(a)) K(σ, p)

which by construction mirror the KLT relations.

Note these are not actually physical kets because a) all-in formalism has negative energy states, and b) they describe a non-normalized branch

K(1)(2)(3) = α1

( 3pt amplitude ) Assign an independent ansatz function for each conjugacy class of .S3

K(12)(3) = α2 K(123) = α3

The 3pt graviton amplitude is then

M3 = A(1a2b3c)[ α1

6 A(1a2b3c)+α1

2 A(1a3b2c)+α1

3 A(3a1b2c)]+ perm

∝ (α1 − 3α2 + 2α3)A(123)A(123)

which is correct but completely unsurprising since 3pt is fixed by Poincare symmetry.

K(1)(2)(3)(4) = 0

( 4pt amplitude ) Ansatz has 3 parameters, with remaining kernels zero by perm invariance.

The 4pt amplitude is effectively a color-dressed and permutation invariant double copy formula:

M4 = sA(1a2b3c4d)[ β1A(1a2b4c3d) + β1A(2a1b4c3d) + β2A(2a1b4c3d)

K(12)(34) = β2s K(1234) = β3uK(12)(3)(4) = β1s

+ β3A(3a4b2c1d) + β3A(4a3b1c2d)]+ t-channel + u-channel

K(123)(4) = 0

All choices of ansatz coefficients match gravity!

M4 ∝ (β1 + β3)[ sA(1234)A(1234) − tA(1324)A(1234)

− tA(1234)A(1324) + uA(1324)A(1324) ]

This expression is a somewhat more symmetric form of the usual KLT relation.

So any ansatz of entangled gluons scatters like gravitons. More generally, any permutation invariant KLT-like product 4pt amplitudes will be proportional to the correct graviton amplitude.

( 5pt amplitude ) Ansatz has 22 parameters, of which only a 17 dimensional subspace matches the correct graviton amplitude.

Dilaton parity or duality are sufficient to fix to the desired 17 dimensional space.

U(1)

M5 ∝ A(1a2b3c4d5e)[− 120 (s12 + s13 + s24)(s12 + s14 + s23)A(2a1b4c3d5e)

+ 112 s2

12A(2a1b3c4d5e)]+ perm

One can extract a nice-looking perm invariant, color-dressed form of the 5pt graviton amplitude:

A simple corollary is that we can recast entangled pions as Galileons, and so on and so forth.

Amusingly, while entropy is way too difficult to compute, the “purity” is not,

S = − Tr[ρ log ρ]

P = Tr[ρ2] ∼ N−2n

so at large the ensemble is maximally mixed. This is not surprising since there is high entropy in the number of ways rewiring color at large .

N

N

conclusions

• Conformal invariance appears as a common feature of theories in the double copy. DBI and special Galileon are uniquely fixed.

• Gluons, pions, and Galileons can be recast as strangely polarized, higher-dimensional gravitons. This amplitudes-level connection follows from a peculiar version of dimensional reduction.

• A state-to-state double copy mapping would be a powerful tool. Some preliminary steps taken to recast gravitons as “entangled” gluons.

thank you!

entanglement, conformal invariance, and the double copy€¦ · scattering amplitudes have revealed...

Documents