Lessons from massive S-matrix

Lessons from massive S-matrix

Yu-tin Huang (National Taiwan University)

with Nima Arkani-Hamed, Tzu-Chen huang

Arxiv:1709.04891 ⊕ to appear

NTU-Nov-9-2017

Often by studying consistency conditions directly on physical observables yield model independent constraint:

• Tree-level unitarity constraint for WLWL→WLWL

• Positivity for higher dimensional operators in EFT:A. Adams, N. Arkani-Hamed, S. Dubovsky, A. Nicolis and R. Rattazzi

A(s, t)|t→0=X

i

cis²ⁱ ci>0

• Bounds on gap for operator dimensions in CFTSheer El-Showk, Miguel F. Paulos, David Poland, Slava Rychkov, David Simmons-Duffin, Alessandro Vichi

→

• Consistent massive amplitudes in the IR

• Towards the spectrum for large N QCD

• Constraints for higher dimension operators

s t= (p+p )^2

1 4

s= (p+p )^2

12

Consistent QFTs from unitarity and locality of S-matrix

Kinematics:

• Massless:

p²=0 → det[p^{α ˙α}] =0, p^{α ˙α}= λ^α˜λ^α˙

Particle species are labelled by their representation under the U(1) little group

λ^α→ e^iθ2λ^α, λ˜^α˙ → e^−iθ2λ˜^α˙

• Massive:

p²=m²→ det[p^{α ˙α}] =m², p^{α ˙α}= λ^αIλ˜^α_I^˙

Particle species are labelled by their representation under the SU(2) little group λ^αI→ g^IJλ^αJ, ˜λ^αI˙ → g^IJλ˜^αJ˙

Consistent QFTs from unitarity and locality of S-matrix

The S-matrix is a Lorentz invariant function that transforms covariantly under the little group

• Massless:

M_n^h1h2···hn(e^iθ2λi,e^−iθ2λ˜i) = Y

i

e^{ihi θi2}

!

M_n^h1h2···hn(λi, ˜λi)

• Massive: if the i-th leg is spin-L

M_n^{h1···{I1I2,···I2L}···hn}(λ^I_i, ˜λ^I_i, λi, ˜λj)

Massless warm up:

• Kinematics: the three point amplitude for arbitrary massless particles is fixed by its helicity (h1,h2,h3)

where h12i = λ^α₁λ2α, pi= λiλ˜i.

• Dynamics: the residue of the four point function is fixed

Massless warm up:

• Dynamics: the residue of the four point function is fixed

Consistency of the four-point function in all channels impose theory constraints:

The solution:

An interaction vector theory has a global Lie-2 algebra structure

Further constraints from coupling mixed spins:







Rs=^{(h13i[24])2Sh3|p}_u2^{1−p4|2]4−2S} Ru=^{(h13i[24])2Sh3|p1−p4|2]4−2S}

s²

which leads to the following universal form of coupling to gravity

Note that for S > 2 there is unphysical poles →there can be no fundamental massless particle with S > 2 in flat space

Now for massive:

Find two vectors (vαuβ)to span the SL(2,C) space. All possible massive amplitudes are just all possible polynomial function of (vαuβ)

• 2 massless 1 massive

Unique!

• 1 massless 2 massive

For unequal mass:

• 1 massless 2 massive

For equal mass: v , u becomes colinear

v^αuα=0 →

Exposes the simplicity of the interactions:

Minimal coupling is just x !

Consistent factorisation at four-point leads to similar theory constraints:

Compton scattering for spin-3/2

Leads to

At high energies:

There are no isolated fundamental charged particles beyond spin-1! No fundamental particles beyond spin-2

Higgs mechanism as an IR unification

If particles are fundamental, the IR massive amplitude must be consistent with massless UV ones → recovers all features of the Higgs mechanism

Shows unambiguously an obstruction for spin-2

At high energies

Applications

Streamline loop calculations:

Lets consider the e⁺,e⁻→ γ at one loop. The diagram we want to build is:

+

+ + −

−

p₁ p₂

q

a b

c ∼ e³m³xaεαβ

"

ε^βγxb

xc

 ε +xc

λ`λ`

m

αδ

+ ε^αδxc

xb

 ε −xb

λ`λ`

m

βγ#

The pure ε piece is identical to a charged scalar. The magnetic moment is then e²m²xa(xb− xc)λ^δ_{`</sub>λ<sup>γ</sup><sub>`}= −mxaq^δα˙`^αβ˙

This is a simple linear integrand:

−mxa

Z d⁴` (2π)⁴

q^δα˙`^αβ˙

`<sup>2</sup>((` −p2)²− m²)((` +p1)²− m²) = e² (4π)²2xa

q^δα˙p^αβ₁^˙

m = α

2πx^aλ^γ_qλ^δ_q

Applications

W⁺,W⁻→ γ fits on a page:

+

+ + −

− p₁ q

a b c

p₂

2 2 1 1

∼ e³m³xaε_(α1α2ε_β1)β2

"

ε^β1(γ1ε^α1δ1)xb

xc

 ε +xc

λ`λ`

m

α₂(δ₂ ε +xc

λ`λ`

m

β₂γ₂)

+ε^β2(γ2ε^α2δ2)xc

xb



ε −x_bλ`λ`

m

α₁(δ₁

ε −x_bλ`λ`

m

β₁γ₁)# .

Leaving behind the electric coupling:

−4e²xam h

ε^δ1(δ2q^γ1α˙`<sup>αγ˙ 2)</sup>+ ε<sup>γ1(δ2</sup>q<sup>δ1</sup>α˙`^{αγ˙ 2)} i

f₁(q)

+2e²xa

3m h

(p1 ˙αδ₁`<sup>αγ˙ 1</sup>)(p2 ˙αδ<sub>2</sub>`^{αγ˙ 2}) +perm i

f₂(q)

Which leads to:

F1(q) = I3[f1(q)] = 4α 2πx^a



ε^δ1(δ2λ^γ_q¹λ^γ_q²⁾+ ε^γ1(δ2λ^δ_q¹λ^γ_q²⁾

 .

F2(q) = I3[f2(q)] = α (4π)9m³

O^δ_1,2^1γ1O^δ_1,2^2γ2+perm : δ1, γ1, δ2, γ2

 ,

where we’ve defined O_i,j^αβ≡ pi ˙ααp^αβ_j^˙ .

Applications

Correlation functions:

hφ(x1) · · · φ(xn)i In real measurements we are really considering

h ˜φ(k1) · · · ˜φ(kn)i ≡Fourier [hφ(x1) · · · φ(xn)i]

For arbitrary k ! (This is why local operators cannot exist in gravity)

Instead of Fourier transform position space calculations, we can compute massive S-matrix with ˜φ(k1)as a massive higher spin particle!

Can one learn anything from S-matrix constraint of consistent UV completions?

• UV completion of gravity

• Spectrum of bound states in SU(N) YM

For SU(N) QCD the spectrum contains stable glueballs and mesons

• From analyticity of the S-matrix if ¹

s^LA(s, t)|s→∞=0 for some t and L, then after subtraction A⁰

A⁰(s, t) = I

C

dv

v − sA⁰(v , t) = −

I

C⁰

dv v − sA⁰(v , t)

This implies an alternative representation:

A⁰(s, t) =X

i=1

n(mi,t) s − m²_i

To reproduce the poles in t we must let the spectrum tend to infinity!

Unitarity requires that

→

The residue must take the simple form:

[(p1− p2) · (p3− p4)]²ⁿ= (t − u)²ⁿ→ n(mi,t) = P`(1 + 2t/m_i²) In other words

A⁰(s, t) =X

i=1

c²_iP`_i(1 + 2t/m²_i) s − m²_i Using this representation, at 1 << s < t

A(s, t)|s→∞s^2+J(t)

the Regge trajectory must be linear!Simon Caron-Huot, Zohar Komargodski, Amit Sever, and Alexander Zhiboedov

Can we find a function that behaves string like at t > 0 but polynomial at t < 0 ? String in AdS! Polchinski and Strassler (Siegel and Andreev):

We can do even better!

A^π(s, t) = (s + t)Γ[−s]Γ[−t]

Γ[1 − s − t]+1 s +1

t We would obtain

Reproduces linear at t > 0, and hard scattering at t < 0!

Can we gather information with respect to the spectrum from the IR ?

In the IR again these UV degrees of freedom are encoded in the higher dimensional operators:

A(s, t) =X

i,j

gi,jsⁱt^j

There are already known constraints from unitarityA. Adams, N. Arkani-Hamed, S. Dubovsky, A.

Nicolis and R. Rattazzi

For t = 0, we can write

gi,0= I ds

sⁱ⁺¹A(s, 0) = =

Z∞ s₀

Im[A(s, 0)] i ∈ even

But from the optical theorem we know that Im[A(s, 0)] = sσ > 0 →gi,0>0

There are more!

For fixed t we have

a a

b b

1 4

2 3

A(s, t) = − 1

s − m²_i + 1 u − m²_i

! c²_iG_`^D

i 1 + 2t

m²_i

!

G^D_`

i are Gegenbauer polynomials with G_`^D

i(1) > 0.

The low energy expression can be expanded in a way that reflects the s ↔ u symmetry

s = −t

2+z, u = −t 2− z Then

A(z, t) =X

i,j

g˜i,jz²ⁱt^j

A(z, t) =X

i,j

g˜_i,jz²ⁱt^j

For a fixed mass dimension L, the space of possible higher dimensional operators has dimension that correspond to the number of (i, j)s that satisfies 2i + j = L. Any particular theory lives on a particularpointin this subspace:

Exp: for L = 4 we have

˜g_0,4t⁴+g˜_1,2z²t⁴+g˜_2,0z⁴

A given theory is represented as a specific point(˜g0,4, ˜g1,2, ˜g2,0)in this three-dimensional space

Within this space, what is the subspace that has a UV completion of the form?

A(z, t) = −X

i

1

−₂^t +z − m_i²+ 1

−₂^t − z − m²_i

! c²_i,`

iG^D_`i 1 + 2t m²_i

!

We can also expand in low energy. For fixed L the coefficient for z^kt^L−k

z^kt^L−k : X

i

c_i,`²

i







L−k

X

q=0

 L − k k



2^2q−L+k(−)^L−qG_`,q^D







=X

i

c_i,`²

i

Gˆ^D_`,k

where G^D_{`,q</sub>= <sub>∂x</sub><sup>∂q</sup>qG<sup>D</sup><sub>`}(1 + x )  

x =0. For L = 4 we have infinite set of three-dimensional vectors

V_`=





 Gˆ^D_{`,4</sub> Gˆ<sup>D</sup><sub>`,2} Gˆ^D_`,0







The allowed space is spanned by the convex hull of V`s.

V`=





 Gˆ^D_{`,4</sub> Gˆ<sup>D</sup><sub>`,2} Gˆ^D_`,0





 The allowed space is spanned by the convex hull of V`s.

Remarkably, it does not span the full three-dimensional space ! Let’s consider the space projectively. Taking

v`= V<sub>`</sub>¹− V_`²

V_{`</sub><sup>3</sup> ,V<sub>`}²− V_{`</sub><sup>3</sup> V<sub>`}³

!

2 5

6

7

8 910 2

5

6

7

8

910 2

5 6

7 8

9 10

-400 -300 -200 -100 100 200

50 100 150 200

L = 6 is spanned by a four-dimensional space: the projective three-dimensional polytope:

v_{`</sub>= V<sub>`}¹− V_`²

V_{`</sub><sup>4</sup> ,V<sub>`}²− V_`³

V_{`</sub><sup>4</sup> ,V<sub>`}³− V_{`</sub><sup>4</sup> V<sub>`}⁴

!

1 52346

7 8

9

10

-10 000

-5000

0

5000 0 5000 10 000

05001000

We can ask where does super string theory lies:

2 5

6

7

8910 2

5 6

7

8

910 2

5 6

7 8

9 10

-400 -300 -200 -100 100 200

50 100 150 200

1 5234 6

7 8

9

10

-10 000

-5000

0

5000

0 5000 10 000

0 5001000

What about spinning external state?

In four-dimensions we also have a universal polynomial, G^D<sub>`</sub> → J (` + 4h, 0, −4h, cos θ)

Conclusion

Consistency conditions (Lorentz InvarianceandUnitarity) on the S-matrix imposes stringent constraint

• The general structure of renormalizable interactions (no need for UV completion) are fully determined

• Higher dimensional operators are constrained to live with in a small region of available coupling.