Polynomial Shift Symmetry: from Naturalness to Graph Theory · Kevin T. Grosvenor Polynomial Shift...

Kevin T. Grosvenor Polynomial Shift Symmetry: from Naturalness to Graph Theory 1

Polynomial Shift Symmetry:from Naturalness to Graph Theory

Kevin T. Grosvenor

Berkeley Center for Theoretical Physics

String Theory Group Meeting

September 24, 2014


Collaborators:

Tom Griffin, Kevin T. Grosvenor, Petr Horava, and Ziqi Yan.

Based on:

Multicritical Symmetry Breaking and Naturalness of SlowNambu-Goldstone Bosons (1308.5967)

and work on graphs to appear.


OutlinePart 1: Naturalness

• Effective field theories (EFTs) of Nambu-Goldstone Bosons (NGBs).

• Technical naturalness of slow NGBs and the polynomial shift symmetry.

Transition: the Search for Invariant Lagrangians

• Linear shift invariants.

• Higher-degree polynomials.

Part 2: Graph Theory

• The graph theoretic approach and some new invariants.

• Outlook


Part 1: Naturalness


EFTs for NGBs: Symmetries• [Murayama and Watanabe, 1203.0609]

• Spacetime RD+1 with coordinates t, xi, i = 1, . . . , D.

• Impose isometries of spatial RD and time translations.

• Enhanced by anisotropic scaling (t, xi)→ (bzt, bxi) at renormalization group (RG) fixed points.

• Some symmetry group G is spontaneously broken to H ⊂ G.


EFTs for NGBs: the Action• The NG field components πA form coordinates on the space of vacua M = G/H .

• EFT derivative expansion: S = 12

∫dt dDx

[ΩA(π) πA + gAB(π)πAπB − hAB(π) ∂iπ

A ∂iπB + · · ·

].

• The subspace of πA’s that get their canonical momentum from ΩA can be decomposed into canonically

conjugate pairs: two broken generators giving only one “Type-B” NGB with z = 2.

• The rest are “Type-A” and have z = 1.

• Higher values of z are only possible if hAB vanishes accidentally.


Technical Naturalness of Slow NGBs: the Theory• O(N) z = 2 linear sigma model (LSM) in the broken phase:

SLSM =1

2

∫dt d3x

[φIφI − ∂2φI ∂2φI + m4φIφI − λ

2(φIφI)2

].

• φI = (ΠA, v + σ) where v = m2/√λ.

• Feynman rules (same as relativistic case, except for propagators):

σ σ =i

ω2 − |k|4 − 2m4 + iεΠA ΠB =

iδAB

ω2 − |k|4 + iε

= −6iλvA B

= −2iλvδAB

= −6iλA B

= −2iλδAB

A B

C D= −2iλ

[δABδCD + δACδBD + δADδBC

].


Technical Naturalness of Slow NGBs: the Result• Do the interactions generate the relevant −c2 ∂iφ

I ∂iφI term?

• One-loop contribution comes from the (non-divergent) diagram

• The result is

δc2 =27/4 · 563π5/2

[Γ(

54

)]2 λm≈ 0.0125

λ

m.

• The same scaling δc2 ∼ λ/m is obtained in the non-linear sigma model (NLSM).

• Large-N limit: δc2 = 0 to all orders in λN .

• δc is small compared to possible large scales, m or a cut-off Λ.


Symmetry Argument• Suppose a symmetry keeps λ and c2 naturally small:

λ = O(εµ3), c2 = O(εµ2).

• µ is the naturalness scale, at which the symmetry is broken.

• m is naturally of order µ

• δc2 = O(λ/µ) = O(λ/m), as confirmed by explicit calculation.

• Same argument in unbroken phase, NLSM, or most general LSM.


Quadratic Shift Symmetry• Quadratic shift symmetry :

ΠA → ΠA + aAijxixj + aix

I + a.

• This symmetry is broken by λ and c2.

• Allows for technically natural Type-A NGBs with z = 2 instead of z = 1 and Type-B NGBs with

z = 4 instead of z = 2.


Polynomial Shift Symmetry: Guardians of Naturalness• Why stop at quadratic shift?

• A polynomial shift symmetry of degree 2z− 2 protects Type-A NGBs with z ≥ 1 and Type-B NGBs

with 2z ≥ 2.

• Coleman-Hohenberg-Mermin-Wagner (CHMW) theorem: z < D (spatial dimension).


Transition: the Search for Invariant Lagrangians


Classification of Polynomial Shift Invariant Lagrangians• Previous examples satisfied the polynomial shift symmetry only at the UV free field fixed point.

• Can we find interacting field theories that satisfy the symmetry?

• To simplify the problem, consider the theory of one real scalar field.

• Goal: find the most relevant P -invariant n-point terms (the one with the fewest derivatives).

• Complication: we want terms (up to total derivatives) which are invariant (up to total derivatives).

• Warm up: exact P -invariants are given by putting at least P + 1 derivatives on every φ.


The Galileon• [Nicolis et. al., 0811.21972; Goon et. al., 1203.3191]

• Polynomial shift symmetry generalizes the linear shift transformation of the Galileon.

• Unique minimal (fewest derivatives) 1-invariant n-pt term is

Ln-pt ∼ δ[j1i1· · · δjn−1]

in−1φ ∂i1∂j1φ · · · ∂in−1∂jn−1φ.

• Two fewer derivatives than minimal exact invariants.

Scalar Field with Extended Shift Symmetry• [Hinterbichler and Joyce, 1404.4047]

• The generalization to degree P polynomial shifts is basically of the form

Ln-pt ∼ δ[j1i1· · · δjn−1]

in−1∂P−1φ ∂P−1∂i1∂j1φ · · · ∂

P−1∂in−1∂jn−1φ.

• Take away message: still two fewer derivatives than the minimal exact invariants.


Part 2: Graph Theory


The Graphs

• We have developed a different approach involving graphs.

• We have discovered families of new invariants.

• Ingredients: φ’s and properly pair-wise contracted ∂i’s.

• The graph • represents φ.

• The graph represents ∂iφ ∂iφ.

• The graph represents ∂iφ ∂i∂jφ ∂jφ.

• The graph represents ∂iφ ∂i∂jφ ∂j∂kφ ∂kφ.

• The graph represents ∂iφ ∂jφ ∂kφ ∂i∂j∂kφ.


Rewriting the 1-invariants

• φ ∂2φ = = − = −∂iφ ∂iφ.

• δ[j1i1δj2]i2φ ∂i1∂j1φ ∂i2∂j2φ = φ ∂2φ ∂2φ− φ ∂i∂jφ ∂i∂jφ = −

= − − + +

= + +

= 3

• For n = 4, we get 12 + 4

• For n = 5, we get 60 + 60 + 5


It’s all about the “Trees”• A tree is a connected graph with no closed paths.

• For n = 2, there is only one tree:

• For n = 3, if we label the vertices, there are three trees: , , and

• For n = 4, there are 16 trees; 12 are equivalent to (itself, , and their four rotations),

and 4 are equivalent to (its four rotations).

• For n = 5, there are 125 trees; 60 are equivalent to , 60 to , and 5 to

• For general n, there are nn−2 trees (Cayley’s formula).

• The degeneracies of the trees are the same as the coefficients with which they appear in the 1-invariants!


We proved that the trees generalize to arbitrary n

The unique minimal n-pt 1-invariant is represented graphically asan equal-weight sum of all possible trees with n vertices.


Some highlights of the proof

• If a graph in a loopless 1-invariant contains more than one connected component, then none of these

connected components is a tree.

• The minimum number of edges in a loopless 1-invariant is ≥ n− 1.

• A nonzero minimal loopless 1-invariant with n− 1 edges must contain all trees.

• It is therefore unique.

• The equal-weight sum of the trees is 1-invariant and has n− 1 edges.

• Therefore, it is the unique minimal 1-invariant.

• Independently, we prove that it is equal to the usual Galileon-like expression, up to a total derivative.


Mo Spaces Mo Graphs

• L = linear combinations of possible Lagrangian terms.

• R = total derivative relations among Lagrangian terms.

• L× = linear combinations of variations of Lagrangian terms.

• R× = total derivative relations among variations of Lagrangian terms.

• We need graphical representations of all of these spaces.


Families of Graphs

• L is just the graphs we have discussed so far. We call these plain graphs.

• L× are represented by ×-graphs, where one •-vertex is replaced with ⊗ (a ×-vertex)

• Plain relations in R are given by ?-ed plain graphs.

• An example of a plain relation in R is F = + + +

• Note that a ?-ed plain graph can have multiple ?’s attached to various different •-vertices.

• ×-relations in R× are given by ?-ed ×-graphs.

• An example is the above plain relation with one •-vertex replaced with ⊗.

• There can be multiple ?’s and they need not be attached to the ×-vertex.

• A ×-graph with a ×-vertex of degree > P vanishes identically.

δ2 =×

+×

+×

+×


The Loopless Basis

• P -invariants form the kernel of the map δ:

L

δ //L×

L/R δ //L×/R×

• The set of loopless plain graphs forms a basis for L/R.

• Therefore, we can restrict L× and R× to loopless elements as well.

• We need a similarly nice way to parametrize L×/R×.


Unexpected help from a Greek mythical character

• An important class of ?-ed ×-graphs is the Medusa. It is loopless, the ?’s are attached only to the

×-vertex, and, counting twice the edges ending on ?’s, the degree of the ×-vertex is P + 1.

• Two examples of Medusas for n = 4 and P = 3 with six edges total (∆ = 6) are

F

×FF

×

• The ×-relations given by reconnecting the ?-ed ends in Medusas excluding loops forms a basis of

loopless ×-relations for R×loopless.

• For example,

F

×

=×

+×

+×

+×


Minimal Degree of Vertices

• The vertices of a plain graph in a P -invariant are of degree ≥ 12(P + 1).

• The variation of a P -invariant is a sum of loopless ×-relations arising from Medusas whose •-vertices

are of degree ≥ 12(P + 1).


Example 1: (n, P,∆) = (3, 3, 4)

• Graphs with 4 edges and vertices of degree ≥ 12(P + 1) = 2: A = and B =

• Three ×-graphs: A1 =×

, B1 =×

and B2 =×

• δA = 2A1 and δB = B1 + 2B2.

• Possible Medusas: M1 = F F×

and M2 = F

×

• M1 relation is 2A1 + 2B1 = 0 and M2 relation is 2B2 = 0.

• Eliminate B1 and B2 to get δA = 2A1 and δB = −A1.

• The matrix form of δ is δ =(2 −1

).

• The nullspace is spanned by the vector(

12

), or A + 2B.

• No valid Medusa with 3 or fewer edges exists.

• The unique minimal 3-pt 3-invariant is

+ 2


Example 2: (n, P,∆) = (4, 2, 5)

• No valid Medusa with ∆ ≤ 4.

• Possible Medusas: M1 =

F

×

and M2 =

F

×

• Five ×-graphs: A1 =×

, D2 =×

, B1 =×

, C1 =×

and D1 =×

• M1 relation is 2A1 + D2 = 0 and M2 relation is B1 + C1 + D1 = 0.

• Graphs with 5 edges and vertices of degree ≥ 2:

A = B = C = D = E =

• E does not give ×-graphs in the list. We can drop E.

• δA = 2A1, δB = 2B1, δC = 2C1 and δD = 2D1 + D2 = −2(A1 + B1 + C1).

• Unique minimal 4-pt 2-invariant is

+ + +


Superposing Graphs to form Higher-P Invariants

• Superposing two 3-pt 1-invariants gives 3(

+ 2)

, the 3-pt 3-invariant:

• Superposing the exact 0-invariant on one 4-pt 1-invariant gives 4(

+ + +),

the 4-pt 2-invariant:


We proved that superpositions generalize

For fixed n, the superposition of any expact PE-invariant with the superposition ofp minimal loopless 1-invariants results in a P -invariant, provided PE + 2p ≥ P .


Summary

• Polynomial shifts protect Type-A NGBs with z > 1 and Type-B NGBs with z > 2 (and even).

• The search for polynomial shift invariants has led us to graphs.

• The unique minimal n-pt 1-invariant is represented graphically as an equal-weight sum of trees.

• Superposing invariants for lower degree polynomials yields invariants for higher degree polynomials.

Outlook

• Clarify the connection to the traditional method of generating invariants via the coset construction

• Prove minimality and/or uniqueness in general.

• Apply polynomial shifts to other phenomena (e.g., high-Tc superconductivity, Higgs physics and the

hierarchy problem).


4-point 5-invariant

4

(+ 3 + 27 + 108 + 72

+18 + 54 + 54 + 9 + 18

+18 + 36 + 36 + 153 + 36

+54 + 18 + 18 + 108 + 18

+18 + 18 + 54 + 48 + 27

)

... is unique and minimal!


Medusas for the 4-point 5-invariant

12

FFF

×+ 72

FFF

×+ 24

FFF

×

+72

FF×

+ 72

FF×

+ 144

FF×

+ 144

FF×

+ 216

FF×

+72

F×

+ 432

F×

+ 72

F×

+ 72

F×

+ 288

F×

+108

F×

+ 216

F×

+ 144

F×

+ 216

F×


Thank you

Polynomial Shift Symmetry: from Naturalness to Graph Theory · Kevin T. Grosvenor Polynomial Shift...

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