Lost in an EFT? You are here! The path to You are here! the - - PowerPoint PPT Presentation

SLIDE 1

Lost in an EFT?

SLIDE 2

You are here!

SLIDE 3

The path to the pagoda You are here!

SLIDE 4

Current LHC exploration.. …no obvious signposts so far

SLIDE 5

What can we understand about the maze in general?

SLIDE 6

A first glance at some deep underlying structure of EFTs

Ongoing work to understand mathematical structure of quantum EFTs on general grounds

with Brian Henning, Xiaochuan Lu, and Hitoshi Murayama

The importance and ubiquity of EFTs has been understood for decades — remarkable that very basic questions about their structure are (were) unknown

SLIDE 7

M=SO(4) inv. M=SO(d) G linear G non-linear IR free EFTs invariant under M(spacetime)xG(gauge,global)

SLIDE 8

M=SO(4) inv. M=SO(d) G linear G non-linear IR free

Brian Henning, Xiaochuan Lu, TM, and Hitoshi Murayama, to appear

EFTs invariant under M(spacetime)xG(gauge,global)

SLIDE 9

M=SO(4) inv. M=SO(d) G linear G non-linear IR free

Mostly touch on this for this talk

EFTs invariant under M(spacetime)xG(gauge,global)

SLIDE 10

Practical problem

Redundancies between operators from equations of motion and integration by parts

L = X

i

ciOi

O1 ∼ O2 + dO3 O1 ∼ O2 + δS δφi O3 EOM IBP

SLIDE 11

EFT of a single real scalar field

4 7 10 16 25 39 61 98 155 254 410 687 1137 1933 3304 5691 9966 17456 30973 55133 98613 2 3 4 6 8 12 16 25 35 55 80 127 194 312 495 810 1316 2196 3624 6109 10213 1 1 1 2 2 3 3 5 6 9 11 17 22 34 47 74 106 169 256 412 643

SOH4L SOH4L + EOM SOH4L + EOM + IBP

5 10 15 20 25 1 10 102 103 104 105 Mass Dimension

• No. of Operators

SLIDE 12 4 7 10 16 25 39 61 98 155 254 410 687 1137 1933 3304 5691 9966 17456 30973 55133 98613 2 3 4 6 8 12 16 25 35 55 80 127 194 312 495 810 1316 2196 3624 6109 10213 1 1 1 2 2 3 3 5 6 9 11 17 22 34 47 74 106 169 256 412 643

SOH4L SOH4L + EOM SOH4L + EOM + IBP

5 10 15 20 25 1 10 102 103 104 105 Mass Dimension

• No. of Operators

‘naive’=SO(∞) is interesting (but hard)

∂µ∂νφ ∂µ∂νφ φ2 ∂µφ ∂µ∂νφ ∂νφ φ ∂µφ ∂µφ ∂νφ ∂νφ

Graph theory:

?

SLIDE 13

It turns out the structure of an

• perator basis is controlled by

the conformal algebra Organize into irreps. of the conformal group — the basis is spanned by primary operators

SLIDE 14

Why the conformal algebra?

SU(2) toy: build ‘operators’ out of two spin 1/2 states + + + Spin A

• r

Spin B

• r

The ‘Lagrangian’ =

SLIDE 15

Why the conformal algebra?

SU(2) toy: build ‘operators’ out of two spin 1/2 states +

• 2 x 2 = 3 + 1

x = +

SLIDE 16

Why the conformal algebra?

x.. x..

φ ∂µφ ∂{µ∂ν}φ ∂{µ∂ν∂ρ}φ Fµν ∂ρFµν ψα ∂βψα ∂δ∂βψα ∂σ∂ρFµν R[ 3

2 ,( 1 2 ,0)]

R[2,(1,0)] = X

∆,j1,j2

c∆,j1,j2 O[∆,(j1,j2)] ∂µO[∆,(j1,j2)] . . . . . . . . . . . .

repr. = R[∆,(j1,j2)]

R[1,(0,0)]

SLIDE 17

Why the conformal algebra?

x.. x..

φ ∂µφ ∂{µ∂ν}φ ∂{µ∂ν∂ρ}φ Fµν ∂ρFµν ψα ∂βψα ∂δ∂βψα ∂σ∂ρFµν R[ 3

2 ,( 1 2 ,0)]

R[2,(1,0)]

repr. = R[∆,(j1,j2)]

= X

∆,j1,j2

c∆,j1,j2 O[∆,(j1,j2)] ∂µO[∆,(j1,j2)]

All total derivatives

. . . . . . . . . . . .

Keep only scalar primary

• perators

R[1,(0,0)]

SLIDE 18

Why the conformal algebra? We have an alternate picture in terms of scattering amplitudes EOM: on-shell particles IBP: momentum conservation

SLIDE 19

A recipe for constructing a Hilbert series for 4d phenomenological theories

SLIDE 20

Application to the SM

On this operator basis we defined a generating function — Hilbert series Evaluate to count the number of independent

• perators at a given mass dimension in the SM

Buchmuller, Wyler 1986 Grzadkowski et. al. 2010 Manohar et. al. 2013 Lehman, Martin 2014 dim 6, 1 gen. dim 6, 1 gen, corrected dim 6, Nf gen. dim 7, Nf gen. dim 8, 1 gen.

SLIDE 21

What a Hilbert series looks like

Hilbert series

H(D, Q, u, d, L, e, H, FL, ..)

= Z

SO(6)

Z

SU(3)×SU(2)×U(1)

PE(χQ, χu, χd, χL, χe, χH, χFL, ..)

SLIDE 22

What a Hilbert series looks like

Plethystic exponential: generate all possible operators

= Z

SO(6)

Z

SU(3)×SU(2)×U(1)

PE(χQ, χu, χd, χL, χe, χH, χFL, ..)

H(D, Q, u, d, L, e, H, FL, ..)

PE(χL) = exp ∞ X

r=1

Lr χ[ 3

2 ,( 1 2 ,0)](αr, βr, qr) χ2,SU(2)(yr)

!

e.g.

χR(g) = trR(g) χ2,SU(2)(g) = tr

• diag(eiθ, e−iθ)
• = y + y−1

Characters e.g.

hgh−1 = eiθaHa

SLIDE 23

What a Hilbert series looks like

Haar measures: project out scalar, gauge inv operators

e.g.

= Z

SO(6)

Z

SU(3)×SU(2)×U(1)

PE(χQ, χu, χd, χL, χe, χH, χFL, ..)

Z dµSU(2)(y) = I dy 2πi 1 y (1 − y2)(1 − y−2)

H(D, Q, u, d, L, e, H, FL, ..)

SLIDE 24

= I dα 2πi I dβ 2πi

• 1 − α22

1 − β22 4α3β3 I dx 2πi 1 x I dy 2πi

• 1 − y22

2y3 I dz1 2πi I dz2 2πi

• z2

1 − z2

2 (1 − z1z2)2 z1 − z2

2

2 6z5

1z5 2

 H2L2

• y4 + y2 + 1

⇣ α2 + α2y4 +

• α2 + 1

2 y2⌘ α2y4 + (H†)2(L†)2

• y4 + y2 + 1

⇣ β2 + β2y4 +

• β2 + 1

2 y2⌘ β2y4

+ . . .

• = H2L2 + H† 2L† 2

Neutrino mass

HS(D, Q, u, d, L, e, H, FL, ..)

• dim 5

What a Hilbert series looks like

SLIDE 25

What a Hilbert series looks like

HS(D, Q, u, d, L, e, H, FL, ..)

• dim 6

b6 = H3H† 3 + u†Q†HH† 2 + 2Q2Q† 2 + Q† 3L† + Q3L + 2QQ†LL† + L2L† 2 + uQH2H† + 2uu†QQ† + uu†LL† + u2u† 2 + e†u†Q2 + e†L†H2H† + 2e†u†Q†L† + eLHH† 2 + euQ† 2 + 2euQL + ee†QQ† + ee†LL† + ee†uu† + e2e† 2 + d†Q†H2H† + 2d†u†Q† 2 + d†u†QL + d†e†u† 2 + d†eQ†L + dQHH† 2 + 2duQ2 + duQ†L† + de†QL† + deu2 + 2dd†QQ† + dd†LL† + 2dd†uu† + dd†ee† + d2d† 2 + u†Q†H†GR + d†Q†HGR + HH†G2

R + G3 R + uQHGL

+ dQH†GL + HH†G2

L + G3 L + u†Q†H†WR + e†L†HWR + d†Q†HWR + HH†W 2 R + W 3 R

+ uQHWL + eLH†WL + dQH†WL + HH†W 2

L + W 3 L + u†Q†H†BR + e†L†HBR

+ d†Q†HBR + HH†BRWR + HH†B2

R + uQHBL + eLH†BL + dQH†BL + HH†BLWL

+ HH†B2

L + 2QQ†HH†D + 2LL†HH†D + uu†HH†D + ee†HH†D + d†uH2D + du†H† 2D

+ dd†HH†D + 2H2H† 2D2 . (3.16)

First systematic procedure (one generation)

SLIDE 26

Counting in the SM EFT

2 30 560 11962 257378 5474170 84 993 15456 261485 4614554 12 1542 90456 3472266 175373592 7557369962 3045 44807 2092441 75577476 2795173575 5 6 7 8 9 10 11 12 13 14 15 1 10 100 1000 10000 100000 1000000 10000000 100000000 1000000000 10000000000

Mass dimension

• No. of independent ops

Nf=1 Nf=3

SLIDE 27

Never have to count ops again….

Hilbert series for your theory

= Z

SO(6)

Z

gauge, global

PE(χψ1, χψ2, . . .) H(D, ψ1, ψ2, . . .)

Characters for the repr. of each field Gauge and global symmetries of the EFT

SLIDE 28

Explicit example of a particular maze…

SLIDE 29

Explicit example of a particular maze…

H(φ1, .., φN, t) = Z dµSO(d) 1 P(t; x)

N

Y

i=1

exp ∞ X

r=1

φi(1 − t2)P(t; x) !

in four dimensions

H(φ, t)|φ4 = φ4 1 (1 − t4)(1 − t6)

SLIDE 30

Explicit example of a particular maze…

H(φ1, .., φN, t) = Z dµSO(d) 1 P(t; x)

N

Y

i=1

exp ∞ X

r=1

φi(1 − t2)P(t; x) !

in four dimensions

H(φ, t)|φ4 = φ4 1 (1 − t4)(1 − t6)

n=4

SLIDE 31

Explicit example of a particular maze…

H(φ1, .., φN, t) = Z dµSO(d) 1 P(t; x)

N

Y

i=1

exp ∞ X

r=1

φi(1 − t2)P(t; x) !

in four dimensions

n=4 n=5 …

H(φ, t)|φ5 = φ5 t28 + t22 + t16 + t14 + t12 + t8 − t6 + t4 − t2 + 1 (1 − t2)(1 − t6)(1 − t8)(1 − t10)(1 − t12)

SLIDE 32

Explicit example of a particular maze…

H(φ, t)|φ4 = φ4 1 (1 − t4)(1 − t6)

n=4 n=5 …

H(φ1, .., φN, t) = Z dµSO(d) 1 P(t; x)

N

Y

i=1

exp ∞ X

r=1

φi(1 − t2)P(t; x) !

in four dimensions

fixed powers of t …

SLIDE 33

Explicit example of a particular maze…

H(φ1, φ2, t) = 1 (1 − φ1)(1 − φ2)(1 − tφ1φ2)

in one dimension

SLIDE 34

Explicit example of a particular maze…

H(φ1, φ2, t) = 1 (1 − φ1)(1 − φ2)(1 − tφ1φ2)

in one dimension

H(φ1, φ2, φ3, t) = 1 − tφ1φ2φ3 (1 − φ1)(1 − φ2)(1 − φ3)(1 − tφ1φ2)(1 − tφ1φ3)(1 − tφ2φ3)

SLIDE 35

Explicit example of a particular maze…

H(φ1, φ2, t) = 1 (1 − φ1)(1 − φ2)(1 − tφ1φ2)

in one dimension

H(φ1, φ2, φ3, t) = 1 − tφ1φ2φ3 (1 − φ1)(1 − φ2)(1 − φ3)(1 − tφ1φ2)(1 − tφ1φ3)(1 − tφ2φ3)

HN+1 (¯ u0, ¯ u1, · · · , ¯ uN+1) =

• |x|=1

dx 2πi 1 xHN (¯ u0, · · · , ¯ uN−1, x) H2

• x−1, ¯

uN, ¯ uN+1

• .

u0 u1 u2

u1 u0 u2 u3 x x−1 → u1 u0 u2 u3

Flavour recursion

SLIDE 36

Explicit example of a particular maze…

H(φ1, φ2, φ3, t) = 1 − tφ1φ2φ3 (1 − φ1)(1 − φ2)(1 − φ3)(1 − tφ1φ2)(1 − tφ1φ3)(1 − tφ2φ3)

Information/structure not seen at any perturbative order This object contains physics — counts number of independent ‘measurements’

SLIDE 37

Some dreaming…

HSM = ***All order result???***

How best to interpret the information? Can it provide hints to possible paths? Non-renormalization info?

• J. Elias-Miro, J. R. Espinosa and A. Pomarol
• C. Cheung and C.-H. Shen
• R. Alonso, E. E. Jenkins, and A. V. Manohar

…

SLIDE 38

Who knows what we might discover if we go far enough…

SLIDE 39

Extra slides

SLIDE 40

n=3 n=4 n=… n=5

L = X

n,k

cn,kφn∂k

SLIDE 41

Operator Basis

SLIDE 42

C4,4 C4,6 C4,8 C4,10 C4,12 C4,12

1 2

C5,4 C5,6 C5,8 C5,8

1 2

Searching for theories ? ? ? ? ? ?

L = X

n,k

cn,kφn∂k

SLIDE 43

C4,4 C4,6 C4,8 C4,10 C4,12 C4,12

1 2

C5,4 C5,6 C5,8 C5,8

1 2

Searching for theories ? ? ? ? ? Look for enhanced soft limits in amplitudes

Cheung, Kampf, Novotny, Trnka 2015 Cachazo, Cha, Mizera 2016

SLIDE 44

Tom Melia SLAC Theory Seminar 29th Jan ‘16

SU(2)

44

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Tom Melia SLAC Theory Seminar 29th Jan ‘16

Warm-up with SU(2)

• Characters
• Plethystic Exponential

45

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Tom Melia SLAC Theory Seminar 29th Jan ‘16

Warm-up with SU(2)

Characters

• Plethystic Exponential

46

SLIDE 47

Tom Melia SLAC Theory Seminar 29th Jan ‘16

Characters

χR(g) = TrR(g) g ∈ G R G

Repr

• f

= TrR(hgh−1) hgh−1 = eiθaHa

where can always write

Cartan generators

χR(x1, . . . , xr)

Rank of G (# Cartan)

47

SLIDE 48

Tom Melia SLAC Theory Seminar 29th Jan ‘16

Characters of SU(2)

Spin j irreps. labelled by their dimensions (2j+1)

Rank 1, Cartan generator T3

eiθT3 = diag(eijθ, ei(j−1)θ, . . . , ei(−j)θ)

y = eiθ

χ(2j+1) = Tr(eiθT3) = y2j + y2j−2 + . . . + y−2j

48

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Tom Melia SLAC Theory Seminar 29th Jan ‘16

Characters of SU(2)

• tensor products, decomposition

2 × 2 = 3 + 1

χ2χ2 = χ3 + χ1

(y + y−1)(y + y−1) = (y2 + 1 + y−2) + 1

49

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Tom Melia SLAC Theory Seminar 29th Jan ‘16

Characters of SU(2)

• character orthogonality

Z dµ({xr}) χ∗

i ({xr})χj({xr}) = δij

Z

G

dg χ∗

i (g)χj(g) = δij

For SU(2)

Z dµSU(2) = I

|y|=1

dy 2πi (1 − y2)(1 − y−2) y

50

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Tom Melia SLAC Theory Seminar 29th Jan ‘16

Warm-up with SU(2)

Plethystic Exponential

• Characters

51

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Tom Melia SLAC Theory Seminar 29th Jan ‘16

Plethystic Exponential

• generating function for sym tensor products

1 detR(1 − ug) = X

n

unsymnχR

1 detR(1 − ug) = exp [−TrR log(1 − ug)] = exp " ∞ X

n=1

1 nunTrR(gn) #

= exp " ∞ X

n=1

1 nunχ(xn

1, . . . , xn r )

# = PE[uχR]

52

SLIDE 53

Tom Melia SLAC Theory Seminar 29th Jan ‘16

• for SU(2)

1 det2(1 − ug) = 1 det ✓ 1 1 ◆ − ✓ uy u/y ◆ = 1 (1 − uy)(1 − u/y) = 1 + (y + y−1)u + (y2 + 1 + y−2)u2 + (y3 + y + y−1 + y−3)u3 + . . .

Plethystic Exponential

53

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Tom Melia SLAC Theory Seminar 29th Jan ‘16

• for SU(2)

1 det2(1 − ug) = 1 det ✓ 1 1 ◆ − ✓ uy u/y ◆ = 1 (1 − uy)(1 − u/y) = 1 + (y + y−1)u + (y2 + 1 + y−2)u2 + (y3 + y + y−1 + y−3)u3 + . . .

1 2(χ2(y2) + χ2(y)2)

sym2χ2 =

Plethystic Exponential

54

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Tom Melia SLAC Theory Seminar 29th Jan ‘16

• for SU(2)

1 det2(1 − ug) = 1 det ✓ 1 1 ◆ − ✓ uy u/y ◆ = 1 (1 − uy)(1 − u/y) = 1 + (y + y−1)u + (y2 + 1 + y−2)u2 + (y3 + y + y−1 + y−3)u3 + . . .

1 2(χ2(y2) + χ2(y)2) = χ3(y)

2 2 = 3

sym2χ2 =

Plethystic Exponential

55

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Tom Melia SLAC Theory Seminar 29th Jan ‘16

SO(d+2,C)

56

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Tom Melia SLAC Theory Seminar 29th Jan ‘16

Characters for SO(d+2,C)

χ[∆,l](q, x1, . . . , xr) = q∆ P(q; x)χl(x)

• Irrep. of conformal group

R[∆,l] =      Φl ∂µΦl ∂{µ1∂µ2}Φl . . .     

“Character formulae and partition functions in higher dimensional conformal field theory” arXiv:hep-th/0508031

• F. A. Dolan

57

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Tom Melia SLAC Theory Seminar 29th Jan ‘16

Characters for SO(d+2,C)

χ[∆,l](q, x1, . . . , xr) = q∆ P(q; x)χl(x) R[∆,l] =      Φl ∂µΦl ∂{µ1∂µ2}Φl . . .     

P(q; x) = 1 det⇤(1 − qg) = 1 Qr

i=1(1 − qxi)(1 − q/xi)

1 1 − qδd,odd

Irrep of SO(d)

• Irrep. of conformal group

l = (l1, . . . , lr)

spin scaling dim

r = d 2 ⌫

58

SLIDE 59

Tom Melia SLAC Theory Seminar 29th Jan ‘16

counting in Nf

# Dim 5 = ✓ Nf + N 2

f

◆ # Dim 6 = ✓ 15 + 135 4 N 2

f + 1

2N 3

f + 107

4 N 4

f

◆ + ✓2 3N 2

f + N 3 f + 19

3 N 4

f

◆ # Dim 7 = ✓ 2Nf + 26 3 N 2

f + N 3 f + 31

3 N 4

f

◆ + ✓ N 3

f + 7N 4 f

◆ # Dim 8 = ✓ 89 + 789 2 N 2

f + 823

2 N 4

f

◆ + ✓2 3N 2

f + N 3 f + 289

3 N 4

f

◆ # Dim 9 = ✓ 9Nf + 83N 2

f + 49

12N 3

f + 2587

12 N 4

f − 1

12N 5

f + 437

12 N 6

f

◆ + ✓ − 4 3N 2

f + 29

3 N 3

f + 463

3 N 4

f + 1

3N 5

f + 41N 6 f

◆ + ✓1 4N 2

f + 61

24N 3

f + 29

24N 4

f + 11

24N 5

f + 85

24N 6

f

◆ # Dim 10 = ✓ 530 + 53927 12 N 2

f − 17

2 N 3

f + 82127

12 N 4

f − 6N 5 f + 3776

3 N 6

f

◆ + ✓ − 10 9 N 2

f + 155

3 N 3

f + 30169

18 N 4

f + 37

3 N 5

f + 10891

18 N 6

f

◆ # Dim 11 = ✓ 18Nf + 2812 3 N 2

f − 152

3 N 3

f + 11689

3 N 4

f − 58

3 N 5

f + 5551

3 N 6

f

◆ + ✓ − 2N 2

f + 443

3 N 3

f + 8830

3 N 4

f + 352

3 N 5

f + 5855

3 N 6

f

◆ + ✓3 4N 2

f + 307

24 N 3

f + 7

24N 4

f + 197

24 N 5

f + 3599

24 N 6

f

◆ # Dim 12 = ✓ 4481 + 1 2Nf + 613247 12 N 2

f − 5381

24 N 3

f + 7846991

72 N 4

f − 8927

24 N 5

f + 3181709

72 N 6

f − 35

6 N 7

f + 50947

36 N 8

f

◆ + ✓28 9 N 2

f + 1954

3 N 3

f + 27779N 4 f + 6823

12 N 5

f + 131429

6 N 6

f + 169

12 N 7

f + 17803

18 N 8

f

◆ + ✓11 24N 3

f + 1483

144 N 4

f + 19

12N 5

f + 149

72 N 6

f + 47

24N 7

f + 4555

144 N 8

f

◆

+ full Hilbert series provided

SLIDE 60

Tom Melia SLAC Theory Seminar 29th Jan ‘16

counting in Nf

# Dim 5 = ✓ Nf + N 2

f

◆ # Dim 6 = ✓ 15 + 135 4 N 2

f + 1

2N 3

f + 107

4 N 4

f

◆ + ✓2 3N 2

f + N 3 f + 19

3 N 4

f

◆ # Dim 7 = ✓ 2Nf + 26 3 N 2

f + N 3 f + 31

3 N 4

f

◆ + ✓ N 3

f + 7N 4 f

◆ # Dim 8 = ✓ 89 + 789 2 N 2

f + 823

2 N 4

f

◆ + ✓2 3N 2

f + N 3 f + 289

3 N 4

f

◆ # Dim 9 = ✓ 9Nf + 83N 2

f + 49

12N 3

f + 2587

12 N 4

f − 1

12N 5

f + 437

12 N 6

f

◆ + ✓ − 4 3N 2

f + 29

3 N 3

f + 463

3 N 4

f + 1

3N 5

f + 41N 6 f

◆ + ✓1 4N 2

f + 61

24N 3

f + 29

24N 4

f + 11

24N 5

f + 85

24N 6

f

◆ # Dim 10 = ✓ 530 + 53927 12 N 2

f − 17

2 N 3

f + 82127

12 N 4

f − 6N 5 f + 3776

3 N 6

f

◆ + ✓ − 10 9 N 2

f + 155

3 N 3

f + 30169

18 N 4

f + 37

3 N 5

f + 10891

18 N 6

f

◆ # Dim 11 = ✓ 18Nf + 2812 3 N 2

f − 152

3 N 3

f + 11689

3 N 4

f − 58

3 N 5

f + 5551

3 N 6

f

◆ + ✓ − 2N 2

f + 443

3 N 3

f + 8830

3 N 4

f + 352

3 N 5

f + 5855

3 N 6

f

◆ + ✓3 4N 2

f + 307

24 N 3

f + 7

24N 4

f + 197

24 N 5

f + 3599

24 N 6

f

◆ # Dim 12 = ✓ 4481 + 1 2Nf + 613247 12 N 2

f − 5381

24 N 3

f + 7846991

72 N 4

f − 8927

24 N 5

f + 3181709

72 N 6

f − 35

6 N 7

f + 50947

36 N 8

f

◆ + ✓28 9 N 2

f + 1954

3 N 3

f + 27779N 4 f + 6823

12 N 5

f + 131429

6 N 6

f + 169

12 N 7

f + 17803

18 N 8

f

◆ + ✓11 24N 3

f + 1483

144 N 4

f + 19

12N 5

f + 149

72 N 6

f + 47

24N 7

f + 4555

144 N 8

f

◆

some large primes… + full Hilbert series provided

SLIDE 61

Tom Melia SLAC Theory Seminar 29th Jan ‘16

some larger primes…

# Dim 13 = −109Nf + 159296 15 N 2

f + 32063

90 N 3

f + 5140756

45 N 4

f + 78253

72 N 5

f + 42846881

360 N 6

f + 68723

360 N 7

f

+ 4311047 360 N 8

f

# Dim 14 = 40715 − 2Nf + 105860297 180 N 2

f + 89759

18 N 3

f + 1513774187

720 N 4

f + 63971

72 N 5

f + 299553293

180 N 6

f

− 117979 72 N 7

f + 51562231

240 N 8

f

# Dim 15 = −2427Nf + 21647887 180 N 2

f − 114619

20 N 3

f + 387130705

216 N 4

f − 10026269

1440 N 5

f + 456200951

160 N 6

f

− 3717991 720 N 7

f + 103741331

144 N 8

f − 534941

1440 N 9

f + 9163865

864 N 10

f

counting in Nf

61