Beyond standard model physics as a source of lattice projects

Beyond standard model physics as a source of lattice projects

Tom DeGrand

University of Colorado at Boulder Sign Workshop, Seattle, March 2017

Outline

• Very general overview of the landscape

• Alternatives to the Higgs

– Nearly conformal systems (technicolor) – Dilatons

– Partial compositeness

• Composite dark matter

• An attempt at a summary

Supported by U. S. Department of Energy; updated/condensed version of 1510.05018

Beyond standard model physics with a lattice literature

1. Alternatives to the Higgs mechanism

• Technicolor

• Composite Higgs 2. Composite dark matter

3. Systems in D > 4 spacetime dimensions 4. Theorists’ models

• N = 1 super Yang Mills

• N = 4 super Yang Mills

• Chiral gauge theories

• Large N^c QCD

Ii will focus on item 1, a few words on item 2 Bottom line, up front

• Many systems, very interesting to study in themselves

• I don’t believe that any of them have anything to do with “real” physics beyond the standard model

Can I make this talk relevant for a “sign conference”?

Well, not really.

Q: Really big picture issue: Is the fermion path integral a real, positive number?

ZF = Z

d ¯ψdψ exp ¯ψDψ (1)

A: It depends!

• Most of the time γ⁵Dγ₅ = D^† so Z_F = det D is real

• Not much interest in µ 6= 0

• One Dirac flavor can have a sign problem (like the strange quark in N^f = 2 + 1 QCD)

• N^f = 2n even cures this for Wilson fermions, Nf = 4n cures this for staggered Easy to find “impossible systems” (ex: several flavors of chiral fermions for dark matter) But no one model is compelling.

When they can, people just stay away from systems with a sign problem

There are other, more serious, issues out there (both conceptual and computational)

The Standard Model

Left-handed leptons and quarks form a doublet of weak isospin plus right-handed singlets e_R, u_R, d_R Lagrangian has three parts

L^SM = L^g + L^Φ + L^m. (2)

L^g =

3

X

j=1

( ¯E_L^j(iD/ )E_L^j + ¯Q^j_L(iD/ )Q^j_L + ¯e^j_R(iD/ )e^j_R + ¯u^j_R(iD/ )u^j_R + ¯d^j_R(iD/ )d^j_R)

−1

4F_µν² − 1

4W_µν² − 1 4B_µν²

(3) No question, this is correct...

(Aside: no simulations yet of these “chiral gauge theories”)

By itself, L^g describes massless gauge bosons and fermions.

Gauge symmetry is broken (by hand) using the Higgs mechanism

L^Φ = |D^µΦ|² − V (Φ) (4)

and

V (Φ) = −µ²Φ^†Φ + λ(Φ^†Φ)²) (5)

so

hΦi = 1

√2

 0 v



(6) where v² = µ²/λ.

With the Higgs you get two for the price of one: fermion masses from

L^m = −λ^ij_d Q¯ⁱ_L · Φd^j_R − λ^ijuǫ^abQ¯ⁱ_La · Φ^†_bu^j_R − λ^ij_l E¯_Lⁱ · Φe^j_R + h.c. (7) AND tightly constrained Higgs couplings to fermions, (mi/v)h ¯ψiψi

Alternatives to the Standard Model replace L^Φ and then have to deal with replacing L^m

Issues with the Standard Model

• There is new physics (neutrino masses, baryon asymmetry, dark matter, dark energy...)

• V (Φ) = −µ²Φ^†Φ + λ(Φ^†Φ)² seems contrived (it is!)

• Many parameters in Standard Model

• Hierarchy problem – why is the Higgs so light?

µ² − µ²0 = λ

8π²Λ² − 3y_t²

8π²Λ² + 3(3g² + g^′2)

16π² Λ². (8)

But at the same time

• The Higgs really is the Higgs (at the 20-30 per cent level)

• M^H = 126 GeV and v = 246 GeV says λ = 0.13, µ² = (89 GeV)²

• The rest of the Standard Model works really well

– “precision EW tests” – any new physics scale for flavor observables is ≥ 5 − 10 TeV

• No new states at the LHC, yet

Resolutions of the hierarchy problem

1) None – Nature is fine tuned

2) A symmetry keeps the Higgs light

• Gauge symmetry protects massless gauge bosons – Higgs as extra gauge DoF in D > 4

• Chiral symmetry protects massless fermions – SUSY – scalars in multiplets with fermions

• Shift symmetry protects Goldstone bosons

– Higgs as a Goldstone boson – “partial compositeness”

3) There is some dynamical reason

• and it is QCD like

MH ∼ Λ exp



− 1

cg²(Λ)



. (9)

• or it is not QCD like (“dilaton” or some conformal dynamics) These are not mutually exclusive scenarios!

Electroweak symmetry breaking and composite Goldstones

Eaten Goldstones don’t have to be fundamental (Susskind, Weinberg, 1979)

h0|J^µ5a|π^bi = if^πp^µδ^ab (10)

gh0|J^µ5a|π^biA^µ = igfπp^µAµ (11)

Π^ab_µν = (gf_π

2 )²[g_µν − p_µp_ν

p² ]δ^ab (12)

m_W = gf_π/2 so f_π = 246 GeV sets the scale for new physics (replaces Higgs v) Scenario to evade the hierarchy problem

• Invent some new new dynamics with fundamental or composite Goldstones

• Goldstones get eaten

• Technicolor the classic example But there’s no free lunch

• Here, the Higgs boson itself has nothing to do with EWSB

• What replaces L^m = λ( ¯Q · Φ)d → ???

Fermion masses in composite EW scenarios

Need a whole new dynamics to give fermion masses. The generic solution assumes

• Dynamics at some even higher scale Λ^{ET C}

• Dimension-6 operators at that scale that feed down to an observable lower scale Technicolor version: replace Φ with 4-fermion interaction with new fermions (T )

λ( ¯Q · Φ)d → ( ¯QΓT )( ¯T Γd)

Λ²_{ET C} ∼ ¯QdΣ_{ET C}

Λ²_{ET C} (13)

mq = λv → ΣET C

Λ²_{ET C} = ΣT C

Λ²_{ET C} exp

Z _{ΛET C}

ΛT C

γm(g²)dµ

µ (14)

Want this to be Λ_{T C}×(1 to 1/1000)

In addition, you don’t want flavor changing neutral currents

F CN C ∼ (qΓq)(qΓq)

Λ²_{ET C} (15)

In an ordinary system (QCD) the exponent is O(1), Σ_{T C} ∼ Λ³T C,

m_q = Σ_{T C}/Λ²_{ET C} = Λ_{T C}[Λ_{T C}/Λ_{ET C}]² (16)

The full formula is

mq = λv → ΣET C

Λ²_{ET C} = ΣT C

Λ²_{ET C} exp

Z _{ΛET C}

ΛT C

γm(g²)dµ

µ (17)

• If the coupling runs slowly, exp(. . .) ∼ [Λ^{ET C}/ΛT C]^γm(g∗)2

• If γ^m(g_∗²) ∼ 1 then

m_q = Λ_{T C}[Λ_{T C}/Λ_{ET C}]¹ (18)

A large ratio ΛT C/ΛET C can co-exist with large mq and FCNC ∼ 1/Λ²ET C

This is a generic bug/feature of composite Higgs dynamics

• You have to replace a dimension 4 operator by a dimension 6 one

• The dimension 6 coupling has a scale 1/Λ² where Λ is very large

• Some miracle has to cancel the 1/Λ² and make it O(1).

This is a nonperturbative miracle, which is why lattice people wanted to study it

Lattice technicolor 2008-2014 or so

Goal: are there systems which are

• Chirally broken and confining

• Have slowly running couplings

• Have large anomalous dimensions Methodology

• Filter theories using perturbative beta function

– β(g²) = b1g⁴ + b2g⁶ + . . ., for big Nf/Nc can have b2/b1 < 0 or β(g²) ∼ 0

• Simulate them and measure either – Spectroscopy

– A running coupling constant “derived” from an observable

• Decide if they are viable candidates for BS model physics This turned out to be hard!

Computational issue is easy to see in one-loop beta function 1

g²(s) − 1

g²(1) = b₁

8π² log s. (19)

b₁ = 11 − 2N^f/3 for N_c = 3, N_f fundamental rep Dirac fermions

For Nf = 3, b1 = 9. We know that in s = 10 (0.1 fm to 1 fm) QCD goes from weak to strong For N_f = 12 b₁ = 3. If it were identical to N_f = 3, we’d need s = 1000 to see same change But all lattice simulations are done in finite volume

• If a system has a slowly running coupling, it is either

– Strongly coupled at long distance and hence strongly coupled at short distance – Weakly coupled at short distance and hence weakly coupled at long distance

• It’s very hard to tell slow running from no running or slow running the wrong way (to an “IR fixed point”)

And

• in QCD, as we tune to a → 0 at ever smaller bare coupling, we know what we are doing

• In a slow running theory, strong coupling at short distance means we don’t know what we are doing

Map of the N_c, N_f, representation plane

Most work involved toy models, SU (3) with N_f fundamentals

• Large N^f > 12 to 16 are infrared conformal (Nf > 16 loses AF), no particles!

• Small N^f < 8 are confining, chirally broken, like QCD, but coupling runs fast

• In between it was a mess but most people think – N_f = 10, 12 are “inside the conformal window”

– N_f = 8 is barely confining

• The only large γ^m’s were “right on the border”

Beta function for SU (2) with N_f = 2 adjoints, with a zero (β(g²)/g⁴ vs 1/g²

Beta function for SU (3) with Nf = 12 fundamentals (u = g²), with a zero, from 1610.10004

Dilatonic Higgs

Several lattice calculations of slowly running systems report light scalar states.

Scenario for our Higgs, and new physics far upstream?

Example: scalars with

V (φ) = a₂φ^T · φ + a⁴(φ^T · φ)², (20) for a₂ < 0, massless Goldstones, m²_H ∼ −a², as a₂ → 0, m^H → 0

“dilaton” – PNGB associated with scale symmetry breaking Old idea, fraught phenomenology –

• Our low energy world isn’t conformal

• Any high scale Λ pulls M^H up to gΛ, dilaton or not

• There has to be some scale (v or f^π) set by EW physics, governs mass of other states

• Have to get the dilaton’s branching ratios to be Higgs-like But this is a long way from a Higgs!

Extra: Higgs couplings with dilatonic/composite Higgs

Usually, there are two scales, f and v = 246 GeV and h

v → χ

f = χ v

v

f (21)

L = (2χ

f + χ²

f²)[m²_WW_µ⁺W^µ− + 1

2m²_ZZ_µZ^µ] + χ f

X

ψ

m_ψψψ¯ (22)

And in h → γγ or h → gg

L = χ f[2

e∆β^EMF_µν² + 2

g₃∆β^QCDG²_µν] (23)

Composite Higgs: the Higgs as a pseudo - Nambu - Goldstone boson

In TC, condensate breaks SU (2) × U(1), Σ ∼ v³ = (246 GeV)³

Alternative: global symmetry breaks, condensate preserves SU (2) × U(1) Gauge a subgroup of G to get SU (2) × U(1).

Gauge interactions violate shift symmetry for GB’s, generate masses

Λ → f

G → G/H → SU(2) × U(1)

(24)

Do this carefully to avoid δm² = g²Λ²

v 6= f; phenomenology in terms of ξ = v²/f²

Little Higgs: gauge a product symmetry, G = G1 × G² × . . ., δm² = g₁²g₂²Λ²

Partial compositeness (Kaplan 1991) – linear coupling of top to technibaryons so |ti = |t⁰i + ǫ|Bi

L = λ ¯ψO + h.c. (25)

Higgs mass from unstable Goldstone effective potential L = f²

4 Tr|D^µΣ|² − V (Σ) (26)

where

V (Σ) = α cos h

f − β sin² h

f (27)

α is from gauge bosons

• Calculation is like π⁺ − π⁰ mass difference (from vector-axial correlator integral)

• α cos ^h_f ∼ cg²f²/(16π²)π²

• α > 0 due to vacuum alignment or Witten’s inequality – don’t need a simulation for this

• Simulation can give c (one calculation, so far)

β is from fermion loop, no attempts to compute it yet

Context:

• SM ∈ H makes for large cosets: SU(5) → SO(5), SO(5) → SO(4)

• A fairly large EFT literature (EFT for both G/H and for corrections to the SM)

• The partial compositeness operator is

L =∼ (ψΓχ)(χΓχ)

Λ²_{ET C} (28)

so there is probably another “enhancement argument” needed – not well documented

• “Unusual” patterns of chiral symmetry breaking require higher representation of fermions, special color content, or both

• Often need several representations of fermions at once (one for GB’s, another for top compositeness) These are entertaining systems to simulate. But –

• There are only a tiny number of explicit, published 4-d UV completions, none without issues

A Ferretti model (sort of)

Ferretti and Karateev (1312.5330) have a catalog, Ferretti (1404.7137 and 1604.06467) has a favorite model

• SU(4) gauge group

• N^f = 5 AS2 Majoranas (Q) for SU (5) → SO(5) for the Higgs

• N^f = 3 Dirac’s (q) for mixing with the top through a qqQ baryon

Ayyar, TD, Hackett, Jay, Neil, Shamir, Svetitsky simulating (almost)– N_f = 2 Diracs, each of Q and q Very interesting!

• Two representations: do they have separate chiral symmetry breaking transitions?

• How are the representations’ spectroscopies related (are there large-N^c stories)?

• qqQ baryon spectroscopy – Can’t be a Higgs alternative!

• Beta function for ’t Hooft coupling g²Nc is identical to Nf = 6 QCD

• How can four fermion interactions be anything but small? (quark mass generation, again)

0.116 0.118 0.120 0.122 0.124 0.126 0.128

6

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

am

Baryon masses versus

at fixed

.

Sextet baryons

Fundamental baryons Chimera

Chimera

Chimera

Composite dark matter

We don’t know much about dark matter:

• It is there

• it is dark

(Okay, it is also probably weakly interacting, various astrophysical hints...) Dark matter is either

• Connected to other physics (i.e. LSP)

• Not connected to other physics

Some people like to imagine that it is composite (hidden sector, dark QCD-like) with EW-charged constituents in neutral bound state “neutrons”

See Kribs and Neil 1604.04627 for (long) list of possibilities

A small lattice literature, mostly spectroscopy but some matrix element studies

Dark matter B with dark fermions q interacting with ordinary matter a with quarks Q M_a = y_Qy_q

m²_Higgs X

q

hB|¯qq|BiX

Q

a| ¯QQ|a . (29)

f_q^(B) ≡ m_q M_B

∂M_B

∂m_q = m_q

M_BhB|¯qq|Bi. (30)

I can’t resist showing pictures of large-N_c QCD!

Trying to sum up

Back to the sign problem – mostly, people ignore it (for now) Some systems are not very QCD - like (they are nearly conformal)

• SU(N) gauge theories with many fermion DoF’s

• N = 4 SYM

Here, the big technical issue is dealing with slow running in finite volume Some systems are very QCD - like

• Partial compositeness

• QCD with different N^c, Nf, fermion representations

Technical issues are also QCD-like – lots of QCD-like things to measure (and compare with QCD) BUT–

None of these systems really exist, so

• No single system is interesting by itself

• You need to study many related ones, and look for trends