Synthesis and Phenomenology of large NuclearDark Matter (and Twin Higgs asides)

Robert Lasenby, University of Oxford

Work with E. Hardy, J. March-Russell, S. WestarXiv 1411.3739 and arXiv 1504.05419

and J. March-Russell, I. Garcıa Garcıa (work in progress)

LLNL, April 23rd 2015

Large composite DM states

I Standard model: example of conserved baryon number,attractive interactions leading to multitude of large, stablebound states (nuclei)

I What if a similar thing happens for dark matter?

I Possibilities:

I Number distribution over DM statesI States with large spinI Structure on scales � 1/m — form factors in scattering,

possibility of larger cross sectionsI Coherent enhancement of interactionsI Inelastic processes — fusions, fissions, excited statesI ‘Late-time’ (T � m) synthesis — can achieve very heavy

(& 100TeV) DM from thermal freeze-out

I Earlier example of Q-balls — non-topological solitons of scalarfields

I Related work: Krnjaic et al, Detmold et al, Wise et al

SM nuclei

50 100 150 200A

2

4

6

8

10

12

BE � A

MeV

SM nuclei without Coulomb repulsion

50 100 150 200A

2

4

6

8

10

12

BE � A

MeV

Dark nucleosynthesis

Free energy F = E − T S :large T ⇒ everything dissociatedsmall T ⇒ large states favouredAssume asymmetric

T >> BE T ~ BE T << BE

Freeze-out of fusions

I Equal sizes: A + A→ 2A

Γ

H∼ 〈σv〉nA

H∼ σ1v1n0

HA2/3A−1/2A−1 =

σ1v1n0

HA−5/6

With MA = AM1,

σ1v1n0

H∼ 2×107

(1GeV fm−3

ρb

)2/3(T

1MeV

)3/2( M1

1GeV

)−5/6

so build-up to A ∼ 5× 108 may be possible

I Small + large: 1 + A→ (1 + A). Rate for A of these is

Γ ∼ 〈σv〉nkk

A∼ σ1v1n0

1

k1/2A2/3A−1 =

σ1v1n0

k1/2A−1/3

Aggregation process

dnk

dt+ 3Hnk = −

∑j≥1

〈σv〉j ,k nj nk +1

2

∑i+j=k

〈σv〉i ,j ni nj

k

jk+j k

j

ikk

k

In terms of yields, nk (t)/s(t) ≡ Yk (t) ≡ Y0yk (w(t)), (∑

yk = 1)

where new ‘time’ w is

Then,

dw

dt= Y0σ1s(t)f (Td (t)) = n0σ1v1

dyk

dw= −yk

∑j

Rk,j yj +1

2

∑i+j=k

Ri ,j yi yj

Scaling solution

〈σv〉i ,j ∼ (radiusi + radiusj )2vrel

Ri ,j = (i2/3 + j2/3)(i−1/2 + j−1/2) , Rλi ,λj = λ1/6Ri ,j

1 5 10 50 100 5001000k10

-9

10-7

10-5

0.001

0.1

yk

Number distributions at equally-spaced log w values, up to w = 75.

Scaling solution

I Shape stays the same, average size increases, k(w) ∼ w 6/5.

0.01 0.1 1 10 100

k

k

10-8

10-6

10-4

0.01

1

k2

yk

k2yk versus k/k , where k = 1/∑

k yk .

I Attractor solution, depending only on large-k behaviour ofkernel — reach this form (eventually) independent of initialconditions, small-k kernel.

Real-time behaviour

Td ∝ 1/a ⇒ w(T ) ' 2

3

n0σ1v1

H0

(1−

(T

T0

)3/2)

Most of build-up completes within one Hubble time.

1 5 10 50 100 5001000k10

-9

10-7

10-5

0.001

0.1

yk

Number distributions at half e-folding time intervals

What if there’s a bottleneck at small numbers? (cf. SM)I If Ri ,j for small i , j is low enough, and wmax is small enough,

never reach scaling regimeI Counter-intuitively, this can result in building up larger nuclei,

since small + large fusions are less velocity-suppressedI For 1 + k → (k + 1) fusions,

dk

dw' R1,k y1 ∝ k2/3 ⇒ k ∼

(∫dw y1

)3

0 200 400 600 800 1000 1200 1400k

0.001

0.002

0.003

0.004

k yk

Mass distribution at w = 25 for R1,1 = 4, 10−4, 10−5

Power-law distributionIf we have a bath of small-number states throughout,

k ∼ (wmax − winj)3 ⇒ −dwinj

dk∼ k−2/3 ⇒ yk ∼ k−2/3 (k large)

Leads to power-law number distribution, qualitatively differentfrom scaling solution

1 5 10 50 100 500 1000k10

-10

10-8

10-6

10-4

0.01

1

yk

Number distribution at w = 25 for R1,1 = 1, 10−1, . . . , 10−6

Summary: Synthesis of Nuclear Dark Matter

I Considered DM models with large bound states ofstrongly-interacting constituents

I Properties of sufficiently large ‘dark nuclei’ may obeygeometrical scaling laws — this can determine numberdistribution from Big Bang Dark Nucleosynthesis

I If small-small fusions are fast enough, obtain universal scalingform of number distribution — may have A & 108

I With a bottleneck at small numbers, may build up even largernuclei, with power-law number distribution

I In both cases, most of build-up completes within a Hubbletime

I Have assumed that deviations from geometrical cross sectionsare eventually unimportant — not necessarily the case!

Signatures of Nuclear Dark Matter

Most model-independent consequences:

I Soft scatterings coherently enhanced by A2

I Number density ∝ 1/A, so total direct detection rate ∝ AI For given direct detection rate, production at colliders etc.

suppressed

I Possibility of new momentum-dependent form factors in directdetection

I Low-energy collective excitations may allow coherentlyenhanced inelastic scattering

I Inelastic self-interactions between DM may lead to indirectdetection signals, or modify distribution in halos / captureddistribution in stars

Many other model-dependent possibilities still to be investigated

Twin Higgs

I Proposed solution to ‘little hierarchy’ problem — stabilisingthe EW scale up to collider energies, Λ ∼ 5− 10TeV.

I SM Higgs as PNGB of approximate SU(4) global symmetry,broken down to SU(3):

H = (HA,HB) , V = λ(|H|2 − f 2/2)2

I SU(4) explicitly broken by SM gauge and Yukawa couplings— but, if A, B sectors related by approximate Z2, this givesus back accidental SU(4).

I Since observed light Higgs is SM-like, need to break Z2 sothat PNGB Higgs is mostly aligned with A,

f 2 = v 2A + v 2

B , v 2A � v 2

B

The Minimal (‘Fraternal’) Twin Higgs

I Idea: introduce only those B-sector states we need in order tohave acceptable tuning up to Λ ∼ O(10TeV).

I Main contributions from SM: top, SU(2)L gauge bosons,QCD (two loops).

I B sector:I t with yt ' yt .I SU(2)′L gauge group, g2 ' g2 (and so b partner of t).I SU(3)′ gauge group, roughly similar confinement scale.

I For anomaly cancellation, need to have bR and (τ , ν) leptondoublet.

I As long as yb, yτ � yt , no effect on tuning.

Fraternal Twin Higgs — Nuclear DM?

I Stable states: B = bbb baryons, τ , ν

I In analogy to SM, B asymmetry ⇒ asymmetric relic Bpopulation.

I Possibility of B bounds states? Nuclear matter? Synthesisproblem: de-excitation from small-number fusions.

I Radiative capture via SU(2)′L too slow. Introducing gauged

U(1)′Y , bound state formation of BB suppressed compared tonon-identical fermions — still appears to be too slow.

I Models with additional twin quark generation have a betterchance of forming bound states.

Fraternal Twin Higgs — Cosmology

I B nucleons as viable ADM

I Analogously to Lee-Weinberg bound, τ abundance sub-DMrequires either mτ . eV, or mτ & 70GeV. Formτ ∼ 70GeV, have symmetric τ DM

I Interesting effects related to SU(3)′ phase transition:I If there are light hidden sector states (e.g. ν), does twin sector

entropy end up there or in SM?

hSM

G G

SMvs.

g entropy ⇒ ∆Neff ' 0.5, b + g entropy ⇒ ∆Neff ' 1

SU(3)′ phenomenology

Dependence on glueball/meson spectrum, decay constants,transition matrix elements.

Stable glueball spectrum in pure SU(3) (Morningstar and Peardon)

If there are no light hidden sector states, Higgs mixing portal ⇒possibility of (meta)-stable glueball states.

More SU(3)′ phenomenology

I Pure-glue case appropriate to heavy quarks — light quarksgenerally imply faster energy loss to hidden sector.

I Dynamics of phase transition: only heavy quarks ⇒ first orderphase transtion ⇒ entropy production, gravitational radiation.

I Effect of CP violation in twin sector (effect on SM EDMssmall) — e.g. θ angle?

I Summary:I Fraternal twin Higgs provides motivated, ‘minimal’ example of

strongly-coupled hidden sectorI Demonstrates that assumptions about SM portal may have

important consequences for cosmology of hidden sector phasetransition, in some regions of parameter space

BACKUPS

Freeze-out of dissociations

I Overall forward rate for k + (A− k)↔ A is

〈σv〉(k,A−k)→Ank nA−k − ΓA→(k,A−k)nA

I Fusions dominate over dissociations if

〈σv〉nk nA−k

ΓnA� 1 ⇐ n0Λ3e∆B/T � (const. wrt T )

I Since n0Λ3 � 1, equality is at T � ∆B

I Go from equality to n0Λ3e∆B/T � const. within smallfraction of Hubble time.

Independence of initial conditions

Exponentially-falling ICs,yk (0) ∝ e−k/30

1 5 10 50 100 5001000k

10-10

10-8

10-6

10-4

0.01

1

yk

1 5 10 50 100 5001000k

10-10

10-8

10-6

10-4

0.01

1

yk

‘Broad tail’ ICs,y1(0) = 0.97

Form factors in scatteringIf RDM > (∆p)−1, probe DM form factorSharp boundary ⇒ spherical Bessel function form factor

F (q) =qR cos(qR)− sin(qR)

(qR)3∼ 1

(qR)2

If skin depth etc of DM is smaller than SM nuclear scales, goodapproximation

0.5 1.0 1.5 2.0 2.5 3.0 3.5

q

fm-1

10-8

10-6

10-4

0.01

1

 F HqL¤2

e.g. form factor for nuclear charge distribution of 70Ge.

Coherent enhancement

e.g. dim-6 interactions: σ(q = 0) ∼ A2N2 µ2

Λ4

0 5 10 15 20 25 30

ER

keV

10-8

10-7

10-6

10-5

dR � dER

Hday kg keVL-1

Energy recoil spectrum for state with R = 50 fm, A = 3× 106,each constituent with M1 = 20GeV, σn = 2× 10−13 pb. Blue, redcurves for 20GeV, 1TeV WIMP (σn = 10−9 pb).

Effective form factor from distribution of sizes

Distribution over radii averages outpeaks and troughs

ææàà

0.050.10 0.501.00 5.0010.00k � k0

10-5

10-4

0.001

0.01

0.1

1

dn � dk

Effective formfactor similar tointermediate-massmediator

0.1 0.2 0.3 0.4 0.5 0.6

q

fm-1

0.001

0.01

0.1

1

 F HqL¤2

Dependence on DM velocity distribution

Differential event rate:

dR

dER∝

(∫|v |>vmin

d3vf (v)

v

)FN(q)2FD(q)2

withvmin ∝

√ER

Consequence: ignoring FN(q),FD(q), energy recoil spectrum isnon-increasing with ER .

Rising energy recoil spectrum

6 8 10 12 14 16 18

ER

keV

2

4

6

8

nbin

Samples from recoil spectrum,RDM = 50 fm

p-value CDFs for 30, 50, 100events

0.0 0.2 0.4 0.6 0.8 1.0quantile

0.001

0.01

0.1

1

p - value

Astrophysical consequences

I Self-interaction cross section & DM halo constraints?

σAA

mA' 0.05 barn

GeVA−1/3

(1GeV

M1

)1/3(1GeV fm−3

ρb

)2/3

Cross sections saturate at geometrical value, so can be safefrom elastic-scattering constraints

I Proportion of DM mass density released by fusions:

〈σv〉nAtgal∆BE

MA∼ 10−3A−2/3 ρDM

0.3GeV cm−3

For comparison, annihilating symmetric DM has

〈σv〉nX tgal ∼ 3× 10−8

(100MeV

mX

)(〈σv〉Xpb

)Possibility of detectable annihilation-type signal from fusions:depends on SM injection channels etc.