Dark Interactions and Supercomputers - indico.bnl.gov · Brookhaven Forum 2017 10/12/2017 - BNL Dark Interactions and Supercomputers Enrico Rinaldi RIKEN BNL Research Center

Brookhaven Forum 2017 10/12/2017 - BNL

Dark Interactions and Supercomputers

Enrico Rinaldi

RIKEN BNL Research Center

What is Dark Matter?

What is Dark Matter?

mSUGRA

R-parityConserving

Supersymmetry

pMSSM

R-parityviolating

Gravitino DM

MSSM NMSSM

DiracDM

Extra Dimensions

UED DM

Warped Extra Dimensions

Little Higgs

T-odd DM

5d

6d

Axion-like Particles

QCD Axions

Axion DM

Sterile Neutrinos

LightForce Carriers

Dark Photon

Asymmetric DM

RS DM

Warm DM

?

HiddenSector DM

WIMPless DM

Littlest Higgs

Self-InteractingDM

Q-balls

T Tait

Solitonic DM

QuarkNuggets

Techni-baryons

Dynamical DM

[Planning the Future of U.S. Particle Physics (Snowmass 2013), 1401.6085]

Composite Dark Matter


✦ Dark Matter is a composite object


✦ Dark Matter is a composite object bound state of NEW strong force



✦ Interesting and complicated internal structure

✦ Properties dictated by strong dynamics

✦ Self-interactions are natural

bound state of NEW strong force







Analogous to QCD






✦ Composite object is neutral

✦ Constituents may interact with Standard Model particles


Analogous to QCD









Analogous to QCD

Chance to observe them in experiments and give the

correct relic abundance









Analogous to QCD

Chance to observe them in experiments and give the

correct relic abundance

solved with supercomputers

Gauge Theories on Supercomputers

• Discretize space and time • lattice spacing “a” • lattice size “L”

• Keep all d.o.f. of the theory • not a model! • no simplifications

• Amenable to numerical methods

• Monte Carlo sampling • use supercomputers

• Precisely quantifiable and improvable errors

• Systematic • Statistical

a

L

[KEK-Japan]

Gauge Theories on Supercomputers

• Discretize space and time • lattice spacing “a” • lattice size “L”

• Keep all d.o.f. of the theory • not a model! • no simplifications

• Amenable to numerical methods

• Monte Carlo sampling • use supercomputers

• Precisely quantifiable and improvable errors

• Systematic • Statistical

a

L

[KEK-Japan]

Lattice Gauge Theory

“Stealth Dark Matter” - case study

✦ New strongly-coupled SU(4) gauge sector “like” QCD with a plethora of composite states in the spectrum: all mass scales are technically natural for hadrons

✦ New Dark fermions: have dark color and also have electroweak charges (W/Z,𝛾)

✦ Dark fermions have electroweak breaking masses (Higgs) and electroweak preserving masses (not-Higgs)

✦ A global symmetry naturally stabilizes the dark lightest baryonic composite states (e.g. DM is a stable dark neutron)

[LSD collab., Phys. Rev. D88 (2013) 014502]



[LSD collab., Phys. Rev. Lett. 115 (2015) 171803]

Detection: EM Polarizability

� �

Nucleus Nucleus

p p0

k k0

` `� q

k + `

q = k0 � k = p� p0

cF e2

m3�

�?�Fµ↵F ⌫↵vµv⌫

γγ

[Weiner & Yavin,1206.2910] [Frandsen et al., 1207.3971][Pospelov & Veldhuis, hep-ph/0003010] [Ovanesyan & Vecchi, 1410.0601] [Detmold et al., 0904.1586-1001.1131]

DM DM

SM SM


� �

Nucleus Nucleus

p p0

k k0

` `� q

k + `

q = k0 � k = p� p0Lattice

cF e2

m3�


γγ


DM DM

SM SM


� �

Nucleus Nucleus

p p0

k k0

` `� q

k + `

q = k0 � k = p� p0Lattice

Nuclear Physics

cF e2

m3�


γγ


DM DM

SM SM

�nucleon

(Z,A) =Z4

A2

144⇡↵4µ2

n�(MAF )2

m6

�R2

[cF ]2

Lowest bound from EM polarizability

Electric polarizability from lattice simulations with background fields

γγ

��

��×��-��

��×��-��

��×��-��

��×��-��

��×��-��

��×��-��

��×��-��

�χ (��)

��-��

��(�� )

[LSD collab., Phys. Rev. Lett. 115 (2015) 171803][NEW! PandaX-II, 1607.07400 and LUX-332d,1608.07648]

�nucleon

(Z,A) =Z4

A2

144⇡↵4µ2

n�(MAF )2

m6

�R2

[cF ]2



γγ

��

��×��-��

��×��-��

��×��-��

��×��-��

��×��-��

��×��-��

��×��-��

�χ (��)

��-��

��(�� )

LUX exclusion bound for spin-independent cross

section


�nucleon

(Z,A) =Z4

A2

144⇡↵4µ2

n�(MAF )2

m6

�R2

[cF ]2



γγ

��

��×��-��

��×��-��

��×��-��

��×��-��

��×��-��

��×��-��

��×��-��

�χ (��)

��-��

��(�� )


section

Coherent neutrino scattering

background


�nucleon

(Z,A) =Z4

A2

144⇡↵4µ2

n�(MAF )2

m6

�R2

[cF ]2



γγ

��

��×��-��

��×��-��

��×��-��

��×��-��

��×��-��

��×��-��

��×��-��

�χ (��)

��-��

��(�� )


section


background

LEPII bound on charged mesons


�nucleon

(Z,A) =Z4

A2

144⇡↵4µ2

n�(MAF )2

m6

�R2

[cF ]2



γγ

lowest allowed direct detection cross-section for composite dark matter theories with EW charged constituents

��

��×��-��

��×��-��

��×��-��

��×��-��

��×��-��

��×��-��

��×��-��

�χ (��)

��-��

��(�� )


section


background

LEPII bound on charged mesons


Axion Dark Matter

• Axions were originally proposed to solve the Strong-CP problem

• They are also considered a plausible DM candidate

• The axion energy density at early times requires non-perturbative QCD input

Ωtot = 1.000(7) PDG 2014

[Peccei & Quinn: PRL 38 (1977) 1440, PR D16 (1977) 1791] [Preskill, Wise & Wilczek, Phys. Lett. B 120 (1983) 127-132]

10-10

10-16

10-15

10-14

10-13

10 100 1000

1 10 100

Axio

n Co

uplin

g |g

aaa |

(GeV

-1)

Axion Mass (µeV)

Gen 2 ADMX Projected Sensitivity

Cavity Frequency (GHz)

2015

2016

2017

20182019

"Hadronic" Coupling

Minimum Coupling

Axion

Cold D

ark Ma

tter

Warm D

ark Ma

tter

ADMX Published Limits

Non RF-cavity Techniques (CAST/IAXO)

Too Much Dark Matter

White Dwarf and Supernova Bounds

[ADMX]

Axion Dark Matter




• Being sought in ADMX (LLNL, UW) & CAST-IAXO (CERN) with large discovery potential in the next few years Ωtot = 1.000(7)

PDG 2014


10-10

10-16

10-15

10-14

10-13

10 100 1000

1 10 100

Axio

n Co

uplin

g |g

aaa |

(GeV

-1)

Axion Mass (µeV)



2015

2016

2017

20182019

"Hadronic" Coupling

Minimum Coupling

Axion

Cold D

ark Ma

tter

Warm D

ark Ma

tter





[ADMX]

Axion Dark Matter




• Being sought in ADMX (LLNL, UW) & CAST-IAXO (CERN) with large discovery potential in the next few years

• Requiring Ωaxion ≤ ΩCDM yields a lower bound on the axion mass today

Ωtot = 1.000(7) PDG 2014


10-10

10-16

10-15

10-14

10-13

10 100 1000

1 10 100

Axio

n Co

uplin

g |g

aaa |

(GeV

-1)

Axion Mass (µeV)



2015

2016

2017

20182019

"Hadronic" Coupling

Minimum Coupling

Axion

Cold D

ark Ma

tter

Warm D

ark Ma

tter





[ADMX]

Axion Dark Matter




• Being sought in ADMX (LLNL, UW) & CAST-IAXO (CERN) with large discovery potential in the next few years

• Requiring Ωaxion ≤ ΩCDM yields a lower bound on the axion mass today

Ωtot = 1.000(7) PDG 2014

solved with supercomputers


10-10

10-16

10-15

10-14

10-13

10 100 1000

1 10 100

Axio

n Co

uplin

g |g

aaa |

(GeV

-1)

Axion Mass (µeV)



2015

2016

2017

20182019

"Hadronic" Coupling

Minimum Coupling

Axion

Cold D

ark Ma

tter

Warm D

ark Ma

tter





[ADMX]

Axion mass from lattice simulations

[Bonati et al., 1512.06746]

[Berkowitz, Buchoff, ER., 1505.07455, PRD 92 (2015)]

m2af

2a =

@2F

@✓2

��✓=0

△△◇◇△△△△△△△△◇◇◇◇★★◇◇

✶✶

1. 2. 3. 4. 5. 6.0.

0.1

0.2

0.3

0.4

0.5

T/Tc

χ1/4 /T c

Nσ

DIGM fit + statistical error △ 64

Systematic fitting error ◇ 80★ 96✶ 144

�

T 4c

=C

(T/Tc)nNon-perturbative calculation of QCD topology

at finite temperature


• Pure gauge SU(3) topological susceptibility ➥ compatible with model predictions (DIGM/IILM), but lattice identifies important non-perturbative effects

[Bonati et al., 1512.06746]


m2af

2a =

@2F

@✓2

��✓=0

△△◇◇△△△△△△△△◇◇◇◇★★◇◇

✶✶

1. 2. 3. 4. 5. 6.0.

0.1

0.2

0.3

0.4

0.5

T/Tc

χ1/4 /T c

Nσ



�

T 4c

=C

(T/Tc)n

[Kitano&Yamada,1506.00370][Borsanyi et al.,1508.06917][Frison et al.,1606.07175]

Non-perturbative calculation of QCD topology at finite temperature



• is QCD topological susceptibility at high-T well described by models? ➥ light fermions importantly affect the vacuum

[Bonati et al., 1512.06746]


m2af

2a =

@2F

@✓2

��✓=0

△△◇◇△△△△△△△△◇◇◇◇★★◇◇

✶✶

1. 2. 3. 4. 5. 6.0.

0.1

0.2

0.3

0.4

0.5

T/Tc

χ1/4 /T c

Nσ



�

T 4c

=C

(T/Tc)n



[Trunin et al.,1510.02265][Petreczky et al.,1606.03145][Borsanyi et al.,1606.07494]



• is QCD topological susceptibility at high-T well described by models? ➥ light fermions importantly affect the vacuum

[Bonati et al., 1512.06746]


m2af

2a =

@2F

@✓2

��✓=0

△△◇◇△△△△△△△△◇◇◇◇★★◇◇

✶✶

1. 2. 3. 4. 5. 6.0.

0.1

0.2

0.3

0.4

0.5

T/Tc

χ1/4 /T c

Nσ



�

T 4c

=C

(T/Tc)n



[Trunin et al.,1510.02265][Petreczky et al.,1606.03145][Borsanyi et al.,1606.07494]

Great effort to control all systematic lattice effects in order to guide experiments.

Challenging state-of-the art simulations.

Axion mass lower bound

[ADMX Website]

[Berkowitz, Buchoff, ER, PRD 92 (2015) 034507]


Lattice SU(3) Pure Glue

fa < (4.10±0.04) 1011 GeV ma > (14.6±0.1) μeV

[ADMX Website]




fa < (4.10±0.04) 1011 GeV ma > (14.6±0.1) μeV

[ADMX Website]


axions < 100% of DM

smaller χ



fa < (4.10±0.04) 1011 GeV ma > (14.6±0.1) μeV

[ADMX Website]


axions < 100% of DM

smaller χ

control systematics of the non-perturbative physics through lattice methods

be able to derive the bound from first-principle results


Lattice QCD with physical quarks

ma > (28±2) μeV

[ADMX Website]

m2af

2a =

@2F

@✓2

��✓=0

[Borsanyi et al., 1606.07494, Nature 539]


Lattice QCD with physical quarks

ma > (28±2) μeV

[ADMX Website]

m2af

2a =

@2F

@✓2

��✓=0

[Borsanyi et al., 1606.07494, Nature 539]

[Lombardo, Nature News & Views]

Concluding remarks

✦Composite dark matter and Axion dark matter are viable interesting possibility with rich phenomenology

✦Lattice methods can help in calculating direct detection cross sections, production rates at colliders, self-interaction cross sections and the axion mass bound. Direct phenomenological relevance and guide to experimental searches.

✦Dark matter constituents can carry electroweak charges and still the stable composites are currently undetectable. Stealth cross section.

✦Lowest bound for composite dark matter models: ~300 GeV (colliders+direct detection+lattice)

✦Axions from QCD dynamics have a lowest mass bound ~20-50 μeV (cosmology+lattice)

5

��

��×��-��

��×��-��

��×��-��

��×��-��

��×��-��

��×��-��

��×��-��

�� (��)

�-��

��

��(�� )

MB(GeV)

mB(GeV)

FIG. 2. The DM spin-independent scattering cross section per nu-cleon evaluated for xenon is shown as the purple band obtainedfrom the SU(4) polarizability, where the width of the band cor-responds to 1/3 < MA

F < 3 from low to high. The blue curveand the light blue region above it is excluded by the LUX con-straints [1]. The vertical, darker shaded region is excluded bythe LEP II bound on charged mesons [23]. The orange regionrepresents the limit at which direct detection experiments willbe unable to discriminate DM events from coherent neutrino re-coil [39]. We emphasize that this plot is applicable for xenon, andwould require calculating Eq. (17) to apply to other nuclei.

would have form factor suppression. This implies the stan-dard missing energy signals that arise from DM productionand escape from the detector are rare.

Finally, there are many avenues for further investiga-tion of stealth dark matter, detailed in [23]. One vital is-sue is to better estimate the abundance. In the DM massregime where stealth DM is detectable at direct detectionexperiments, the abundance of stealth dark matter can arisenaturally from an asymmetric production mechanism [23]that was considered long ago [7–9] and more recently re-viewed in [40]. If there is indeed an asymmetric abundanceof bosonic dark matter, there are additional astrophysicalconsequences [41–43] that warrant further investigation toconstrain or probe stealth DM.

Acknowledgements – MIB would like to thank SilasBeane, Paulo Bedaque, Tom Cohen, Jon Engel, Wick Hax-ton, David B. Kaplan, Jerry Miller, Sanjay Reddy, MartinSavage, Achim Schwenk, and Luca Vecchi for enlighteningdiscussions on the full complexity of the nuclear physicscalculation required for an accurate cross section predic-tion. We are also indebted to Brian Tiburzi for access to in-formation that allowed for very useful checks of our back-ground field code. We thank the Lawrence Livermore Na-tional Laboratory (LLNL) Multiprogrammatic and Institu-tional Computing program for Grand Challenge allocationsand time on the LLNL BlueGene/Q (rzuseq and vulcan) su-percomputer. We thank LLNL for funding from LDRD 13-ERD-023 “Illuminating the Dark Universe with PetaFlopsSupercomputing”. Computations for this work were car-ried out in part from the LLNL Institutional ComputingGrand Challenge program and from the USQCD Collab-

oration, which is funded by the Office of Science of theUS Department of Energy. MIB thanks the University ofOregon and University of Maryland, College Park for hos-pitality during the course of this work.

This work has been supported by the U. S. Depart-ment of Energy under Grant Nos. de-sc0008669 and de-sc0009998 (D.S.), de-sc0010025 (R.C.B., C.R., E.W.),DE-FG02-92ER-40704 (T.A.), de-sc0011640 (G.D.K.),DE-FG02-00ER41132 (M.I.B.), and Contracts DE-AC52-07NA27344 (LLNL), DE-AC02-06CH11357 (ArgonneLeadership Computing Facility), and by the National Sci-ence Foundation under Grant Nos. NSF PHY11-00905(G.F.), OCI-0749300 (O.W.). Brookhaven National Labo-ratory is supported by the U. S. Department of Energy un-der contract de-sc0012704. S.N.S was supported by the Of-fice of Nuclear Physics in the U. S. Department of Energy’sOffice of Science under Contract DE-AC02-05CH11231.

⇤ Present address: Higgs Centre for Theoretical Physics,School of Physics & Astronomy, The University of Edin-burgh, EH9 3FD, UK

[1] D. Akerib et al. (LUX Collaboration), (2013),arXiv:1310.8214 [astro-ph.CO].

[2] R. Agnese et al. (SuperCDMS Collaboration),Phys.Rev.Lett. 112, 241302 (2014), arXiv:1402.7137[hep-ex].

[3] P. Cushman, C. Galbiati, D. McKinsey, H. Robertson,T. Tait, et al., (2013), arXiv:1310.8327 [hep-ex].

[4] M. Xiao et al. (PandaX Collaboration), Sci.ChinaPhys.Mech.Astron. 57, 2024 (2014), arXiv:1408.5114[hep-ex].

[5] S. Nussinov, Phys.Lett. B165, 55 (1985).[6] R. S. Chivukula and T. P. Walker, Nucl.Phys. B329, 445

(1990).[7] S. M. Barr, R. S. Chivukula, and E. Farhi, Phys.Lett. B241,

387 (1990).[8] S. M. Barr, Phys.Rev. D44, 3062 (1991).[9] D. B. Kaplan, Phys.Rev.Lett. 68, 741 (1992).

[10] R. Lewis, C. Pica, and F. Sannino, Phys.Rev. D85, 014504(2012), arXiv:1109.3513 [hep-ph].

[11] A. Hietanen, C. Pica, F. Sannino, and U. I. Sondergaard,Phys.Rev. D87, 034508 (2013), arXiv:1211.5021 [hep-lat].

[12] T. Appelquist, R. C. Brower, M. I. Buchoff, M. Cheng, S. D.Cohen, G. T. Fleming, J. Kiskis, M. F. Lin, E. T. Neil, J. C.Osborn, C. Rebbi, D. Schaich, C. Schroeder, S. N. Syrit-syn, G. Voronov, P. Vranas, and J. Wasem (Lattice StrongDynamics (LSD) Collaboration), Phys.Rev. D88, 014502(2013), arXiv:1301.1693 [hep-ph].

[13] A. Hietanen, R. Lewis, C. Pica, and F. Sannino, JHEP 1412,130 (2014), arXiv:1308.4130 [hep-ph].

[14] T. Appelquist, E. Berkowitz, R. C. Brower, M. I. Bu-choff, G. T. Fleming, J. Kiskis, G. D. Kribs, M. Lin,E. T. Neil, J. C. Osborn, C. Rebbi, E. Rinaldi, D. Schaich,C. Schroeder, S. Syritsyn, G. Voronov, P. Vranas,E. Weinberg, and O. Witzel (Lattice Strong Dynam-ics (LSD) Collaboration), Phys.Rev. D89, 094508 (2014),


extra

The darkness of Composite Dark Matter

[Wikipedia]

[Bagnasco et al., hep-ph/9310290]


[Wikipedia]


Lowest dimensional operators:★ magnetic dipole (5) ★ charge radius (6) ★ polarizability (7)


[Wikipedia]


Most relevant interaction if constituents have Yukawa couplings!

Lowest dimensional operators:★ magnetic dipole (5) ★ charge radius (6) ★ polarizability (7)

—

+

Higgs exchange

—

+

Uncorrelated Paired

Electric field

QCD++

— —

—++

—

Dark Matter

+ +

— —

—+

—+

PRL Editors’ Suggestion: Polarizability

PRD Editors’ Suggestion: Higgs exchange[LSD collab., Phys. Rev. D92 (2015) 075030]


https://www.llnl.gov/news/new-stealth-dark-matter-theory-may-explain-mystery-universes-missing-mass

http://www.lescienze.it/news/2015/09/28/news/materia_oscura_stealth_matter_lhc-2779983

http://www.media.inaf.it/2015/09/25/materia-oscura-stealth/



http://www.media.inaf.it/2015/09/25/materia-oscura-stealth/

“Stealth Dark Matter” model

• The field content of the model consists in 8 Weyl fermions

• Dark fermions interact with the SM Higgs and obtain current/chiral masses

• Introduce vector-like masses for dark fermions that do not break EW symmetry

• Diagonalizing in the mass eigenbasis gives 4 Dirac fermions

• Assume custodial SU(2) symmetry arising when u ↔ d

3

Field SU(N)D

(SU(2)L

, Y ) Q

F1

=

Fu

1

F d

1

!N (2, 0)

+1/2

�1/2

!

F2

=

Fu

2

F d

2

!N (2, 0)

+1/2

�1/2

!

Fu

3

N (1,+1/2) +1/2

F d

3

N (1,�1/2) �1/2

Fu

4

N (1,+1/2) +1/2

F d

4

N (1,�1/2) �1/2

TABLE I. Fermion particle content of the composite dark mattermodel. All fields are two-component (Weyl) spinors. SU(2)

L

refers to the standard model electroweak gauge group, and Y isthe hypercharge. The electric charge Q = T

3

+Y for the fermioncomponents is shown for completeness.

yet have the ability to simulate on the lattice. Naive di-mensional analysis applied to the annihilation rate suggeststhe dark matter mass scale should be >⇠ 10-100 TeV, but amore precise estimate is not possible at this time. In anycase, for dark matter with mass below this value, there isan underproduction of dark matter through the symmet-ric thermal relic mechanism, and so this does not restrictconsideration of dark matter mass scales between the elec-troweak scale up to this thermal abundance bound.

CONSTRUCTING A VIABLE MODEL

[placeholder for a description of how a viable modelwith interactions with the Higgs can be constructed whilesatisfying the various (gross) experimental constraints]

We consider a new, strongly-coupled SU(N)

D

gaugegroup with fermionic matter in the vector-like representa-tions shown in Table I.

This is not the only possible choice for the charges, butthe requirement for the presence of Higgs Yukawa cou-plings, along with extremely strong bounds on the ex-istence of stable fractionally-charged particles based onsearches for rare isotopes [? ], greatly constrains the num-ber of possible models.

DARK FERMION INTERACTIONS AND MASSES

The fermions F u,d

i

transform under a global U(4) ⇥U(4) flavor symmetry that is broken to [SU(2) ⇥ U(1)]4by the weak gauging of the electroweak symmetry. Fromthis large global symmetry, one SU(2) (diagonal) sub-group will be identified with SU(2)

L

, one U(1) subgroup

will be identified with U(1)

Y

, and one U(1) will be iden-tified with dark baryon number. The total fermionic con-tent of the model is therefore 8 Weyl fermions that pairup to become 4 Dirac fermions in the fundamental oranti-fundamental representation of SU(N)

D

with electriccharges of Q ⌘ T

3,L

+ Y = ±1/2. We use the notationwhere the superscript u and d (as in F u, F d and later u, d, u, d) to denote a fermion with electric charge ofQ = 1/2 and Q = �1/2 respectively.

The fermion kinetic terms in the Lagrangian are givenby

L =

X

i=1,2

iF †i

�̄µDi,µ

Fi

+

X

i=3,4;j=u,d

iF j

i

†�̄µDj

i,µ

F j

i

,

(1)where the covariant derivatives are

D1,µ

⌘ @µ

� igW a

µ

�a/2 � igD

Gb

µ

tb (2)

D2,µ

⌘ @µ

� igW a

µ

�a/2 + igD

Gb

µ

tb

⇤(3)

Dj

3,µ

⌘ @µ

� ig0Y jBµ

� igD

Gb

µ

tb (4)

Dj

4,µ

⌘ @µ

� ig0Y jBµ

+ igD

Gb

µ

tb

⇤(5)

with the interactions among the electroweak group and thenew SU(N)

D

. Here Y u

= 1/2, Y d

= �1/2 and tb

are the representation matrices for the fundamental N ofSU(N)

D

.The vector-like mass terms allowed by the gauge sym-

metries are

L � M12

✏ij

F i

1

F j

2

�Mu

34

F u

3

F d

4

+Md

34

F d

3

F u

4

+h.c., (6)

where ✏12

⌘ ✏ud

= �1 = �✏12 and the relative minussigns between the mass terms have been chosen for laterconvenience. The mass term M

12

explicitly breaks the[SU(2) ⇥ U(1)]2 global symmetry down to the diagonalSU(2)

diag

⇥ U(1) where the SU(2)

diag

is identified withSU(2)

L

. The mass terms Mu,d

34

explicitly break the re-maining [SU(2)⇥U(1)]2 down to U(1)⇥U(1) where oneof the U(1)’s is identified with U(1)

Y

. (In the special casewhen Mu

34

= Md

34

, the global symmetry is accidentally en-hanced to SU(2)⇥U(1), where the global SU(2) acts as acustodial symmetry.) Thus, after weakly gauging the elec-troweak symmetry and writing arbitrary vector-like massterms, the unbroken flavor symmetry is thus U(1)⇥U(1).

Electroweak symmetry breaking mass terms arise fromcoupling to the Higgs field H that we take to be in the(2, +1/2) representation. They are given by

L � yu

14

✏ij

F i

1

HjF d

4

+ yd

14

F1

· H†F u

4

� yd

23

✏ij

F i

2

HjF d

3

� yu

23

F2

· H†F u

3

+ h.c., (7)

where again the relative minus signs are chosen for laterconvenience. After electroweak symmetry breaking, H =

(0 v/p

2)

T , with v ' 246 GeV. Inserting the vev

L �+ yu14✏ijFi1H

jF d4 + yd14F1 ·H†Fu

4

� yd23✏ijFi2H

jF d3 � yu23F2 ·H†Fu

3 + h.c.

L � M12✏ijFi1F

j2 �Mu

34Fu3 F

d4 +Md

34Fd3 F

u4 + h.c.

yu14 = yd14 yu23 = yd23 Mu34 = Md

34


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Dark Interactions and Supercomputers - indico.bnl.gov · Brookhaven Forum 2017 10/12/2017 - BNL Dark Interactions and Supercomputers Enrico Rinaldi RIKEN BNL Research Center