Effective Higgs interactions of composite dark matter†

E. T. Neil∗1,∗2 for the LSD Collaboration

The recent discovery of the Higgs boson has placedsignificant and direct constraints on models which pre-dict new physics near the electroweak scale. At thesame time, recent dark matter direct-detection experi-ments have greatly improved their sensitivity, pushinginto the region of parameter space where dark matterinteracts with the standard model through exchangeof a Higgs boson.

These new constraints are particularly interesting inthe context of composite dark matter models (e.g.1)).In such a model, the lightest stable bound state of astrongly-coupled gauge sector provides a dark mattercandidate. Although the bound state itself must be netelectroweak neutral due to direct-detection experimen-tal constraints, its constituents may carry electroweakcharges; the resulting interactions with the standardmodel in the early universe can explain the observeddark matter abundance, while scattering from present-day direct detection experiments is through suppressedhigher-dimensional operators2).If the dark gauge interaction spontaneously breaks

a continuous symmetry, then some of the bound stateswill be massless Nambu-Goldstone bosons, whichare ruled out by experiment if they are electroweakcharged. A realistic composite dark matter modelthus requires a mechanism for mass generation ofthe constituents. In most models the constituentsare fermions, and they can be given mass throughthe Higgs mechanism; depending on the assignmentof charges, “vector-like” mass terms may also be al-lowed3). The Higgs couplings required for mass gener-ation are directly constrained by direct-detection ex-periment, in terms of the Higgs coupling of the darkmatter bound state itself.

We consider for concreteness a model consisting ofan SU(4) dark gauge force, and a set of degeneratefermions mf carrying electric charge Q = ±1/2. Thedark matter candidate is a neutral baryon-like boundstate of four such fermions, and has total mass mB.The coupling of the Higgs boson to this baryon-likestate is given by the formula

gh,B =mB

v

f

v

mf

∂mf(h)

∂h

h=v

f

(B)f (1)

where v = 246 GeV is the vacuum expectation value

of the Higgs field. The factor f(B)f contains the scalar

matrix element of the fermions inside the baryon-like

† Condensed from the article in Phys. Rev. D89, 094508(2014)

∗1 RIKEN Nishina Center∗2 Department of Physics, University of Colorado, Boulder

0 1000 2000 3000 4000

0.2

0.1

0.05

0.02

0.01

mB GeV

Αmax

Fig. 1. From3), upper bound on the effective Higgs cou-

pling α based on three lattice simulations at various

values of the fermion mass (green curves) and in the

heavy quark limit (red dotted curve). The filled region

on the left shows values of mB ruled out by experimen-

tal bounds on light charged particles from LEP.

state, B|f f |B. This matrix element can be ex-tracted using the Feynman-Hellmann theorem fromthe slope of the baryon mass, ∂mB/∂mf ; this is anon-perturbative quantity, and is calculated using lat-tice simulations of the SU(4) theory. We find 0.15

f(B)f 0.34 in the range of fermion masses simulated.Finally, the term in parentheses,

α =v

mf

∂mf(h)

∂h

h=v

, (2)

is entirely model-dependent, and parameterizes thefraction of the fermion massmf which is generated dueto the Higgs mechanism; if there are no “vector-like”mass terms, then α = 1. The current upper boundon α in this model, based on our lattice calculation of

f(B)f and the latest experimental results from LUX4),is shown in Fig. 1. For the entire range of modelsconsidered, this constraint is significantly less than 1,indicating that generation of mf in this model requiresmore than the Higgs mechanism alone.

References1) G. D. Kribs, T. S. Roy, J. Terning, and K. M. Zurek,

Phys. Rev. D81 (2010) 0950012) T. Appelquist et al. (LSD Collaboration): Phys. Rev.D88

(2013), 0145023) T. Appelquist et al. (LSD Collaboration): Phys. Rev.D89

(2014), 0945084) D. S. Akerib et al. (LUX Collaboration): Phys. Rev. Lett. 112

(2014) 9, 091303

Effective field theory for spacetime symmetry breaking†

Y. Hidaka,∗1 T. Noumi,∗1 and G. Shiu∗2,∗3

Symmetry and its spontaneous breaking play an im-portant role in various areas of physics. In particular,the low-energy effective field theory (EFT) based onthe underlying symmetry structures provides a pow-erful framework for understanding the low-energy dy-namics in the symmetry broken phase.For internal symmetry breaking in Lorentz invari-

ant systems, the EFT based on coset construction hadbeen established in 1960’s1,2). When a global symme-try group G is broken to a residual symmetry groupH, the corresponding Nambu-Goldstone (NG) fieldsπ(x) are introduced as the coordinates of the cosetspace G/H and the general effective action can be con-structed from the Maurer-Cartan one form,

Jµ = Ω−1∂µΩ with Ω(x) = eπ(x) ∈ G/H . (1)

Such a coset construction was also extended to space-time symmetry breaking3,4) accompanied by the in-verse Higgs constraints5) and has been applied to var-ious systems. Although the coset construction cap-tures certain aspects of spacetime symmetry breaking,its understanding seems incomplete compared to theinternal symmetry case.For example, a naive counting of broken spacetime

symmetries based on the global symmetry picture con-tains redundant fields and causes a wrong counting ofNG modes. For conformal symmetry breaking, it isknown that the inverse Higgs constraints compensatesuch a mismatch of NG mode counting. It is also ar-gued recently that the inverse Higgs constraints elimi-nate not only the redundant fields but also the massivemodes, which nonlinearly transform under the brokensymmetries (see, e.g.,6–8)). In addition to the masslessmodes, such massive modes associated with the sym-metry breaking can be relevant in the construction ofphenomenological models (e.g. massive fields with aHubble scale mass are nonnegligible in cosmology).In this work, we discussed the effective field the-

ory for spacetime symmetry breaking from the localsymmetry point of view. By gauging spacetime sym-metries, the identification of NG fields and the con-struction of the effective action are performed based onthe breaking pattern of diffeomorphism, local Lorentz,and (an)isotropic Weyl symmetries as well as the in-ternal symmetries including possible central extensionsin nonrelativistic systems. Such a local picture dis-tinguishes, e.g., whether the symmetry breaking con-densations have spins and provides a correct identifica-

† Condensed from the article in arXiv:1412.5601.∗1 RIKEN Nishina Center∗2 Department of Physics, University of Wisconsin, Madison∗3 Center for Fundamental Physics and Institute for Advanced

Study, Hong Kong University of Science and Technology

tion of the physical NG fields, while the standard cosetconstruction based on global symmetry breaking doesnot. We illustrated that the local picture becomes im-portant in particular when we take into account mas-sive modes associated with symmetry breaking, whosemasses are not necessarily high.

We also revisited the coset construction for space-time symmetry breaking. Based on the relation be-tween the Maurer-Cartan one form and connections forspacetime symmetries, we classify the physical mean-ings of the inverse Higgs constraints by the coordinatedimension of broken symmetries. Inverse Higgs con-straints for spacetime symmetries with a higher dimen-sion remove the redundant NG fields, whereas those fordimensionless symmetries can be further classified bythe local symmetry breaking pattern.

We are now working on several applications of ourapproaches for spacetime symmetry breaking. For ex-ample, there are some recent discussions that inhomo-geneous chiral condensations may appear in the QCDphase diagram. Using our EFT framework, we discussthe dispersion relation of the NG field in such a phase.Another ongoing application is inflation. We are e.g.trying to classify the source of primordial gravitationalwaves (which potentially affect the B-mode polariza-tion of cosmic microwave backgrounds) by the symme-try breaking point of view. We hope to report thoseapplications in near future.

References1) S. R. Coleman, J. Wess and B. Zumino, Phys. Rev.

177, 2239 (1969).2) C. G. Callan, Jr., S. R. Coleman, J. Wess and B. Zu-

mino, Phys. Rev. 177, 2247 (1969).3) D. V. Volkov, Sov. J. Part. Nucl. 4, 3 (1973) .4) V. I. Ogievetsky, Proc. of X-th Winter School of The-

oretical Physics in Karpacz 1, 117 (1974).5) E. A. Ivanov and V. I. Ogievetsky, Teor. Mat. Fiz. 25,

164 (1975).6) A. Nicolis, R. Penco, F. Piazza and R. A. Rosen, JHEP

1311, 055 (2013) [arXiv:1306.1240 [hep-th]].7) S. Endlich, A. Nicolis and R. Penco, Phys. Rev. D 89,

no. 6, 065006 (2014) [arXiv:1311.6491 [hep-th]].8) T. Brauner and H. Watanabe, Phys. Rev. D 89, no. 8,

085004 (2014) [arXiv:1401.5596 [hep-ph]].

- 156 - - 157 -

RIKEN Accel. Prog. Rep. 48 (2015) Ⅱ-6. Particle Physics Ⅱ-6. Particle Physics RIKEN Accel. Prog. Rep. 48 (2015)

完全版2014_本文.indd 157 15/10/16 17:50