Power corrections to the universal heavy WIMP-nucleon cross section

Citation for this paper:

Chen, C., Hill, R.J., Solon, M.P. & Wijangoco, A.M. (2018). Power corrections to the

universal heavy WIMP-nucleon cross section. Physics Letters B, 781, 473-479.

https://doi.org/10.1016/j.physletb.2018.04.021

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Power corrections to the universal heavy WIMP-nucleon cross section

Chien-Yi Chen, Richard J. Hill, Mikhail P. Solon, Alexander M. Wijangco

2018

© 2018 The Authors. Published by Elsevier B.V. This is an open access article under

the CC BY-NC-ND license (

http://creativecommons.org/licenses/BY-NC-ND/4.0/

).

This article was originally published at:

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Power

corrections

to

the

universal

heavy

WIMP-nucleon

cross

section

Chien-Yi Chen

a

,

b

,

Richard J. Hill

c

,

d

,

∗

,

Mikhail P. Solon

e

,

Alexander M. Wijangco

f

a_{DepartmentofPhysicsandAstronomy,UniversityofVictoria,Victoria,BCV8P5C2,Canada} b_{PerimeterInstituteforTheoreticalPhysics,Waterloo,ON,N2L2Y5,Canada}

c_{DepartmentofPhysicsandAstronomy,UniversityofKentucky,Lexington,KY40506,USA} d_{Fermilab,Batavia,IL60510,USA}

e_{WalterBurkeInstituteforTheoreticalPhysics,CaliforniaInstituteofTechnology,Pasadena,CA 91125,USA} f_{TRIUMF,Vancouver,BC,V6T2A3,Canada}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received17February2018

Receivedinrevisedform22March2018 Accepted11April2018

Availableonline13April2018 Editor: B.Grinstein

WIMP-nucleon scattering isanalyzed atorder1/M inHeavyWIMPEffectiveTheory.The 1/M power

corrections, where M_mW is the WIMPmass,distinguish betweendifferent underlyingUV models with the same universal limit and their impact on direct detection rates can be enhanced relative to naive expectations due to generic amplitude-level cancellations at leading order. The necessary one- and two-loop matchingcalculations ontothe low-energyeffectivetheory forWIMP interactions with Standard Model quarks and gluons are performed for the case of an electroweak SU(2) triplet WIMP, considering both the cases of elementary fermions and composite scalars. The low-velocity WIMP-nucleon scattering cross section is evaluated and compared with current experimental limits andprojectedfuturesensitivities.Ourresultsprovidethemostrobustpredictionforelectroweaktriplet Majoranafermiondarkmatterdirectdetectionrates;forthiscase,acancellationbetweentwosources ofpowercorrectionsyieldsasmalltotal1/M correction,andatotalcrosssectionclosetotheuniversal limitforMfew×100GeV.FortheSU(2)compositescalar,the1/M correctionsintroducedependence onunderlyingstrongdynamics.Usingaleadingchirallogarithmevaluation,thetotal1/M correctionhas alargermagnitudeand uncertaintythaninthefermioniccase,withasignthatfurthersuppressesthe totalcrosssection.Theseexamplesprovidedefinitetargetsforfuturedirectdetectionexperimentsand motivatelargescaledetectorscapableofprobingtotheneutrinofloorintheTeVmassregime.

©2018TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3_.

1. Introduction

The WIMP paradigm remains a leading explanation for astro-physicaldarkmatter [1–7].NullresultsattheLHC [8–11] suggest thatnewphysics isheavy comparedtomassesofweakscale par-ticles,

∼

100GeV.Thissituationpresentsexperimentalchallenges. Forexample,athigh-energycollidersitisdifficulttoproduceand detecton-shellheavy statesthat arecoupledweakly tothe Stan-dardModel.Production crosssectionsare smallandnovel search strategiesare requiredtodistinguish signalfrombackground.For theSU

(

2

)

W

×

U

(

1

)

Y chargedWIMPsconsideredinthispaper,with

masses above the electroweak scale, detection prospects remain challenging at foreseeablecolliders [12–17]. Indirect searchesfor WIMPannihilationsignalspresentacomplementary setof oppor-tunities and experimental challenges, and introduce dependence

*

Correspondingauthor.

E-mailaddress:richard.hill@uky.edu(R.J. Hill).

onastrophysical modeling [18–25]. Heavyparticletechniquescan besimilarlyappliedtothiscase [22,26–28].

TheheavyWIMPregimeisalsochallengingfordirectdetection prospects.First,since theabundance ofastrophysicaldarkmatter particles fora given local energy densityscales inversely as the particle mass, WIMPs are less abundant and detection rates for a given cross section are smaller. Second, asthe mass spectrum of new physics states becomes stretched above the weak scale, theabsence ofaccessibleintermediate statesforbidsthe simplest higgs-mediatedinteractionsofWIMPswithnucleons,causingcross sectionstobesmaller.

However, although the interaction rates between WIMPs and nucleons may become smaller, they also become more certain. HeavyWIMPsymmetryemergesinthelimitthattheWIMPmass,

M, is large compared to the electroweak scale, i.e., M



mW.

Scattering crosssectionsbecome universal forgivenWIMPgauge quantumnumbers,independentofthedetailedUVphysics[29,30]. For example, the cross section in this limit is independent of whethertheparticleisscalarorfermion,compositeor

fundamen-https://doi.org/10.1016/j.physletb.2018.04.021

0370-2693/©2018TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3_.

474 C.-Y. Chen et al. / Physics Letters B 781 (2018) 473–479

tal. Thisuniversality provides robust sensitivity targetsfor ambi-tiousnextgenerationdirectdetectionexperiments,andwillbekey tointerpretinganyconfirmedsignal.

In previous work, two of the authors (RJH and MPS) ana-lyzed the universal heavy WIMP limit for WIMP-nucleon scat-tering [29–32]. In this limit a generic amplitude-level cancella-tion [29,30,33] was shown to suppress the low-velocity WIMP-nucleon cross section to the level of

∼

10−47 cm2 for wino-like WIMPs(i.e.,self-conjugateelectroweaktriplets),andhiggsino-like crosssectionstoanevensmallervalue.Itisnaturaltoaskwhether inthe presence ofsuch cancellations, formally subleadingeffects can become numerically relevant beyond naive dimensional es-timates. For example, focusing on the electroweak triplet case, the cancellationresults ina total amplitude whose magnitudeis

∼

20%thesizeofthecomponentsubamplitudes [32],anda WIMP-nucleon cross section that is therefore suppressed by more than anorderofmagnitude. ForTeV scaleWIMPs,correctionsoforder

mW

/

M could potentially enter ata similar numerical level.Here

we analyzesuch 1

/

M power corrections, andquantifythe corre-spondingviolationsofheavyWIMPuniversality.

Theremainder ofthepaperis structuredasfollows.Section 2 extendsHeavyWIMPEffectiveTheory(HWET)toincorporate1

/

M

power corrections, and Sec. 3 matches to the low energy effec-tive theory after integrating out weak-scale particles. Section 4 computes the low-velocity scattering cross section of WIMPs on nucleons.Section5providesasummaryandoutlook.

2. HeavyWIMPeffectivetheoryatorder1

/

M

HeavyparticleeffectivetheorycanbeusedtoanalyzeStandard Model(SM)extensionsconsistingofelectroweakmultipletswhose mass M is large comparedto SM particlemasses, M



mW.

Ad-ditionalheavy multiplets, of mass M, maybe integrated out for genericmasssplittingM

−

M

=

O(

M

)

.ThespecialcaseM

−

M

=

O(

mW

)

requiresthat theadditionalmultiplet appearexplicitlyin

the HWET [30].1 _{Here we focus on a single multiplet of}

self-conjugate heavy particle fields with arbitrary spin, transforming underirreduciblerepresentationsofelectroweakSU

(

2

)

W

×

U

(

1

)

Y.

Wherea specificrepresentation isrequired,weillustrate withan electroweaktriplet.

Working throughorder1

/

M, thegauge- and Lorentz-invariant lagrangian in the one-heavy-particle sector (i.e., bilinear in hv)

is [29]

L

=

hv



i v

·

D

− δ

m

−

D 2 ⊥ 2M

+

cH H†_H M

+

cW 1

σ

μνWμν M

+

cW 2



μνρσ

σ

μνWρσ M

+ . . .



hv

,

(1)

where the timelike unit vector vμ defines the heavy WIMP ve-locity, Dμ

= ∂

μ

−

ig1Y Bμ

−

ig2Waμta is the covariant derivative, W μν

=

i

[

Dμ

,

Dν

]/

g2

=

Wμν ta a is the field strength, and D

μ

⊥

=

Dμ

−

vμ v

·

D. Theheavy particlefield hv satisfiesprojection

re-lationsasdiscussedindetailinRef. [35];forexample,afermionic heavy particle field obeys

/

vhv

=

hv.The self-conjugate condition

is enforced inthe effectivetheory by requiringinvariance ofthe lagrangianunder

vμ

→ −

vμ

,

hv

→

hcv

,

(2)

1 _{Forarelatedapplicationofheavyparticle effectivetheorytothecaseofan} electroweaksingletbinothatisnearlydegeneratewithastop,seeRef. [34].

wherehc

v denoteschargeconjugation.Foranirreducible

represen-tation of a self-conjugate field, we necessarily have zero hyper-charge and integer isospin. The interactions labeled by cW 1 and cW 2 are present for the fermionic case. They contribute only to

spin-dependentinteractionsatlowvelocityandwillbeignoredin thefollowing.

The coefficient,

−

1

/

2,of thekinetic term D2_⊥

/

M in Eq. (1) is fixed by relativistic invariance [35,36]. The residual mass,

δ

m in

Eq. (1),maybechosen forconvenience. Inatheorywithout elec-troweaksymmetrybreaking,taking

δ

m

=

0 wouldenforcethat M

is the physical particle (pole)mass. For matching calculationsat theelectroweakscale,itisconvenienttochoose

δ

m

=

cH

|

H

|

2

/

M

tocancelthemasscontributionfromelectroweaksymmetry break-ing.

The parameter cH encodes ultravioletphysics above the scale M, and can be determined by a matching computation between a specifiedUV theory andHWET,described by Eq. (1). Asan ex-ample, let usconsiderthe casewherethe UV theory isgivenby theSMandanelectroweaktripletofMajoranafermions.Matching ontoHWETisillustratedinFig.1.Thematchingcanbeperformed intheelectroweaksymmetrictheory.AfterexpandingintheHiggs mass parameter, theEFT diagrams arescaleless butdimensionful and thus vanish in dimensional regularization. Evaluation of the fulltheorydiagramsyieldsthematchingcondition,

cH

(

Majorana fermion

)

= −

3

α

22

.

(3)

As a simple renormalizable extension of this case, consider an additionalelectroweakmultiplettransformingwithhiggsino quan-tum numbers (SU(2)W doublet, hypercharge Y

=

1

/

2) withmass MD.Forgenericdoublet-tripletmasssplitting,MD

−

MT

=

O(

MT

)

,

thematchingcoefficientbecomes cH

(

doublet-triplet

)

= −

3

α

22

+

4

π α

2

κ

2

MT MD

−

MT

,

(4)

where

κ

is the renormalizable trilinear coupling between the triplet and doublet fermions and the SM Higgs field [30,31]. As expected, when MD

/

MT

→ ∞

,theresult(4) reduces tothe pure

tripletresult(3).

As an example involving scalar versus fermionic WIMP, con-sider the pseudo-Goldstone bosons that emerge froma QCD-like SM extension with vector-like SU(2)W couplings to underlying

fermions [37,38].Recallthatthelightestsuchstatesforman elec-troweaktriplet,regardlessofthefermionicSU(2)W representation,

and these “weakly interacting stable pions” are stabilized by a discretesymmetry(theunbrokenanalogofStandardModelG par-ity) [38].ThematchingisagainillustratedinFig.1,wherenowthe full theory diagramsinvolve relativistic scalars, andalso a coun-tertermfourpointfunctionbetweentheWIMPandSMHiggsfield. Theone-loopdiagramsareUVdivergentasafunctionofthecutoff



h representing thenewstrong interactionscale. Thedivergence

is canceled by the counterterm contribution. For the composite theory under consideration,the divergencecorresponds toa log-arithmically enhanced term in the matching. Taking this “chiral” logarithmasanestimate,wehave

cH

(

composite scalar

)

=

α

22log



_h2 M2

+ · · · ≈

α

2 2log 1

α

2

+ · · · ,

(5) where the ellipsis denotes

O(

1

)

terms that are not logarithmi-cally enhanced. The last equalitycorresponds toa chiral symme-try breaking mass M induced by SU(2)W radiative corrections: M2

_/

2

h

∼

α

2 [38]. The precise matchingcondition could in

prin-ciplebecomputedusingstronginteractionmethodsinthechosen UVtheory.

Fig. 1. MatchingconditionforthecoefficientcHforUVtheoryconsistingoftheStandardModelplusSU(2)W-tripletMajoranafermion.SolidlinesdenoteMajoranafermion, dashedlinesdenoteSMHiggsdoublet,zigzaglinesdenoteSU(2)Wgaugefields.Matchingisperformedintheelectroweaksymmetrictheory.DoublelinesontheRHSdenote heavyWIMPsandtheencircledcrossdenotesinsertionofa1/M effectivetheoryvertex.ForUVtheoryconsistingofacompositerealscalartransformingasatripletunder

SU(2)W,theadditionalbracketedtermsappearontheLHS,includingthecountertermcontributiondenotedbythesolidsquare.

Thecases(3),(4),and(5) establishtherangeofcH encountered

inavarietyofweaklycoupledUVmodels,involvingfermionsand scalars, composite and elementary particles, and both pure-state andmulti-component models. Before investigating the impact of thesedifferencesondirectdetectioncrosssections,letusperform theremaining stepof matchingHWETonto effectiveQCD opera-tors.

3. Effectivetheorybelowtheweakscale

The scale separation mW

 

QCD, is exploited by matching ontoa heavy particleeffectivetheory fortherelevant electrically neutralcomponentoftheWIMP,interactingwithfiveflavorQCD:

L

=

h(_v0)h(v0)



_

q=u,d,s,c,b



cq(0)O(q0)

+

cq(2)vμvνO(q2)μν



+

c(g0)O(g0)

+

c(g2)vμvνO(g2)μν



+ · · · .

(6)

Thismatchingstepiscommonto differentUV realizationsofthe electroweaktripletWIMP.InEq. (6),hv(0)istheneutralWIMP,and

the spin-0 and spin-2 QCD operators for quarks and gluons are givenby Oq(0)

=

mqqq

¯

,

O(q2)μν

=

1 2q

¯



γ

{μi Dν₋}

−

g μν d i

/

D−



q

,

O(g0)

= (

GμνA

)

2

,

O(g2)μν

= −

GAμλGλAν

+

1 dg μν

₍

_GA αβ

)

2

,

(7)

whered

=

4

−

2



isthespacetimedimension, D₋

≡

−

→

D

−

←

D ,

−

and curlybraces denotesymmetrization, A{μ Bν}

≡ (

Aμ Bν

+

Aν Bμ

)/

2. The ellipsis in Eq. (6) denotes higher dimension operators sup-pressedby



QCD

/

mW,andspin-dependentoperators.

By restricting to dimension seven operators in Eq. (6), we are neglecting contributions suppressed by additional powers of



2_low-energy

/

m2_W, where



low-energy denotes any scale below mW

(e.g.,mb,or



QCD).However,wewillaccountforcorrectionsof or-dermW

/

M inthecoefficientfunctionsappearinginEq. (6) inour

analysisof HWET power corrections. This power counting is ap-propriatefordarkmatter massesinthe fewhundred GeVtoTeV range,afocusforcurrentandnextgenerationdirectdetection ex-periments.

We now proceed to match the theory (1) to the theory (6). By integrating out weak scale particles (the Higgs boson, elec-troweak gauge bosons, and the top quark), we obtain a solution for the twelve effective theory coefficients (cq(0) and cq(2) with q

=

u

,

d

,

s

,

c

,

b,aswellasc(g0)andc(g2))thatspecifytheinteractions

ofDMwithfiveflavorQCD.Weneglectsubleadingcorrections in-volvinglightquarkmasses,anduseCKMunitaritytosimplifysums over quark flavors. Approximating

|

Vtb

|

≈

1,these simplifications

imply that c(uS)

=

c_d(S)

=

cs(S)

=

c(cS) for S

=

0

,

2,leaving six

inde-pendentcoefficients. Inthefollowing,we denotegeneric up- and down-typequarksinfive-flavorQCDbyU andD,respectively,and anarbitraryquarkflavorbyq.

Feynmandiagramscontributingtothematchingat

O(

1

/

M

)

for the quark and gluon coefficientsare shown in Figs. 2 and3, re-spectively.Diagramsforgluonoperatorscontainanadditionalloop comparedtodiagramsforquark operators.However,owingtothe large gluon matrixelements of thenucleons, theseoperators are numericallyofsimilarsize,ordominant.Wecompute eachofthe operator coefficients in Eq. (6) to leading order in electroweak couplings, andhencewe neglect one-loop diagramsinvolving cH

forquark matchingandtwo-loopdiagramsinvolvingcH forgluon

matching. The impact of higherorder contributions is estimated in the numerical analysis by varying the factorization scale. The techniquesforelectroweakscalematchingdetailedinRef. [31] can beappliedtothepresentcalculation.Wedescribesomepertinent detailshere.Comparedtotheleadingpoweranalysisconsideredin Ref. [31],computationofthe1

/

M correctionsrequiresanextended masterintegralbasis,anddifferentcomponentsoftheelectroweak polarizationtensorforthebackgroundfieldgluonmatching.

In performing the gluon matching, it is convenient to distin-guish betweenamplitudeswithone or two bosons exchanged in the t-channel. One-boson exchange amplitudesare shown in the toprowofFig.3,whiletwo-bosonexchangeamplitudesareshown in the bottom row. The one-boson exchange amplitudes factor-ize into the one-boson exchange amplitudes for quark matching (top row of Fig. 2) times the quark loop, and contribute only to thescalar coefficient.For thetwo-boson exchange amplitudes, we employ electroweak polarization tensors,



μν , induced by a loopofquarksinabackgroundfieldofexternalgluons [31,39,40]. The temporal components, vμ vν



μν , are sufficient for the lead-ingpoweranalysis,whileforthe1

/

M correctionswerequirealso thespatialcomponents;thesemaybeextractedfromRef. [31].The renormalizationofWilsoncoefficientsforthequarkandgluon op-eratorsisdiscussedinRef. [32].

476 C.-Y. Chen et al. / Physics Letters B 781 (2018) 473–479

Fig. 2. Diagrams contributing to 1/M quark matching, with the same notation as in Fig.1. Diagrams with crossed W lines are not displayed.

Fig. 3. Diagramscontributingto1/M gluonmatching,withthesamenotationasinFig.1.Curlylinesdenotegluons.Diagramswithbothgluonsattachedtotheupperquark lineorwithonegluonattachedtoeachoftheupperandlowerquarklinesarenotshown.

Fromthe sumofoneandtwoloop diagramsinFigs. 2and3, weobtainthefinal resultsforcoefficientsrenormalizedintheMS scheme:

ˆ

c(_U0)

(

μ

)

= −

1 x2_h

−

mW

π

M cH

α

2 2x2h

,

ˆ

c(_D0)

(

μ

)

= −

1 x2_h

− δ

Db xt 4

(

xt

+

1

)

3

−

mW

π

M cH

α

2 2x2h

,

ˆ

c(g0)

(

μ

)

=

α

s 4

π



1 3x2_h

+

N 6

+

1 6

(

xt

+

1

)

2

+

mW

π

M cH 3

α

2 2x2h



,

ˆ

c(_U2)

(

μ

)

=

2 3

−

mW

π

M

,

ˆ

c(_D2)

(

μ

)

=

2 3

+ δ

Db



3xt

+

2 3

(

xt

+

1

)

3

−

2 3



+

mW

π

M



−

1

+ δ

Db



−

x2 t

+

xt6

−

4x4tlog xt

(

x2 t

−

1

)

3



,

ˆ

c(g2)

(

μ

)

=

α

s 4

π



N



−

16 9 log

μ

mW

−

2



−

4

(

2

+

3xt

)

9

(

1

+

xt

)

3 log

μ

mW

(

1

+

xt

)

−

4

(

12x5t

−

36x4t

+

36xt3

−

12xt2

+

3xt

−

2

)

9

(

xt

−

1

)

3 log xt 1

+

xt

−

8xt

(

−

3

+

7x2t

)

9

(

x2_t

−

1

)

3 log 2

−

48x6t

+

24x5t

−

104x4t

−

35x3t

+

20x2t

+

13xt

+

18 9

(

x2 t

−

1

)

2

(

1

+

xt

)

+

mW

π

M



N



8 3log

μ

mW

−

1 3



+

16xt4 3

(

x2 t

−

1

)

3 log xtlog

μ

mW

−

4

(

3x2 t

−

1

)

3

(

x2 t

−

1

)

2 log

μ

mW

+

16x2t 3 log 2_x t

−

4

(

4x_t6

−

16x4_t

+

6x2_t

+

1

)

3

(

x2_t

−

1

)

3 log xt

+

8xt2

(

xt6

−

3x4t

+

4xt2

−

1

)

3

(

x2 t

−

1

)

3 Li2

(

1

−

xt2

)

+

4

π

2_x2 t 9

−

8xt4

−

7x2t

+

1 3

(

x_t2

−

1

)

2



.

(8)

Here Li2

(

z

)

≡

∞k=1zk

/

k2 isthepolylogarithmoforder 2.Wealso introduce theshorthand notation ci

= (

π α

22

/

m3W

)

c

ˆ

i forthe

effec-tive operator coefficients, xi

=

mi

/

mW for masses expressed in

units of mW, subscripts U and D denote arbitraryup-type (u, c

or t) or down-type (d, s or b) quarks, respectively (so that the Kronecker delta,

δ

Db,isequalto unityforD

=

b andvanishesfor D

=

d,s), and N

=

2 is thenumber ofmasslessStandard Model generations.Theleadingpowerresults,representedby M

→ ∞

in Eq. (8),wereobtainedinRef. [29].2 _{Letusremarkthatourresults}

(8) obeythecorrectformallimitatsmallxt: [29]

c(g0)

|

xt→0

=

c (0) g

(

nf

=

6

)

−

α

s 12

π

c (0) q

(

nf

=

6

)

+

O

(

α

2s

) ,

c(g2)

|

xt→0

=

c (2) g

(

nf

=

6

)

−

α

s 3

π

log mt

μ

c (2) q

(

nf

=

6

)

+

O

(

α

s2

) ,

(9)

where c

(

nf

=

6

)

denotes the coefficient in six-flavor QCD

com-puted withthree massless generations (i.e.,mt



mW).3 At large

2 _{Inobtainingtheresults (8),itisimportanttoevaluateallintegralsandbare} coefficientsind=4−2dimensions [29,31].ForarelateddiscussionseeRef. [41].

3 _{Inparticular,thequarkmatchingcoefficientsarecˆ}(0)

q (nf=6)= −_x12 h− mW πM cH α2 2x2h andcˆ(q2)(nf=6)=23− mW

Fig. 4. TheWIMP-protonscatteringcrosssectionasafunctionofWIMPmassM foraMajoranaWIMP(leftpanel)andascalarWIMP(rightpanel),whichcorrespondtothe

cHvaluesinEqs. (3) and(5),respectively.TheinnerbandisthecrosssectionobtainedfromthescalarandtensoramplitudescomputedthroughO(1/M).Theouterband includesanestimatefortheO(1/M2

)contributions.TheneutrinofloorforbothArgonandXenondirectdetectionexperimentsarefromRef. [48],andareshownbyblack solidlines;ourextrapolationtolargermassesisdenotedwithblackdashedlines.AlsoshownwithsolidlinesarethecurrentboundsfromLUX [49],XENON1T [50],and PandaX-II [51].Projectedsensitivitiesoffutureexperimentsareshownwithdottedlines:DEAP-3600 [52],XENON1TandXENONnT [53],LZ [54],andDARWIN [55].

xt, mt



mW, the top quark contributions tothe coefficients are

of order

∼

m2

W

/

mt2. For the special case of a Majorana fermion

(cH

= −

3

α

22),the1

/

M correctionsforc

(0)

q,gandc(q2)arereproduced

by an expansion of expressions in Ref. [42]. However, already at leading power theexpression in Ref. [42] for c(g2) disagrees with

thecorrespondingresultsinRef. [29] andEq. (8).Wenotethatthe expressionforc(g2) in Ref. [42] does not havethecorrectmt

→

0

limit.

4. Crosssections

Let us consider the standard benchmark process for direct detection: the zero velocity limit of (spin-independent) WIMP-nucleonscattering. Thecrosssection isdetermined by thespin-0 andspin-2matrix elements,

M

(_N0) and

M

(_N2), oftheoperators in Eq. (7),

M

(S) N

=



i=q,g c(_iS)

(

μ

0

)



N

|

O(_iS)

(

μ

0

)

|

N

.

(10) Inordertoevaluatethehadronicmatrixelements usingavailable low energy inputs, the five flavor QCD theory must be matched to the appropriate three or four flavor theory, accounting for heavy quark threshold matching corrections and renormalization group evolution from electroweak to hadronic scales. Details of thismatchingcanbe foundinRef. [32]. Forthespin-0matrix el-ements, we match to the three flavor theory with NNNLO QCD corrections,4_{andfollowingRef. [32] makethedefaultscalechoices}

μ

t

= (

mt

+

mW

)/

2

=

126GeV,

μ

b

=

4

.

75GeV,

μ

c

=

1

.

4GeV,and

μ

0

=

1

.

2GeV.Forthespin-2matrixelements,weuseNLOrunning andmatching,andcheckthatourevaluationisconsistentwithan evaluation at the weak scale, in the five flavor theory. The im-pactofhigherorderperturbativeQCD correctionsisestimatedby varyingfactorizationscalesm2_W

/

2

≤

μ

2t

≤

2m2t,m2b

/

2

≤

μ

2

b

≤

2m

2

b, m2_c

/

2

≤

μ

2

c

≤

2m2c, and 1

.

0 GeV

≤

μ

0

≤

1

.

4 GeV. There are ad-ditional uncertainties associated with the hadronic form factors

obtainedbyomittingthetopquarkloopcontributionsinEq. (8) andsettingN=3: ˆ c(g0)(nf=6)=8απs andcˆ (2) g (nf=6)=4απs  −16 3 log μ mW−6+ mW πM  8 logmμW−1  . 4 _{Fortheleadingpoweranalysis,thiscorrespondstoamplitude“5”discussedin} Figure 2andSection 6.2.3ofRef. [32].

that characterizethe overlapbetweenthenucleon statesandthe quarkandgluonoperators.Weemploytheformfactorcentral val-ues and uncertainties from Ref. [32], which were adapted from Refs. [43–46] (seealsoRef. [47]).Errorsfromallsourcesareadded inquadraturetoobtainthetotalcrosssectionerror.

NeglectingnumericallysmallCKMfactorsandisospinviolation innucleon matrixelements [32],the crosssectionsforscattering onprotonsorneutronsareidentical5:

σ

p

≈

σ

n

=

m2_r

π

|

M

(0)

p

+

M

(p2)

|

2

,

(11)

where mr

=

mpM

/(

mp

+

M

)

≈

mp is the reduced mass of the

WIMP-nucleon system. In Fig. 4 we show the cross section in-cluding first order power corrections as a function of M for a fundamental fermion,Eq. (3), andfora compositescalar, Eq. (5). Thecentralvalueamplitudes,inunitswith

M

p(2)

|

M→∞

=

1,are

M

(2) p

=

1

−

0

.

52 mW M

,

M

(0) p

= −

0

.

81

−

0

.

50 cH 3

α

2 2 mW M

.

(12) The numericalevaluation (12) exhibitsthe partial cancellationof theuniversalM

→ ∞

result.FortheMajoranafermioncase,where

cH

= −

3

α

22,the mW

/

M power correction alsoexhibitsa

surpris-ing cancellation.Theimpact ofneglected higher-orderpower cor-rections is estimated by including an uncertainty in the tensor amplitude as

M

(p2)

∝

M

(

2)

p

|

M→∞



1

± (

mW

/

M

)

2



. At large mass, thepowercorrectionsvanish,andtheuniversalresultwithcentral valueanduncertaintyfromRef. [32] isreproduced.AtfiniteWIMP mass,the dependenceofthecross section onthe Higgscoupling

cH differentiatesthefermionandscalarcases.

Fig.4comparestoexistinglimitsfromLUX[49],XENON1T[50], and PandaX-II [51],6 _{andto projected sensitivities forthe Xenon}

5 _{TheWilsoncoefficientsc}(S)

u andc( S)

d inEq. (8) areidentical. Thelightquark operatorsinEq. (6) thusappearinthecombinationsO(uS)+O(

S)

d ,whoseproton andneutronmatrixelementsareidenticaluptoisospinviolatingcorrections.These percentlevelcorrections,proportionaltoα≈1/137 or(mu−md)/QCD,are sub-dominantintheerrorbudgetforM(S)

N .SeeRef. [32] fordetails.

6 _{Formasseslargerthantherangesreportedinthesereferences,wehave} dis-played anextrapolationassumingsimple scaling with theWIMP number abun-dance,σlimit∝M.

478 C.-Y. Chen et al. / Physics Letters B 781 (2018) 473–479

basedexperimentsXENONnT [53],LZ [54],andDARWIN [55],and the Argon based experiment DEAP-3600 [52]. Alsoshown is the “discoverylimit” forbothXenonandArgonduetoneutrino back-grounds,takenfromRef. [48].

5. Summary

The scattering of atomic nuclei from approximately static sources of electroweak SU(2) is a well posed but intricate field theoryproblemthatfindsapplicationinthesearchforWIMPdark matter in our local halo. LHC bounds have pushed the scale of newphysicsintoaregimeoflargemasswheredirectdetectionis morechallenging;howeveratthesametime,universalpredictions emerge in this regime and provide well-defined targets for next generationsearches.

Generic amplitude level cancellations imply a potentially en-hanced sensitivity of direct detection rate predictions to naively power suppressed interactions. In this paper we considered the general framework to analyze these power corrections, and an-alyzed the canonical case of a self-conjugate electroweak-triplet WIMP through order 1

/

M. Owing to heavy particle universality, the leading cross section prediction is identical whether such a WIMPisfermionorscalar,elementaryorcomposite,andwhether theWIMPisaccompaniedbyother,heavier,particlesinthe Stan-dardModelextension.Powercorrectionsdifferentiatethese scenar-ios,asillustratedinFig.4forthebenchmarklow-velocity WIMP-nucleoncrosssection. Fortheelementaryfermioncase, two con-tributions to the power correction largely cancel, resulting in a smalldeviationfromtheuniversalM

→ ∞

limit.Ourresult repre-sentsthemostcompletecalculationofthecrosssectionfor wino-like dark matter in the TeV regime. A standard thermal cosmol-ogy,consistentwiththeobserveddarkmatterabundance,predicts

M

∼

2–3TeV forsuchelectroweakchargedWIMPs [56–60].The el-ementaryMajoranafermioncaseinvolvesnofreeparameters,and a predictionM

≈

2

.

9TeV isobtainedaftercareful accountingfor nonperturbativeenhancements [61].Forthescalarcase,theprecise annihilationcrosssection,andhencecosmologicalmassconstraint, depends on internal structure. At the TeV mass scales indicated bycosmologicalarguments,thepredictedWIMP-nucleusscattering rateis comparableto therate forneutrino-inducedbackgrounds. Thiscrosssectionbenchmarkmotivatesverylargescaledetectors, andtechniques to understand and probe into the so-called neu-trinofloor [62].

Anumberofinvestigationsaresuggestedbyourresults.Besides its computationalpower, the heavy WIMPexpansion provides an excellent classification scheme for WIMP direct detection in the increasinglyimportantheavyWIMPregime.TheSU(2)triplet(i.e., wino-like)caserepresentsacanonicalbenchmark.Otherquantum numbers such asthehiggsino-like casemaybe similarly investi-gated. The proximity of the triplet cross section in Fig. 4 to the neutrinofloormakesthepreciseWIMPmassofparticularinterest. Forthecompositescalarcase,newnonperturbativephysicsenters in two key places: the Higgs coupling parameter cH that

deter-minesthesizeofthedirectdetectioncrosssection;andthe anni-hilationprocess thatdeterminesthecosmologicalmassconstraint within a specified cosmologicalmodel.This physics could be ac-cessedbylatticefieldtheory [63] and/orchirallagrangiananalysis forthenewstronglycoupled sector.Nuclear effectssuch as two-body correlationscould potentially have differing impacts onthe spin-0 andspin-2 operators inEq. (6). Like the 1

/

M corrections,

theexistence ofa severecancellationintheleading crosssection canpotentiallyenhance theimpactofsuch naivelysubleading ef-fects. Existing estimates forsuch nuclear effects, focused on the spin-0sector, indicate asmallimpact relativeto other uncertain-ties [64–66],howeveramoresystematicanalysisiswarranted.

Acknowledgements

CYCwouldliketothankC.Burgessforhelpfuldiscussions.RJH thanks TRIUMF and Perimeter Institute for hospitality. Work of MPS was supported by the Office of High Energy Physics of the U.S. DOE under Contract Numbers DE-AC02-05CH11231 and DE-SC0011632.ResearchatthePerimeterInstituteissupportedinpart bytheGovernmentofCanadathroughNSERCandbytheProvince of Ontario through MEDT. TRIUMF receives federal funding via a contribution agreement with the National Research Council of Canada.FermilabisoperatedbyFermiResearchAlliance,LLCunder ContractNo. DE-AC02-07CH11359 withtheUnitedStates Depart-mentofEnergy.

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