indico.ph.tum.de...Mikhail P. Solon - University of Chicago Universal behavior in heavy WIMP-nucleon scattering Theories in this class ﬂow to the same low energy theory-- universal

Mikhail P. Solon - University of Chicago Universal behavior in heavy WIMP-nucleon scattering

Universal behavior in the scattering of heavy, weakly interacting dark matter on

nuclear targets

Physics Letters B 707, 2012, 539-545 [1111.0016]with Richard Hill

Mikhail P. SolonUniversity of Chicago

1

Confinement X, Munich, 10-11-12

/ 22


Space of models admits or requires heavy dark matter (~TeV and above).

?

?...

susy

minimal dark matter

weakly interacting stable pions

electroweak baryons

Model space of electroweak charged DM

2

Is there a way to parametrize theory space efficiently?

Dark sector is unlikely to be so simple.


Theories in this class flow to the same low energy theory-- universal behavior in the heavy particle limit!

can be described using Heavy Particle Effective Theory (HPET).

Idea: The class of models with dark matter that is

M >> mWelectroweak SU(2) charged,

L = φ∗v(iv · D + ...)φv

spin, charge radius, magnetic moment, polarizabilities, UV physics, etc.

3


~~

M

µb

µc

µt

µ0

?

} Integrate out weak scale:+ = c1 + . . .

Figure 1: Matching condition for quark operators. Double lines denote heavy scalars, zigzaglines denote W bosons, dashed lines denote Higgs bosons, single lines with arrows denotequarks, and the solid square denotes an effective theory vertex. Diagrams with crossed Wlines are not displayed.

with derivatives acting on φv or involving γ5, since these lead to spin-dependent interactionsthat are suppressed for low-velocity scattering. The basis of operators is then

Lφ0,SM =1

m3Wφ∗vφv

{ ∑

q

[c(0)1q O

(0)1q + c

(2)1q vµvνO

(2)µν1q

]+ c(0)2 O

(0)2 + c

(2)2 vµvνO

(2)µν2

}+ . . . , (19)

where we have chosen QCD operators of definite spin,

O(0)1q = mq q̄q , O(0)2 = (G

Aµν)

2 ,

O(2)µν1q = q̄

(γ{µiDν} − 1

dgµνiD/

)q , O(2)µν2 = −GAµλGAνλ +

1

dgµν(GAαβ)

2 . (20)

Here A{µBν} ≡ (AµBν + AνBµ)/2 denotes symmetrization. We employ dimensional regu-larization with d = 4 − 2# the spacetime dimension. We use the background field methodfor gluons in the effective theory thus ignoring gauge-variant operators, and assume that ap-propriate field redefinitions are employed to eliminate operators that vanish by leading orderequations of motion. The matrix elements of the gluonic operators, O(S)2 , are numericallylarge, representing a substantial contribution of gluons to the energy and momentum of thenucleon. To account for the leading contributions from both quark and gluon operators, wecompute the coefficients c(S)2 through O(αs) and c

(S)1q through O(α0s).

4 Weak scale matching

The matching conditions for quark operators in the nf = 5 flavor theory at renormalizationscale µ = µt ∼ mt ∼ mW ∼ mh are obtained from the diagrams in Fig. (1):

c(0)1U(µt) = C[− 1

x2h

], c(0)1D(µt) = C

[− 1

x2h− |VtD|2

xt4(1 + xt)3

],

c(2)1U(µt) = C[2

3

], c(2)1D(µt) = C

[2

3− |VtD|2

xt(3 + 6xt + 2x2t )

3(1 + xt)3

], (21)

where subscript U denotes u or c and subscript D denotes d, s or b. Here C = [πα22(µt)][J(J +1)/2], xh ≡ mh/mW and xt ≡ mt/mW . We ignore corrections of order mq/mW for q =u, d, s, c, b, and have used CKM unitarity to simplify the results.

6

+ + +

= c2 + c1

[+

]+ . . .

Figure 2: Matching condition onto gluon operators. The notation is as in Fig. 1.

Matching conditions onto gluon operators are from the diagrams of Fig. (2):

c(0)2 (µt) = Cαs(µt)

4π

[1

3x2h+

3 + 4xt + 2x2t6(1 + xt)2

],


4π

[− 32

9log

µtmW

− 4− 4(2 + 3xt)9(1 + xt)3

logµt

mW (1 + xt)

− 4(12x5t − 36x4t + 36x3t − 12x2t + 3xt − 2)

9(xt − 1)3log

xt1 + xt

− 8xt(−3 + 7x2t )

9(x2t − 1)3log 2

− 48x6t + 24x

5t − 104x4t − 35x3t + 20x2t + 13xt + 18

9(x2t − 1)2(1 + xt)

]. (22)

There is no dependence of c(0)2 or c(2)2 on CKM matrix elements in the limit of vanishing

d, s, b quark masses. The renormalized coefficients are computed in the MS scheme. We haveemployed Fock-Schwinger (x · A = 0) gauge [10] to compute the full-theory amplitudes forgluonic operators in Fig. 2. The effective theory subtractions are efficiently performed ina scheme with massless light quarks, using dimensional regularization as infrared regulator.We have verified that the same results are obtained using finite masses and taking the limitmq/mW → 0. Details of this computation will be presented elsewhere.

5 RG evolution to hadronic scales

To account for perturbative corrections involving large logarithms, e.g. αs(µ0) log mt/µ0, weemploy renormalization group evolution to sum leading logarithms to all orders.

7

QCD anatomy of direct detection.}

} Heavy particle effective theory for electroweak charged DM.

4

UV theory: don’t know, don’t care.}

Universal cross section}σ(SM) +O(mW /M, mb/Mw,Λ2QCD/m2c)


Heavy field transformations under Lorentz are given by the little group for massive particles:

φv(x)→W (B, iD)φv(B−1x)

How to construct HPETs from scratch?

enters as an arbitrary time-like reference vector.vµ

No reference to a full theory (e.g. field redefinitions or strategy of regions).

5

Heinonen, Hill, Solon [1208.0601]How is Lorentz invariance realized?


Gauge, spacetime translations and rotations are OK. Get constraints from boosts.

Write everything possible with building blocks (φv, vµ, Dµ, . . . )

Demand zero variation under field

transformations.

Construct invariant operators

Φv = Γ(v, iD)φv

self-conjugate fields lead to a parity:φv(x)→ Cφ∗v(x), vµ → −vµ

little group ~ SO(3): 0,1/2,1,...

use spinor-vectors to construct spin representations vµ1φ

µ1...µnv = 0, /vφ

µ1...µnv = φ

µ1...µnv , γµ1φ

µ1...µnv = 0

6


~~

M

µb

µc

µt

µ0

Boosts: c1 = c2 = 1, cM = cDReal scalar: everything is even under the parity

Included P and T odd operators: additional polarizabilities

7

Lφ = φ∗v{

iv · D − c1D2⊥2M

+ c2D4⊥8M3

+ g2cDvα[Dβ⊥, Wαβ ]

8M2+ ig2cM

{Dα⊥, [Dβ⊥, Wαβ ]}

16M3

+ g22cA1WαβWαβ

16M3+ g22cA2

vαvβWµαWµβ16M3

+ g22cA3Tr(WαβWαβ)

16M3

+ g22cA4vαvβTr(WµαWµβ)

16M3+ g22c

′A1

"µνρσWµνWρσ16M3

+ g22c′A2

"µνρσvαvµWναWρσ16M3

+ g22c′A3

"µνρσTr(WµνWρσ)16M3

+ g22c′A4

"µνρσvαvµTr(WναWρσ)16M3

+ . . .}

φv

Dµ⊥ = Dµ − vµv · D

Scalar HPET

Mikhail P. Solon - University of Chicago Universal behavior in heavy WIMP-nucleon scattering 8

Real scalar parity enforced, e.g. omittedcQ = cXBoosts:

Lφ,SM = φ∗v{

cHH†H

M+ · · · + cQ

taJQ̄Lτa/vQL

M2+ cX

iQ̄LτaγµQL{taJ , Dµ}2M3

+ cDQQ̄L/viv · DQL

M3

+ cDuūR/viv · DuR

M3+ cDd

d̄R/viv · DdRM3

+ cHdQ̄LHdR + h.c.

M3+ cHu

Q̄LH̃uR + h.c.M3

+ g23c(G)A1

GA αβGAαβ16M3

+ g23c(G)A2

vαvβGA µαGAµβ16M3

+ g23c(G) ′A1

$µνρσGAµνGAρσ

16M3+ g23c

(G) ′A2

$µνρσvαvµGAναGAρσ

16M3

+ . . .}

φv

Interactions with other SM fields

L = φ∗v(iv · D + ...)φv

Focus on universal limit. No non-trivial matching.

ig2vµtaJ

i

v · p

W

φ∗vQ̄L/vQLφvM2


~~

M

µb

µc

µt

µ0

Lφ0,SM =1

m3Wφ∗vφv

{ ∑

q

[c(0)1q O

(0)1q + c

(2)1q vµvνO

(2)µν1q

]+ c(0)2 O

(0)2 + c

(2)2 vµvνO

(2)µν2

}+ . . .

O(0)1q = mq q̄q , O(0)2 = (G

Aµν)

2 ,

O(2)µν1q = q̄(

γ{µiDν} − 1dgµνi /D


1dgµν(GAαβ)

2

operator basis of QCD operators with definite spin

9

Integrate out weak scaleµt ∼ mt ∼ mW ∼ mh

LSMLφ±charged states, SM:

c1 c2Lφ0,SM = +

+ + +

= c2 + c1

[+

]+ . . .




4π

[1

3x2h+

3 + 4xt + 2x2t6(1 + xt)2

],


4π

[− 32

9log

µtmW

− 4− 4(2 + 3xt)9(1 + xt)3

logµt

mW (1 + xt)

− 4(12x5t − 36x4t + 36x3t − 12x2t + 3xt − 2)

9(xt − 1)3log

xt1 + xt

− 8xt(−3 + 7x2t )

9(x2t − 1)3log 2

− 48x6t + 24x

5t − 104x4t − 35x3t + 20x2t + 13xt + 18

9(x2t − 1)2(1 + xt)

]. (22)





7

+ = c1 + . . .



Lφ0,SM =1

m3Wφ∗vφv

{ ∑

q

[c(0)1q O

(0)1q + c

(2)1q vµvνO

(2)µν1q

]+ c(0)2 O

(0)2 + c

(2)2 vµvνO

(2)µν2

}+ . . . , (19)


O(0)1q = mq q̄q , O(0)2 = (G

Aµν)

2 ,

O(2)µν1q = q̄

(γ{µiDν} − 1

dgµνiD/


1

dgµν(GAαβ)

2 . (20)





c(0)1U(µt) = C[− 1

x2h

], c(0)1D(µt) = C

[− 1

x2h− |VtD|2

xt4(1 + xt)3

],

c(2)1U(µt) = C[2

3

], c(2)1D(µt) = C

[2

3− |VtD|2

xt(3 + 6xt + 2x2t )

3(1 + xt)3

], (21)


6

Lφ0 = φ∗v {iv · ∂ + . . . }φvcD = cM = 0


+ = c1 + . . .



Lφ0,SM =1

m3Wφ∗vφv

{ ∑

q

[c(0)1q O

(0)1q + c

(2)1q vµvνO

(2)µν1q

]+ c(0)2 O

(0)2 + c

(2)2 vµvνO

(2)µν2

}+ . . . , (19)


O(0)1q = mq q̄q , O(0)2 = (G

Aµν)

2 ,

O(2)µν1q = q̄

(γ{µiDν} − 1

dgµνiD/


1

dgµν(GAαβ)

2 . (20)





c(0)1U(µt) = C[− 1

x2h

], c(0)1D(µt) = C

[− 1

x2h− |VtD|2

xt4(1 + xt)3

],

c(2)1U(µt) = C[2

3

], c(2)1D(µt) = C

[2

3− |VtD|2

xt(3 + 6xt + 2x2t )

3(1 + xt)3

], (21)


6

Quark operator matching through

c(0)1U (µt) = C[− 1

x2h

], c(0)1D(µt) = C

[− 1

x2h− |VtD|2

xt4(1 + xt)3

],

c(2)1U (µt) = C[23

], c(2)1D(µt) = C

[23− |VtD|2

xt(3 + 6xt + 2x2t )3(1 + xt)3

]

10

O(α0s

)

C ≡ πα22(µt)J(J + 1)/2, xH ≡ mh/mW , xt ≡ mt/mW , U = u, c D = d, s, bxh ≡ mh/mW


+ + +

= c2 + c1

[+

]+ . . .




4π

[1

3x2h+

3 + 4xt + 2x2t6(1 + xt)2

],


4π

[− 32

9log

µtmW

− 4− 4(2 + 3xt)9(1 + xt)3

logµt

mW (1 + xt)

− 4(12x5t − 36x4t + 36x3t − 12x2t + 3xt − 2)

9(xt − 1)3log

xt1 + xt

− 8xt(−3 + 7x2t )

9(x2t − 1)3log 2

− 48x6t + 24x

5t − 104x4t − 35x3t + 20x2t + 13xt + 18

9(x2t − 1)2(1 + xt)

]. (22)





7

Gluon operator matching through


4π

[1

3x2h+

3 + 4xt + 2x2t6(1 + xt)2

],


4π

[− 32

9log

µtmW

− 4− 4(2 + 3xt)9(1 + xt)3

logµt

mW (1 + xt)

− 4(12x5t − 36x4t + 36x3t − 12x2t + 3xt − 2)

9(xt − 1)3log

xt1 + xt

− 8xt(−3 + 7x2t )

9(x2t − 1)3log 2

− 48x6t + 24x5t − 104x4t − 35x3t + 20x2t + 13xt + 18

9(x2t − 1)2(1 + xt)

]

our spin-2 gluon matching is new

11

O(α1s

)

Hisano, Ishiwata, Nagata, Takesako [1104.0228]


Heavy particle Feynman rules simplify calculation.∫

(dl)1

v · l1

l2 −m2WvµvνΠµν(l)

iS(p) = iS0(p)+gs∫

iS0(p)i /A(k)d4k

(2π)4iS0(p− k)

+g2s

∫iS0(p)i /A(k1)

d4k1(2π)4

iS0(p− k1)i /A(k2)d4k2(2π)4

iS0(p− k1 − k2) + ...

Fock-Schwinger gauge is efficient.

Regulated using dimreg or finite quark masses.

+ + +

= c2 + c1

[+

]+ . . .




4π

[1

3x2h+

3 + 4xt + 2x2t6(1 + xt)2

],


4π

[− 32

9log

µtmW

− 4− 4(2 + 3xt)9(1 + xt)3

logµt

mW (1 + xt)

− 4(12x5t − 36x4t + 36x3t − 12x2t + 3xt − 2)

9(xt − 1)3log

xt1 + xt

− 8xt(−3 + 7x2t )

9(x2t − 1)3log 2

− 48x6t + 24x

5t − 104x4t − 35x3t + 20x2t + 13xt + 18

9(x2t − 1)2(1 + xt)

]. (22)





7

A few technical points

Electroweak gauge invariance immediate.


Shift in mass due to EWSB:

Universal in the class.

3.1 Mass correction from electroweak symmetry breaking

We may evaluate the heavy scalar self energy to obtain mass corrections,

−iΣ(p) =W

p+

Z+

γ

+ . . . . (14)

The shift in mass due to electroweak symmetry breaking appears as a nonvanishing value ofΣ(p) at v ·p = 0. We find at leading order in the 1/M expansion, and first order in perturbationtheory,

δM = α2mW

[−1

2J2 + sin2

θW2

J23

]. (15)

In particular, with Q = J3 + Y = J3 for Y = 0, the mass of each charged state is liftedproportional to its squared charge relative to the neutral component,

M(Q) −M(Q=0) = α2Q2mW sin2θW2

+O(1/M) ≈ (170 MeV)Q2 . (16)

Subleading corrections can be similarly evaluated in the effective theory. Since no additionaloperators appear at O(1/M0), the result (16) is model independent.4

3.2 Operator basis

The effective theory after electroweak symmetry breaking will include: the heavy scalar QEDtheory for each of the electric charge eigenstates, with mass determined as in (15);5 theStandard Model lagrangian with W,Z, h, t integrated out; and interactions,

L = Lφ0 + LSM + Lφ0,SM + . . . , (17)

where the ellipsis denotes terms containing electrically charged heavy scalars. For the electri-cally neutral scalar,

Lφ0 = φ∗v,Q=0{

iv · ∂ − ∂2⊥

2M(Q=0)+O(1/m3W )

}φv,Q=0 . (18)

Note that enforcing the reality condition (7) implies the vanishing of cD (= cM).Interactions with Standard Model fields begin at order 1/m3W . We restrict attention to

quark and gluon operators (neglecting lepton and photon operators) and again focus on theneutral φv,Q=0 component, dropping the Q = 0 subscript in the following. Mixing with chargedscalars will become relevant at order 1/m4W in nuclear scattering computations; similarly, werestrict attention to flavor-singlet quark bilinears, since matrix elements of flavor-changingbilinears are suppressed by additional weak coupling factors. Finally, we neglect operators

4The mass splitting (16) appears in limits of particular models, e.g. [1, 7, 8].5We define the pole mass to include the contributions induced by electroweak symmetry breaking, as

opposed to introducing residual mass terms for different charge eigenstates [9].

5

M(Q) −M(Q=0) = α2Q2mW sin2θW2

+O(1/M) ≈ (170 MeV )Q2


~~

M

µb

µc

µt

µ014

QCD anatomy of direct detection: generic component of direct detection

Resum logs . Estimate perturbative uncertainty.αs(µ0) logmtµ0

ci(µt)

ci(µ0)runningmatching

σ

UV

matrixelements


QCD anatomy I: renormalization group and threshold matching

Runningd

d log µO(S)i = −

∑

j

γ(S)ij Oj

γ̂(0) =αs4π

0 0. . .

...0 0

32 · · · 32 −2β0

+ . . .

γ̂(2) =αs4π

649 −

43

. . ....

649 −

43

− 649 · · · −649

4nf3

+ . . .

Tarrach [82]; Grinstein, Randall [89]; Vogt, Moch, Vermaseren [04]; Ovrut, Schnitzer [82]; Inami, Kubota, Okada [83]

(nf = 4)

Matching

c(0)2 (µb) = c̃(0)2 (µb)

(1 +

4ã3

logmbµb

)

− ã3c̃(0)1b (µb)

[1 + ã

(11 +

43

logmbµb

)]+O(ã3),

c(0)1q (µb) = c̃(0)1q (µb) +O(ã2),

c(2)2 (µb) = c̃(2)2 (µb)−

4ã3

logmbµb

c̃(2)1b (µb) +O(ã2),

c(2)1q (µb) = c̃(2)1q (µb) +O(ã)

+= c̃1c̃2c2

+ + +

= c2 + c1

[+

]+ . . .




4π

[1

3x2h+

3 + 4xt + 2x2t6(1 + xt)2

],


4π

[− 32

9log

µtmW

− 4− 4(2 + 3xt)9(1 + xt)3

logµt

mW (1 + xt)

− 4(12x5t − 36x4t + 36x3t − 12x2t + 3xt − 2)

9(xt − 1)3log

xt1 + xt

− 8xt(−3 + 7x2t )

9(x2t − 1)3log 2

− 48x6t + 24x

5t − 104x4t − 35x3t + 20x2t + 13xt + 18

9(x2t − 1)2(1 + xt)

]. (22)





7

+ + +

= c2 + c1

[+

]+ . . .




4π

[1

3x2h+

3 + 4xt + 2x2t6(1 + xt)2

],


4π

[− 32

9log

µtmW

− 4− 4(2 + 3xt)9(1 + xt)3

logµt

mW (1 + xt)

− 4(12x5t − 36x4t + 36x3t − 12x2t + 3xt − 2)

9(xt − 1)3log

xt1 + xt

− 8xt(−3 + 7x2t )

9(x2t − 1)3log 2

− 48x6t + 24x

5t − 104x4t − 35x3t + 20x2t + 13xt + 18

9(x2t − 1)2(1 + xt)

]. (22)





7

+ + +

= c2 + c1

[+

]+ . . .




4π

[1

3x2h+

3 + 4xt + 2x2t6(1 + xt)2

],


4π

[− 32

9log

µtmW

− 4− 4(2 + 3xt)9(1 + xt)3

logµt

mW (1 + xt)

− 4(12x5t − 36x4t + 36x3t − 12x2t + 3xt − 2)

9(xt − 1)3log

xt1 + xt

− 8xt(−3 + 7x2t )

9(x2t − 1)3log 2

− 48x6t + 24x

5t − 104x4t − 35x3t + 20x2t + 13xt + 18

9(x2t − 1)2(1 + xt)

]. (22)





7

(nf = 5)


Σeffs = 366± 142 MeV

Σ0 = 36± 7 MeV

100 120 140 160 180 200

!400

!350

!300

!250

!200

!150

mh!GeV"

mW3

ΠΑ22

! pzero!MeV

"Perturbative corrections areeven more important forsmaller strange form factor(e.g. lattice value)

Σlats = 50± 8 MeV

100 120 140 160 180 200

!250

!200

!150

!100

mh!GeV"mW3

ΠΑ22

! pzero!MeV

"

LO NLO NNLO NNNLO

in the runningin the gluon matrix element

µc → µ0

17

Example: controlof the scale µ0


Strange scalar form factor: lattice vs baryon spectroscopy

u!2"d!2"g!2"s!2"s!0"u!0"+ d!0"

g!0"

0 100 200 300 400 500!250

!200

!150

!100

!50

0

50

100

"s!MeV"

mW3

ΠΑ22

! p!MeV

"

lattice baryon spectroscopy

spin-0 and spin-2 contributions have opposite signs


~~

M

µb

µc

µt

µ0

spin-2: LO, equivalent to high scale evaluation

spin-0: NLO in the running and matchingNNNLO in the runningNNNLO in gluon matrix element

µc → µ0µt → µc

Cirelli, Fornego, Strumia [2006]; Essig [2008]; Hisano, Ishiwata, Nagata, Takesako [2011]

Parameter Value Reference

|Vtd| ∼ 0 -|Vts| ∼ 0 -|Vtb| ∼ 1 -

mu/md 0.49(13) [20]

ms/md 19.5(2.5) [20]

ΣlatπN 0.047(9) GeV [21]

Σlats 0.050(8) GeV [22]

ΣπN 0.064(7) GeV [23]

Σ0 0.036(7) GeV [24]

mW 80.4 GeV [20]

mt 172 GeV [15]

mb 4.75 GeV [15]

mc 1.4 GeV [15]

mN 0.94 GeV -

αs(mZ) 0.118 [20]

α2(mZ) 0.0338 [20]

Table 1: Inputs to the numerical analysis.

We consider “traditional” values ΣπN = 64 ± 7 MeV [23] and Σ0 = 36 ± 7 MeV [24], butinvestigate also the lattice determinations, ΣlatπN = 47±9 MeV [21] and Σlats = 50±8 MeV [22].7We adopt PDG values [20] for light-quark mass ratios. A summary of numerical inputs ispresented in Table 1.

6.1.2 Spin two

The matrix elements of spin-two operators can be identified as

f (2)q,p (µ) =

∫ 1

0

dx x[q(x, µ) + q̄(x, µ)] , (30)

where q(x, µ) and q̄(x, µ) are parton distribution functions evaluated at scale µ. Neglectingpower corrections, the sum of spin two operators in (20) is the traceless part of the QCD

energy momentum tensor, hence independent of scale we have f (2)G,p(µ) ≈ 1−∑

q=u,d,s f(2)q,p (µ).

Using approximate isospin symmetry we set

f (2)u,n = f(2)d,p , f

(2)d,n = f

(2)u,p , f

(2)s,n = f

(2)s,p . (31)

7The latter quantity arises from a naive averaging of Σs = 31 ± 15 MeV [21] and Σs = 59 ± 10 MeV [25].See also [26, 27, 28].

10

µ(GeV) f (2)u,p(µ) f(2)d,p (µ) f

(2)s,p (µ) f

(2)G,p(µ)

1.0 0.404(6) 0.217(4) 0.024(3) 0.36(1)

1.2 0.383(6) 0.208(4) 0.027(2) 0.38(1)

1.4 0.370(5) 0.202(4) 0.030(2) 0.40(1)

Table 2: Operator coefficients derived from MSTW PDF analysis [15] at different values of µ.

Table 2 lists coefficient values for renormalization scales µ = 1 GeV, µ = 1.2 GeV and µ =1.4 GeV determined from the parameterization and analysis of [15].

6.2 Cross section

The low-velocity, spin-independent, cross section for WIMP scattering on a nucleus of massnumber A and charge Z may be written

σA,Z =m2rπ|ZMp + (A− Z)Mn|2 ≈

m2rA2

π|Mp|2 , (32)

whereMp andMn are the matrix elements for scattering on a proton or neutron respectively8,and mr = MmN/(M +mN ) denotes the reduced mass of the dark-matter nucleus system. Asdescribed in Section 6.1, Mn ≈ Mp up to corrections from numerically small CKM factorsand isospin violation in nucleon matrix elements. In the M # mN limit, the cross sectionscales as A4. At finite velocity, a nuclear form factor modifies this behavior [29].

As a numerical benchmark, let us compute the cross section for low-momentum scatteringon a nucleon for a heavy real scalar in the isospin representation J = 1. Figure 3 displaysthe result, as a function of the unknown Higgs boson mass. Using Table 1, we considerseparately the “traditional” inputs ΣπN and Σ0, as well as recent lattice determinations ofΣlatπN and Σ

lats . For each case, separate bands represent the uncertainty due to neglected

perturbative QCD corrections, and due to the hadronic inputs. We estimate the impact ofhigher order perturbative QCD corrections by varying matching scales m2W /2 ≤ µ2t ≤ 2m2t ,m2b/2 ≤ µ2b ≤ 2m2b , m2c/2 ≤ µ2c ≤ 2m2c , 1.0 GeV ≤ µ0 ≤ 1.4 GeV, adding the errors inquadrature.

The renormalization group running and heavy quark matching for spin-2 operators areevaluated at LO.9 For spin-0 operators, we find a large residual uncertainty at LO fromµ0, µc and µb scale variation. The RG running from µc to µ0 from (24) is thus evaluatedwith NNNLO corrections, including contributions to β/g through O(α4s) and γm throughO(α4s). Accordingly, the spin-0 gluonic matrix element from (27) is also evaluated at NNNLO,including contributions to β/g through O(α4s) and γm through O(α3s). The residual µ0 scale

8Explicitly, MN = m−3W 〈N |(∑

q=u,d,s

[c(0)1q O

(0)1q + c

(2)1q vµvνO

(2)µν1q

]+ c(0)2 O

(0)2 + c

(2)2 vµvνO

(2)µν2

)|N〉 .

9Up to power corrections and subleading O(αs) corrections, our evaluation is equivalent to an evaluationin either the nf = 4 or nf = 5 flavors theories, taking the c- and b-quark momentum fractions of the protonas input. We have verified that these results, with the matrix elements taken from [15], are within our errorbudget.

11

Universal cross-sectionMSTW[0901.0002]

σ =m2rπ|ZMp + (A− Z)Mn|2

≈ m2rA

2

π|Mp|2 ∼ A4

small CKM factors, isospin symmetry


!ΠN, !0

!ΠNlat, !slat

100 120 140 160 180 20010#49

10#48

10#47

10#46

mh!GeV"

Σ!cm2 "

uncertainties:dark-perturbative light-hadronic input

Real, isotriplet scalar. Scales with J as σJ =

[J(J + 1)

2

]2σ1

Neglected corrections

O(mW /M, mb/mW ,Λ2QCD/m2c)

σ(mh = 125GeV) = 1.1± 0.2+1.1−0.7 × 10−47 cm2perturbativehadronic inputusing Σ

lats

u!2"d!2"g!2"s!2"s!0"u!0"+ d!0"

g!0"

0 100 200 300 400 500!250

!200

!150

!100

!50

0

50

100

"s!MeV"

mW3

ΠΑ22

! p!MeV

"


LO NLO NNLO NNNLO

in the runningin the gluon matrix element

µc → µ0

124 125 126 127 128

1!10"49

5!10"491!10"48

5!10"481!10"47

5!10"471!10"46

mh(GeV)

σ(cm2)100 120 140 160 180 200

1!10"49

5!10"491!10"48

5!10"481!10"47

5!10"47

mh(GeV)

σ(cm2)

µtPerturbative uncertaintydominated by

Example: controlof the scale µ0


Analysis of QCD corrections provides a systematically improvable method to resum large logarithms, and to estimate perturbative and hadronic input uncertainties.

A precise determination of the strange scalar form factor from lattice is needed.

Heavy particle effective theory can be used to describe electroweak charged, heavy dark matter.

Summary

Universal cross-section of heavy, weakly interacting dark matter is a target for future direct searches:

σ(mh = 125GeV) = 1.1± 0.2+1.1−0.7 × 10−47 cm2

!ΠN, !0

!ΠNlat, !slat

100 120 140 160 180 20010#49

10#48

10#47

10#46

mh!GeV"Σ!cm2 "

Documents

indico.ph.tum.de...Mikhail P. Solon - University of Chicago Universal behavior in heavy WIMP-nucleon scattering Theories in this class ﬂow to the same low energy theory-- universal