Universal behavior in the scattering of heavy, weakly interacting dark matter on nuclear targets
EEFI Preprint 11-28October 31, 2011
Universal behavior in the scattering of heavy, weaklyinteracting dark matter on nuclear targets
Richard J. Hill ∗ and Mikhail P. Solon † Enrico Fermi Institute and Department of PhysicsThe University of Chicago, Chicago, Illinois, 60637, USA
Abstract
Particles that are heavy compared to the electroweak scale ( M (cid:29) m W ), and that arecharged under electroweak SU (2) gauge interactions display universal properties such asa characteristic fine structure in the mass spectrum induced by electroweak symmetrybreaking, and an approximately universal cross section for scattering on nuclear targets.The heavy particle effective theory framework is developed to compute these properties.As illustration, the spin independent cross section for low-velocity scattering on a nu-cleon is evaluated in the limit M (cid:29) m W , including complete leading-order matchingonto quark and gluon operators, renormalization analysis, and systematic treatment ofperturbative and hadronic-input uncertainties. ∗ [email protected] † [email protected] a r X i v : . [ h e p - ph ] J a n Introduction
Cosmological evidence for dark matter consistent with thermal relic Weakly Interacting Mas-sive Particles (WIMPs) motivates laboratory searches for such particles interacting with nu-clear targets. Search strategies and detection potential are highly dependent on the WIMPmass, M , and its interaction strength with nuclear matter. We consider here the class ofmodels where the WIMP belongs to an electroweak SU (2) multiplet. This study is motivatedin part by the observation that an exact discrete parity arises in Standard Model extensionsinvolving confined fermions coupled to electroweak SU (2) [1]. The parity ensures stability ofthe lightest pseudo-Nambu Goldstone mode, which is the electrically neutral component of aLorentz scalar, electroweak SU (2) isotriplet [2].Regardless of the origin for such an SU (2) multiplet, e.g. whether it is a composite orfundamental particle, universal behavior emerges in the limit where the WIMP mass is largecompared to the electroweak scale, M (cid:29) m W . The emergence of these universal properties,and corrections to them, can be systematically analyzed using techniques of heavy particle ef-fective theories [3]. We focus on the case of a real scalar transforming as a triplet of electroweak SU (2), although the results extend straightforwardly to arbitrary SU (2) representations, andto higher spin particles.At energy scales large compared to m W , the new particle is described by an effective heavyparticle SU (2) gauge theory, L eff = φ ∗ v ( iv · D + . . . ) φ v , (1)where φ v is a scalar heavy particle field, v µ is the heavy particle velocity, and D µ is the SU (2)covariant derivative. The leading interactions are thus universal, and corrections dependingon the mass, or other characteristics of the dark matter particle, are suppressed by powersof M . In this paper we determine the general structure of the heavy scalar effective theorythrough O (1 /M ). As an illustrative application, we compute the universal cross section forlow-velocity scattering of SU (2)-charged WIMPs on a nucleon in the limit M (cid:29) m W . Wepresent a complete leading order matching onto gluonic operators, renormalization analysis,and systematic treatment of perturbative and hadronic-input uncertainties.The remainder of the paper is structured as follows. In Section 2, we construct the relevantheavy particle effective theory at scales µ (cid:29) m W , and compute the leading Wilson coefficientsin a simple model. In Section 3 we consider the operator basis and mass corrections in thelow energy theory after integrating out scales µ ∼ m W . In Section 4 we perform explicitmatching computations at the scale µ ∼ m W . In Section 5 we perform renormalization groupevolution from µ ∼ m W down to low scales µ ≈ (cid:38) Λ QCD , where QCD matrix elementsare estimated. Section 6 presents the cross section for low-velocity scattering on a nucleon.Section 7 presents a summary and outlook.
Consider the effective theory and matching conditions for a scalar particle of mass M , chargedunder electroweak SU (2). With obvious modifications, the construction applies to general1auge groups. We start by investigating the effective theory at scales m W (cid:28) µ (cid:28) M , withunbroken electroweak gauge symmetry. We work in terms of an effective heavy scalar field φ v ( x ), in the isospin J representation of SU (2). The covariant derivative is D µ = ∂ µ − ig W aµ t aJ and W µν ≡ i [ D µ , D ν ] /g ≡ W aµν t aJ isthe associated field strength. We let g , g , g ≡ g denote the Standard Model U (1) Y , SU (2) W and SU (3) c gauge coupling constants, respectively. A typical heavy particle momentum canbe decomposed as p µ = M v µ + k µ , (2)where v µ is a velocity, v = 1, and k µ is a residual momentum. The basis of operators involvesthe perpendicular derivative, D µ ⊥ ≡ D µ − v µ v · D . (3)Through O (1 /M ), the scalar heavy particle effective theory in the one-heavy-particle sectortakes the form, L φ = φ ∗ v (cid:26) iv · D − c D ⊥ M + c D ⊥ M + g c D v α [ D β ⊥ , W αβ ]8 M + ig c M { D α ⊥ , [ D β ⊥ , W αβ ] } M + g c A W αβ W αβ M + g c A v α v β W µα W µβ M + g c A Tr( W αβ W αβ )16 M + g c A v α v β Tr( W µα W µβ )16 M + g c (cid:48) A (cid:15) µνρσ W µν W ρσ M + g c (cid:48) A (cid:15) µνρσ v α v µ W να W ρσ M + g c (cid:48) A (cid:15) µνρσ Tr( W µν W ρσ )16 M + g c (cid:48) A (cid:15) µνρσ v α v µ Tr( W να W ρσ )16 M + . . . (cid:27) φ v , (4)where we have employed appropriate field redefinitions to remove possible redundant operatorsinvolving factors of v · D acting on φ v . Note that the operators with coefficients c (cid:48) A through c (cid:48) A violate parity and time reversal symmetries. For the effective theory describing a fundamentalheavy scalar particle, we have c = c = c A = 1 and c D = c M = c A = c A = c A = c (cid:48) A = c (cid:48) A = c (cid:48) A = c (cid:48) A = 0 at tree level [4]. We find that the low-energy manifestation of relativisticinvariance (“reparameterization invariance” [5, 6]) implies the exact relations, c = c = 1 , c M = c D . (5)Section 2.2 provides a nontrivial illustration of the latter relation.The complete lagrangian including Standard Model particles and interactions can be writ-ten L = L φ + L SM + L φ, SM . (6) Additional CPT violating operators at O (1 /M ) and O (1 /M ) are constrained by reparameterizationinvariance to have vanishing coefficient. L SM is the usual Standard Model lagrangian, and by convention we have included interac-tions with W µ in L φ . So far our discussion applies to a general irreducible SU (2) representationfor the heavy scalar field φ v . Specializing to the case of a real scalar field, necessarily withinteger isospin, the effective theory is invariant under v µ ↔ − v µ , φ v ↔ φ ∗ v . (7)It is straightforward to verify that all interactions in L φ are invariant under this transformation.In the one-heavy-particle sector, the remaining terms involving the Higgs field H , gaugefields, and fermions are ( ˜ H ≡ iτ H ∗ ) L φ, SM = φ ∗ v (cid:26) c H H † HM + · · · + c Q t aJ ¯ Q L τ a v/ Q L M + c X i ¯ Q L τ a γ µ Q L { t aJ , D µ } M + c DQ ¯ Q L v/ iv · DQ L M + c Du ¯ u R v/ iv · Du R M + c Dd ¯ d R v/ iv · Dd R M + c Hd ¯ Q L Hd R + h.c.M + c Hu ¯ Q L ˜ Hu R + h.c.M + g c ( G ) A G A αβ G Aαβ M + g c ( G ) A v α v β G A µα G Aµβ M + g c ( G ) (cid:48) A (cid:15) µνρσ G Aµν G Aρσ M + g c ( G ) (cid:48) A (cid:15) µνρσ v α v µ G Aνα G Aρσ M + . . . (cid:27) φ v . (8)Terms odd under (7) have been omitted. Subleading terms containing only H , φ v and theircovariant derivatives are represented by the first ellipsis in (8). Terms bilinear in lepton fields,and terms bilinear in the hypercharge gauge field are also present in L φ, SM but have not beenwritten explicitly. Repeated indices a = 1 .. A = 1 .. c Q = c X . (9) As an illustration of the construction and matching conditions for the heavy particle lagrangian L φ , consider the case of a fundamental scalar, ignoring scalar self interactions (i.e., φ terms).For the matching of the terms containing a single gauge field, we consider the full theory resultfor the W φφ amputated three point function, (cid:1) qp p (cid:48) µ = ig ( p + p (cid:48) ) µ F ( q )( t aJ ) ji , (10)where q = p (cid:48) − p , and F ( q ) is a model-dependent form factor. Setting p = p (cid:48) = M , v µ = (1 , , , µ = 0 or µ = i gauge For a real scalar field, the effective theory is obtained by introducing v µ in the field redefinition φ ( x ) = e − iMv · x φ v ( x ) / √ M = e iMv · x φ ∗ v ( x ) / √ M = φ ∗ ( x ). F (0) − F (cid:48) (0) q + · · · = 1 − c D q M + . . . , ( p + p (cid:48) ) i (cid:20) − F (0) (cid:18) − p + p (cid:48) M (cid:19) + F (cid:48) (0) q + . . . (cid:21) = ( p + p (cid:48) ) i (cid:20) − p + p (cid:48) M + c M q M (cid:21) + q i p (cid:48) − p M ( c D − c M ) + . . . . (11)An explicit computation of one-loop gauge boson corrections, employing dimensional regular-ization in d = 4 − (cid:15) dimensions, yields F ( q ) = 1+ g (4 π ) q M (cid:26) C ( r ) (cid:20) − (cid:15) IR −
1+ 43 log Mµ (cid:21) + C ( G ) (cid:20) − (cid:15) IR + 34 + 112 log Mµ (cid:21)(cid:27) + . . . . (12)The quadratic Casimir coefficients for the isospin- J and adjoint representations of SU (2) are C ( J ) = J ( J + 1) and C ( G ) = 2. From (11) and (12), after effective theory subtractions therenormalized coefficients c D ( µ ), c M ( µ ) in the MS renormalization scheme are found to be c D ( µ ) = c M ( µ ) = α ( µ )4 π (cid:20) − J ( J + 1) + 12 + (cid:18) J ( J + 1)3 + 43 (cid:19) log Mµ (cid:21) . (13)Matching for a general ultraviolet completion model, and for other effective theory coefficientsproceeds similarly.Our focus will be on the limit M (cid:29) m W , where all nontrivial matching conditions at thescale µ ∼ M become irrelevant. We leave a detailed investigation of the model-dependentform factor and subleading 1 /M corrections to future work. Presently we proceed to investigatethe leading order predictions of the effective theory at scales µ (cid:28) M . For scattering phenomena at ∼ keV energy scales of interest to dark matter-nucleus scatteringsearch experiments, we should examine the appropriate effective theory far below the elec-troweak scale. Let us begin by integrating out the degrees of freedom at the scale m W . Fordefiniteness we treat the top quark mass m t and the Higgs boson mass m h as parametricallyof the same order as m W . In following sections, we will renormalize to lower energy scales,integrating out the remaining heavy quark degrees of freedom as we pass the bottom andcharm quark thresholds. The remaining hadronic matrix elements may then be evaluated in n f = 3 flavor QCD to obtain cross section predictions. In particular models with multiple mass scales, 1 /M prefactors can be replaced by inverse powers ofa smaller excitation energy. It is also of interest to investigate whether large anomalous dimensions couldenhance the coefficients of particular subleading operators. .1 Mass correction from electroweak symmetry breaking We may evaluate the heavy scalar self energy to obtain mass corrections, − i Σ( p ) = (cid:2) Wp + (cid:3) Z + (cid:4) 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 = α m W (cid:20) − J + sin θ W J (cid:21) . (15)In particular, with Q = J + Y = J 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) = α Q m W sin θ W O (1 /M ) ≈ (170 MeV) Q . (16)Subleading corrections can be similarly evaluated in the effective theory. Since no additionaloperators appear at O (1 /M ), the result (16) is model independent. 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); theStandard Model lagrangian with W, Z, h, t integrated out; and interactions, L = L φ + L SM + L φ , SM + . . . , (17)where the ellipsis denotes terms containing electrically charged heavy scalars. For the electri-cally neutral scalar, L φ = φ ∗ v,Q =0 (cid:26) iv · ∂ − ∂ ⊥ M ( Q =0) + O (1 /m W ) (cid:27) φ v,Q =0 . (18)Note that enforcing the reality condition (7) implies the vanishing of c D (= c M ).Interactions with Standard Model fields begin at order 1 /m W . We restrict attention toquark and gluon operators (neglecting lepton and photon operators) and again focus on theneutral φ v,Q =0 component, dropping the Q = 0 subscript in the following. Mixing with chargedscalars will become relevant at order 1 /m W 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 The mass splitting (16) appears in limits of particular models, e.g. [1, 7, 8]. We define the pole mass to include the contributions induced by electroweak symmetry breaking, asopposed to introducing residual mass terms for different charge eigenstates [9]. + (cid:6) = c (cid:7) + . . .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 W lines are not displayed.with derivatives acting on φ v or involving γ , since these lead to spin-dependent interactionsthat are suppressed for low-velocity scattering. The basis of operators is then L φ , SM = 1 m W φ ∗ v φ v (cid:26) (cid:88) q (cid:20) c (0)1 q O (0)1 q + c (2)1 q v µ v ν O (2) µν q (cid:21) + c (0)2 O (0)2 + c (2)2 v µ v ν O (2) µν (cid:27) + . . . , (19)where we have chosen QCD operators of definite spin, O (0)1 q = m q ¯ qq , O (0)2 = ( G Aµν ) ,O (2) µν q = ¯ q (cid:18) γ { µ iD ν } − d g µν iD/ (cid:19) q , O (2) µν = − G Aµλ G Aνλ + 1 d g µν ( G Aαβ ) . (20)Here A { µ B ν } ≡ ( A µ B ν + A ν B µ ) / d = 4 − (cid:15) 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 ( α s ) and c ( S )1 q through O ( α s ). The matching conditions for quark operators in the n f = 5 flavor theory at renormalizationscale µ = µ t ∼ m t ∼ m W ∼ m h are obtained from the diagrams in Fig. (1): c (0)1 U ( µ t ) = C (cid:20) − x h (cid:21) , c (0)1 D ( µ t ) = C (cid:20) − x h − | V tD | x t x t ) (cid:21) ,c (2)1 U ( µ t ) = C (cid:20) (cid:21) , c (2)1 D ( µ t ) = C (cid:20) − | V tD | x t (3 + 6 x t + 2 x t )3(1 + x t ) (cid:21) , (21)where subscript U denotes u or c and subscript D denotes d , s or b . Here C = [ πα ( µ t )][ J ( J +1) / x h ≡ m h /m W and x t ≡ m t /m W . We ignore corrections of order m q /m W for q = u, d, s, c, b , and have used CKM unitarity to simplify the results.6 + (cid:9) + (cid:10) + (cid:11) = c (cid:12) + c (cid:34) (cid:13) + (cid:14) (cid:35) + . . .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 ( µ t ) = C α s ( µ t )4 π (cid:20) x h + 3 + 4 x t + 2 x t x t ) (cid:21) ,c (2)2 ( µ t ) = C α s ( µ t )4 π (cid:20) −
329 log µ t m W − − x t )9(1 + x t ) log µ t m W (1 + x t ) − x t − x t + 36 x t − x t + 3 x t − x t − log x t x t − x t ( − x t )9( x t − log 2 − x t + 24 x t − x t − x t + 20 x t + 13 x t + 189( x t − (1 + x t ) (cid:21) . (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 limit m q /m W →
0. Details of this computation will be presented elsewhere.
To account for perturbative corrections involving large logarithms, e.g. α s ( µ ) log m t /µ , weemploy renormalization group evolution to sum leading logarithms to all orders.7 .1 Anomalous dimensions The spin S = 0 and spin S = 2 operators mix amongst themselves, with dd log µ O ( S ) i = − (cid:88) j γ ( S ) ij O j , (23)where γ ( S ) ij are ( n f + 1) × ( n f + 1) anomalous dimension matrices. The leading terms areˆ γ (0) = − γ (cid:48) m · · · − γ (cid:48) m ( β/g ) (cid:48) = α s π · · · − β + . . . , ˆ γ (2) = α s π − . . . ... − − · · · −
649 4 n f + . . . , (24)where β = dg/d log µ ≈ − β α s / π , γ m = d log m q /d log µ ≈ − α s / π , γ (cid:48) m ≡ g∂γ m /∂g ,( β/g ) (cid:48) ≡ g∂ ( β/g ) /∂g , and β = 11 − n f . It is straightforward to include subleading termsfor ˆ γ (0) [11, 12] and ˆ γ (2) [13, 14]. At the scale µ = µ b ∼ m b , we match onto an n f = 4 theory containing u, d, s, c quarks. Thematching equations are c (0)2 ( µ b ) = ˜ c (0)2 ( µ b ) (cid:18) a m b µ b (cid:19) − ˜ a c (0)1 b ( µ b ) (cid:20) a (cid:18)
11 + 43 log m b µ b (cid:19)(cid:21) + O (˜ a ) ,c (0)1 q ( µ b ) = ˜ c (0)1 q ( µ b ) + O (˜ a ) ,c (2)2 ( µ b ) = ˜ c (2)2 ( µ b ) − a m b µ b ˜ c (2)1 b ( µ b ) + O (˜ a ) ,c (2)1 q ( µ b ) = ˜ c (2)1 q ( µ b ) + O (˜ a ) , (25)where q = u, d, s, c and ˜ a = α s ( µ b , n f = 5) / π . Quantities without (with) tilde refer to the n f = 4 ( n f = 5) theory. The scheme dependence for heavy quark masses enters at higher order.For definiteness we use pole masses for m b and m c , with values taken from [15]. Followingour power counting scheme, we consider one less order of α s in the matching for c ( S )1 q relativeto c ( S )2 . For later use in the numerical analysis, we have included NLO QCD corrections inthe spin-0 matching [16, 17]. Similar to above, we evolve coefficients in the n f = 4 theory tothe scale µ = µ c ∼ m c . Finally, we match onto n f = 3 and evolve to a low scale µ ∼ Matrix elements and cross section
Having expressed the lagrangian in terms of operators renormalized at the scale µ ∼ Let us define the zero-momentum matrix elements of renormalized operators (cid:104) N | O (0)1 q | N (cid:105) ≡ m N f (0) q,N , − α s ( µ )8 π (cid:104) N | O (0)2 ( µ ) | N (cid:105) ≡ m N f (0) G,N ( µ ) , (cid:104) N | O (2) µν q ( µ ) | N (cid:105) ≡ m N (cid:18) k µ k ν − g µν m N (cid:19) f (2) q,N ( µ ) , (cid:104) N | O (2) µν ( µ ) | N (cid:105) ≡ m N (cid:18) k µ k ν − g µν m N (cid:19) f (2) G,N ( µ ) , (26)where m N is the nucleon mass. Matrix elements refer to a definite (but arbitrary) spin stateof the nucleon. We recall that the spin-0 operator matrix elements are not independent, being linked by therelation [18] m N = (1 − γ m ) (cid:88) q (cid:104) N | m q ¯ qq | N (cid:105) + β g (cid:104) N | ( G aµν ) | N (cid:105) , (27)derived from the trace of the QCD energy-momentum tensor. Here N = p or n . Neglecting γ m , O ( α s ) contributions to β ( g ), and power corrections in the above formula, the definitions(26) ensure that f (0) G,N ( µ ) ≈ − (cid:80) q = u,d,s f (0) q,N . Corrections arising from (27) are included in thenumerical analysis.For quark operators, define the scale-independent quantities,Σ πN = m u + m d (cid:104) p | (¯ uu + ¯ dd ) | p (cid:105) , Σ = m u + m d (cid:104) p | (¯ uu + ¯ dd − ss ) | p (cid:105) . (28)In the numerical analysis, we will neglect the small contributions proportional to | V td | and | V ts | , so that c (0)1 u = c (0)1 d . Neglecting also the small contribution [19] ( m d − m u ) (cid:104) p | (¯ uu − ¯ dd ) | p (cid:105) ∼ N = p or n , m N ( f (0) u,N + f (0) d,N ) ≈ Σ πN , m N f (0) s,N = m s m u + m d (Σ πN − Σ ) = Σ s . (29) We use nonrelativistic normalization for nucleon states, (cid:104) N ( p ) | N ( p (cid:48) ) (cid:105) = (2 π ) δ ( p − p (cid:48) ). | V td | ∼ | V ts | ∼ | V tb | ∼ m u /m d . m s /m d . .
5) [20]Σ lat πN . lat s . πN . . m W . m t
172 GeV [15] m b m c m N α s ( m Z ) 0.118 [20] α ( m Z ) 0.0338 [20]Table 1: Inputs to the numerical analysis.We consider “traditional” values Σ πN = 64 ± = 36 ± lat πN = 47 ± lat s = 50 ± We adopt PDG values [20] for light-quark mass ratios. A summary of numerical inputs ispresented in Table 1.
The matrix elements of spin-two operators can be identified as f (2) q,p ( µ ) = (cid:90) dx x [ q ( x, µ ) + ¯ q ( x, µ )] , (30)where q ( x, µ ) and ¯ q ( 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 QCDenergy momentum tensor, hence independent of scale we have f (2) G,p ( µ ) ≈ − (cid:80) 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) The latter quantity arises from a naive averaging of Σ s = 31 ±
15 MeV [21] and Σ s = 59 ±
10 MeV [25].See also [26, 27, 28]. (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 . µ =1 . The low-velocity, spin-independent, cross section for WIMP scattering on a nucleus of massnumber A and charge Z may be written σ A,Z = m r π | Z M p + ( A − Z ) M n | ≈ m r A π |M p | , (32)where M p and M n are the matrix elements for scattering on a proton or neutron respectively ,and m r = M m N / ( M + m N ) denotes the reduced mass of the dark-matter nucleus system. Asdescribed in Section 6.1, M n ≈ M p up to corrections from numerically small CKM factorsand isospin violation in nucleon matrix elements. In the M (cid:29) m N limit, the cross sectionscales as A . 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 Σ πN and Σ , as well as recent lattice determinations ofΣ lat πN and Σ lat s . For each case, separate bands represent the uncertainty due to neglectedperturbative QCD corrections, and due to the hadronic inputs. We estimate the impact ofhigher order perturbative QCD corrections by varying matching scales m W / ≤ µ t ≤ m t , m b / ≤ µ b ≤ m b , m c / ≤ µ c ≤ m c , 1 . ≤ µ ≤ . For spin-0 operators, we find a large residual uncertainty at LO from µ , µ c and µ b scale variation. The RG running from µ c to µ from (24) is thus evaluatedwith NNNLO corrections, including contributions to β/g through O ( α s ) and γ m through O ( α s ). Accordingly, the spin-0 gluonic matrix element from (27) is also evaluated at NNNLO, Explicitly, M N = m − W (cid:104) N | (cid:16)(cid:80) q = u,d,s (cid:104) c (0)1 q O (0)1 q + c (2)1 q v µ v ν O (2) µν q (cid:105) + c (0)2 O (0)2 + c (2)2 v µ v ν O (2) µν (cid:17) | N (cid:105) . Up to power corrections and subleading O ( α s ) corrections, our evaluation is equivalent to an evaluationin either the n f = 4 or n f = 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. Π N, (cid:83) (cid:83) Π Nlat , (cid:83) s lat
100 120 140 160 180 20010 (cid:45) (cid:45) (cid:45) (cid:45) m h (cid:72) GeV (cid:76) Σ (cid:72) c m (cid:76) Figure 3: Cross section for low-velocity scattering on a nucleon for a heavy real scalar in theisospin J = 1 representation of SU (2). The dark shaded region represents the 1 σ uncertaintyfrom perturbative QCD, estimated by varying factorization scales. The light shaded regionrepresents the 1 σ uncertainty from hadronic inputs.including contributions to β/g through O ( α s ) and γ m through O ( α s ). The residual µ scalevariation is insignificant compared to other uncertainties. We perform the RG running andheavy quark matching from µ t to µ c at NLO. Hadronic input uncertainties from each sourcein Table 1 and Table 2 are added in quadrature. We have ignored power corrections appearingat relative order α s ( m c )Λ /m c ; typical numerical prefactors appearing in the coefficients ofthe corresponding power-suppressed operators [18] suggest that these effects are small.Due to a partial cancellation between spin-0 and spin-2 matrix elements, the total crosssection and the fractional error depend sensitively on subleading perturbative corrections andon the Higgs mass parameter m h . We find σ p ( m h = 120 GeV) = 0 . ± . +0 . − . × − cm , σ p ( m h = 140 GeV) = 2 . ± . +1 . − . × − cm , (33)where the first error is from hadronic inputs, assuming Σ lat s and Σ lat πN from Table 1, and thesecond error represents the effect of neglected higher order perturbative QCD corrections. Forthe illustrative value m h = 120 GeV, and as a function of the scalar strange-quark matrixelement Σ s , we display the separate contributions of each of the quark and gluon operators inFig. 4. We have presented the effective theory for heavy, weakly interacting dark matter candidatescharged under electroweak SU (2). Having determined the general form of the effective la-12 (cid:72) (cid:76) d (cid:72) (cid:76) g (cid:72) (cid:76) s (cid:72) (cid:76) s (cid:72) (cid:76) u (cid:72) (cid:76) + d (cid:72) (cid:76) g (cid:72) (cid:76) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:83) s (cid:72) MeV (cid:76) m W Π Α (cid:77) p (cid:72) M e V (cid:76) Figure 4: Breakdown of contributions to the matrix element M p using the representativevalues m h = 120 GeV and Σ lat πN = 47(9) MeV. The labels u ( S ) , d ( S ) , s ( S ) and g ( S ) refer to spin-Sup, down, strange and gluon operator contributions, respectively. The thickness representsthe 1 σ uncertainty from perturbative QCD. The left-hand vertical band corresponds to thelattice value Σ lat s = 50(8) MeV and the right-hand vertical band corresponds to the rangeΣ s = 366(142) MeV deduced from Σ πN and Σ in Table 1.grangian (4) through 1 /M , we demonstrated matching conditions for subleading operators ina simple model. Using the effective theory, we demonstrated universality of the mass splittinginduced by electroweak symmetry breaking, and of the cross section for scattering on nuclearmatter. Subleading terms in the 1 /M expansion can be studied systematically using (4).Our focus has been on the case of an isotriplet real scalar [1]. For this case, relic abun-dance estimates [8] indicate that M (cid:46) few TeV in order to not overclose the universe. Thismass range, combined with the universal cross section, provides a target for future searchexperiments.We have presented a complete matching at first nonvanishing order in α s , and at leadingorder in small ratios m W /M , m b /m W and Λ QCD /m c . We performed renormalization groupimprovement to sum leading logarithms to all orders. The residual dependence on the highmatching scale µ t ∼ m t ∼ m W represents uncertainty due to uncalculated higher-order per-turbative corrections. Assuming the hadronic input Σ lat s from Table 1, this scale variation isthe largest remaining uncertainty on the cross section; its reduction would require higher looporder calculations.Our high-scale matching results for quark operators (21) and spin-zero gluon operatorsagree with m W /M → µ t = µ b = µ c , i.e., a one-step matching onto the n f = 3 theory. This approach neglects To make the comparison to the scattering amplitude for a heavy Majorana fermion with χ = χ c , weuse χ = √ e − imv · x ( h v + H v ) = √ e imv · x ( h cv + H cv ), where h v and H v are spinor fields with (1 − v/ ) h v =(1 + v/ ) H v = 0. SU (2) representations(or SU (2) singlets), fermionic WIMPs, and models with additional low-energy field contentbeyond the Standard Model. The treatment of QCD corrections presented here can be ap-plied to compute scattering amplitudes in related models, e.g. models involving coupling tohypercharge, or supersymmetric models [29, 31, 32]. Acknowledgements
We acknowledge discussions with Y. Bai, G. Paz and J. Zupan. Work supported by NSFGrant 0855039.
Note added.
While this paper was in writing, the preprint [33] appeared, mentioning theinvariance (7) for Majorana fermions.
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