Standard Model anatomy of WIMP dark matter direct detection I: weak-scale matching
EEFI Preprint 13-34January 14, 2014
Standard Model anatomy of WIMP dark matter direct detectionI: weak-scale matching
Richard J. Hill ∗ and Mikhail P. Solon † Enrico Fermi Institute and Department of PhysicsThe University of Chicago, Chicago, Illinois, 60637, USA
Abstract
We present formalism necessary to determine weak-scale matching coefficients in the computationof scattering cross sections for putative dark matter candidates interacting with the StandardModel. Particular attention is paid to the heavy-particle limit. A consistent renormalizationscheme in the presence of nontrivial residual masses is implemented. Two-loop diagrams appearingin the matching to gluon operators are evaluated. Details are given for the computation ofmatching coefficients in the universal limit of WIMP-nucleon scattering for pure states of arbitraryquantum numbers, and for singlet-doublet and doublet-triplet mixed states. ∗ [email protected], † [email protected] a r X i v : . [ h e p - ph ] J a n Introduction
The compelling evidence for dark matter (DM) inconsistent with Standard Model (SM) particleshas motivated many theoretical studies and experimental searches to elucidate its particle nature.In particular, the paradigm of Weakly Interacting Massive Particles (WIMPs) continues to play aprominent role, and experiments in the present decade should explore a significant region of remainingWIMP parameter space [1]. Given the multitude of WIMP candidates and search strategies, it isimperative to develop theoretical formalism to delineate the possible interactions of DM with knownparticles, making clear which uncertainties are inherently model dependent and which can, at leastin principle, be improved by further SM analysis.Even in many seemingly simple cases, determination of WIMP-nucleon cross sections demandsan intricate analysis of competing amplitudes mediated by SM particles (see e.g., [2–7]). In thispaper we set out the formalism for electroweak-scale matching computations for application both totheories with specified ultraviolet (UV) completion (e.g., supersymmetric models [8, 9]), and to theheavy WIMP limit where theoretical control is maintained in the absence of a specified UV comple-tion [6]. We review relevant aspects of techniques such as the background field method for matchingto gluon operators [3, 10], the extension of the onshell renormalization scheme for WIMP couplingsto the electroweak SM, and the treatment of effective theory subtractions. Direct detection experi-mental constraints [11,12], together with other phenomenological bounds such as LHC searches, mayplausibly indicate that new particles must have mass somewhat above the mass of electroweak-scaleparticles ( M (cid:29) m W ). In this regime, the prospects for direct detection become more challenging, butin a precise sense more constrained due to heavy particle universality. Extending the particle contentof the SM by one or a few electroweak multiplets, the heavy particle limit implies highly predictivecross sections with minimal parametric input beyond the SM. This limit is thus both physically inter-esting, as well as a useful pedagogical illustration. Within the heavy WIMP framework, we present acomplete reduction of the required one- and two-loop amplitudes into a basis of heavy-particle loopintegrals with nonzero residual mass.Although we aim for generality, for definiteness throughout the paper, we illustrate these methodsfor the case where the lightest, electrically neutral particle of the new sector corresponds to a self-conjugate field (e.g., a Majorana fermion or real scalar) stabilized by a Z symmetry, deriving froma theory consisting of one or two SU (2) W × U (1) Y multiplets beyond the SM particle content [2,5–7, 13–26]. An important simplification occurs when a scale separation exists between SM masses( ∼ m W ) and the lightest new particle mass ( ∼ M ), allowing an expansion in m W /M . We considerin detail the limit M (cid:29) m W where universal behavior appears, and present the necessary heavyparticle effective theory tools for such an analysis. For these SM extensions, we present details ofthe first complete computation of the matching at leading order in perturbation theory onto the fullbasis of operators at the electroweak scale [6].The field of DM direct detection is by now a mature subject. Early treatments of QCD effects inneutralino-nucleon scattering include the works of Drees and Nojiri [27]. Basic aspects of formalismmay be found in the review of Jungman, Griest and Kamionkowski [8]. However, the last fewyears have witnessed the discovery and mass measurement for a SM-like Higgs boson [28, 29], newconstraints on the mass scale of particles beyond the SM [30], and important computational advancesin lattice QCD [31, 32]. A complete description of DM-SM interactions is now possible in many SMextensions but demands the systematic treatment of QCD effects and uncertainties, including theconsideration of loop amplitudes that are typically neglected in the m W ∼ M regime, but which A subset of recent work in the field may be found in the Snowmass review [1].
Heavy particle methods may be used to efficiently describe the interactions of DM, of mass M , withmuch lighter degrees of freedom such as those of n f = 5 flavor QCD (in the case m b (cid:28) M , where m b is the bottom quark mass) or those of the SM electroweak sector (in the case m W (cid:28) M , where m W is the W ± boson mass). Let us briefly review a few aspects of heavy particle effective theoryrelevant for the DM applications in Secs. 3 and 4.A heavy-particle field, h v , is identified with a representation of the little group for massive parti-cles, and carries a label v associated with the time-like unit vector v µ that defines the little group [35].The little group for massive particles is isomorphic to SO (3), and therefore has field representationscarrying spin s = 0 , / , , . . . . We may write such fields in covariant notation using a Dirac spinor-vector with appropriate constraints. For example, a spin-1 / /
2) + 1degrees of freedom and can be written as a Dirac spinor, h v , obeying v/h v = h v as a projectionconstraint. For integer spin we define ¯ h v ≡ h † v , while for half-integer spin h v carries spinor indicesand we define ¯ h v ≡ h † v γ . Additionally, for self-conjugate fields the simultaneous operations v µ → − v µ , h v → h cv ≡ C h ∗ v , (1)where C is the charge conjugation matrix, implement a symmetry of the heavy-particle lagrangian. Having specified the building blocks, interactions with heavy-particle fields can be constructedin the usual way. We write down the most general set of gauge-invariant and Lorentz-covariantoperators in terms of the heavy field h v , the time-like unit vector v µ , and other relativistic degrees offreedom up to a given order in the 1 /M power counting. In the case of a self-conjugate heavy particle, The case of arbitrary spin is discussed in Appendix A.1 of Ref. [35]. A discussion of this invariance is given in Appendix A.2 of Ref. [35]; for a heavy Majorana fermion, see also [36]. /M expansions [35, 37, 38]. In the present paper, we focus on the leadingorder in 1 /M . Let us construct the effective theory of DM with mass M (cid:38) m W interacting with n f = 5 flavorQCD. The hierarchy of scales between the DM mass and the relevant low-energy degrees of freedom,Λ QCD , m c , m b (cid:28) m W , allows us to use heavy particle effective theory to describe the DM field. Themost general lagrangian relevant for spin-independent, low-velocity scattering with nucleons, is thengiven at energies E (cid:28) m W by, L χ v , SM = ¯ χ v χ v (cid:26) (cid:88) q = u,d,s,c,b (cid:20) c (0) q O (0) q + c (2) q v µ v ν O (2) µνq (cid:21) + c (0) g O (0) g + c (2) g v µ v ν O (2) µνg (cid:27) + . . . , (2)where χ v is the lightest, electrically neutral, self-conjugate WIMP of arbitrary spin. The ellipsisin the above equation includes higher-dimension operators suppressed by powers of 1 /m W . Theassumed self-conjugacy of χ v implies that (2) is invariant under (1). The SM component of (2) isexpressed in terms of quark and gluon fields as O (0) q = m q ¯ qq ,O (0) g = ( G Aµν ) ,O (2) µνq = 12 ¯ q (cid:18) γ { µ iD ν }− − d g µν iD/ − (cid:19) q ,O (2) µνg = − G Aµλ G Aνλ + 1 d g µν ( G Aαβ ) . (3)Here D − ≡ −→ D − ←− D , and A { µ B ν } ≡ ( A µ B ν + A ν B µ ) / d = 4 − (cid:15) spacetime dimensions. We use the background field method for gluons in the effectivetheory thus ignoring gauge-variant operators, and assume that appropriate field redefinitions areemployed to eliminate operators that vanish by leading order equations of motion. We ignore flavornon-diagonal operators, whose nucleon matrix elements have an additional weak-scale suppressionrelative to those considered. We will not be concerned here with leptonic interactions.For a self-conjugate WIMP, χ v , with mass M (cid:38) m W and arbitrary spin, Eq. (2) representsthe most general effective lagrangian at leading order in 1 /m W , relevant for spin-independent, low-velocity scattering with nucleons. Details of the UV completion are encoded in the twelve matchingcoefficients c (0) q , c (2) q , c (0) g and c (2) g . Matching onto the effective theory (2) is in general dependent onthe specific SM extension. Although much of the formalism applies more generally, for definitenesswe focus on the heavy WIMP limit, M (cid:29) m W , where universal features appear [6]. General bases including spin-dependent interactions, and non-self-conjugate WIMPs are presented in [34]. Effective theory for one or two heavy electroweak multiplets
In place of a specified UV theory for DM, let us use heavy particle effective theory to describeextensions of the SM consisting of one or two electroweak multiplets with masses large compared tothe mass of electroweak-scale particles,
M, M (cid:48) (cid:29) m W . The extension to more than two multiplets isstraightforward. We will construct the effective theory describing interactions of such heavy WIMPswith the SM in the regime | M (cid:48) − M | , m W (cid:28) M, M (cid:48) . In the case | M (cid:48) − M | (cid:29) m W the effects of theheavier multiplet appear as power corrections in the effective theory for the lighter multiplet. Fornotational clarity, below we omit the subscript v labeling a heavy-particle field.Consider one or two multiplets of heavy-particle fields with arbitrary spin, transforming underirreducible representations of electroweak SU (2) W × U (1) Y . Let us collect the heavy fields in acolumn vector h , and their masses in a diagonal matrix M . The precise specification of M beyondtree level is described in Sec. 5. At leading order in the 1 /M expansion, the most general gauge- andLorentz-invariant lagrangian, bilinear in h , and written in terms of the building-blocks h , v µ , andSM fields, takes the form L = ¯ h [ iv · D − δm − f ( H )] h + O (1 /M ) , (4)where iD µ = i∂ µ + g Y B µ + g W aµ T a , and f ( H ) is a linear matrix function of H (and H ∗ ). For purestates gauge invariance implies f ( H ) = , while for mixed states f ( H ) describes the mixing of thepure-state constituents through the Higgs field. In terms of a reference mass M ref , the residual massmatrix is δm = M − M ref . (5)Note that if the masses composing M are degenerate, as for a single “pure” electroweak multiplet,we may choose M ref appropriately to set δm = . In the case of two “mixed” electroweak multiplets M will have non-degenerate entries in general.Upon accounting for EWSB we may write (4) as L = ¯ h (cid:20) iv · ∂ + eQv · A + g c W v · Z ( T − s W Q )+ g √ v · W + T + + v · W − T − ) − δM ( v wk ) − f ( φ ) (cid:21) h + O (1 /M ) , (6)where T ± = T ± iT , the charge matrix is Q = T + Y in units of the proton charge, and φ denotesthe fluctuation of the Higgs field about (cid:104) H (cid:105) , H = v wk √ (cid:32) (cid:33) + (cid:32) φ + W √ ( h + iφ Z ) (cid:33) . (7)The residual mass matrix now includes EWSB contributions, δM ( v wk ) = δm + f ( (cid:104) H (cid:105) ) , (8)and in the mass eigenstate basis for δM ( v wk ), we will set the residual mass of the lightest, (assumed)electrically neutral WIMP, χ , to zero by appropriate choice of M ref . Other states may have non-vanishing residual masses. In the following, we will suppress the subscript in v wk ; the resulting v isnot to be confused with the velocity v µ . 4he heavy-particle lagrangian (4) can also be obtained at tree level from a manifestly relativisticlagrangian by performing field redefinitions. We illustrate this for the singlet-doublet mixture inAppendix A. Let us now have a detailed look at extensions with one (pure states) or two (mixedstates) electroweak multiplets.
The pure-state heavy-particle lagrangian is completely specified by electroweak quantum numberssince δm = and f ( H ) = . We may proceed in generality, assuming a multiplet of fields in theisospin J representation of SU (2) W with hypercharge Y . The amplitudes for weak-scale matchingin Sec. 6 will be given in terms of Y and the Casimir J ( J + 1). In particular, amplitudes with two W ± bosons or two Z bosons carry the respective factors C W = J ( J + 1) − Y , C Z = Y . (9)For extensions consisting of electroweak multiplets with non-zero hypercharge, we assume that higher-dimension operators cause the mass eigenstates after EWSB to be self-conjugate combinations. Thisforbids a phenomenologically disfavored tree-level vector coupling between the lightest, electricallyneutral state, χ , and Z .As specific illustrations we consider the cases of an SU (2) W triplet ( J = 1) with Y = 0, and a pairof SU (2) W doublets ( J = 1 /
2) with opposite hypercharge Y = ± /
2. In supersymmetric extensions,these represent pure wino and pure higgsino states, respectively. Let us look at these cases in somedetail.
Let the column vector h T = ( h , h , h ), with subscript T for triplet, be a heavy, self-conjugate, SU (2) W triplet with Y = 0. The heavy-particle lagrangian for h T is given by (4) with ( T a ) bc = i(cid:15) bac , f ( H ) = , and δm = . The electric charge eigenbasis is given by h h h ≡ √ √ i √ − i √ h h + h − . (10)In terms of the column vector h = ( h , h + , h − ), where h ≡ χ , the lagrangian is given by (6) with Q = T = diag(0 , , − , T + = √ −√ , T − = −√ √ . (11) We remark that the consistency of an effective description for the one-heavy particle sector for a self-conjugatefield follows from the identification of lowest-lying states odd under a Z symmetry. In contrast, the one-heavy particlesector for a heavy field carrying U (1) global symmetry (e.g., heavy-quark number in a heavy quark effective theory) isidentified by this quantum number. .1.2 Pure doublet Let h ψ and h ψ c be heavy-particle doublets in the ( , /
2) and ( ¯2 , − /
2) representations of SU (2) W × U (1) Y . Anticipating perturbations that cause the mass eigenstates to be self-conjugate fields, let usintroduce the linear combinations h D = h ψ + h ψ c √ (cid:32) h h (cid:33) , h D = i ( h ψ − h ψ c ) √ (cid:32) h h (cid:48) (cid:33) , (12)with subscript D for doublet. The heavy-particle lagrangian for the column vector h = ( h D , h D ) isgiven by (1), with f ( H ) = , and gauge couplings T a = (cid:32) τ a − τ aT − i ( τ a + τ aT )4 i ( τ a + τ aT )4 τ a − τ aT (cid:33) , Y = i (cid:32) − (cid:33) , (13)where τ a are the Pauli isospin matrices. Neglecting the small mass perturbation mentioned above,the tree-level mass eigenstates are degenerate, and we may choose δm = . The charge eigenstatesare given by h h h h (cid:48) ≡ √ √ i √ − i √ h h (cid:48) h + h − . (14)In terms of the column vector h = ( h , h (cid:48) , h + , h − ), where h ≡ χ , the lagrangian is given by (6) with Q = diag( , , −
1) and T = i − i
00 0 0 − , T + = − √ i √ √ − i √ , T − = √
00 0 i √
00 0 0 0 − √ − i √ . (15) As an example of mixed states, let us consider in detail the singlet-doublet admixture. Results forthe triplet-doublet admixture will also be given below.
Let h S , with subscript S for singlet, be a heavy, self-conjugate, SU (2) W singlet with Y = 0 and mass M S . Consider an admixture of h S and the previously defined self-conjugate doublets h D and h D ,with mass M D . At leading order in the 1 /M expansion, the gauge-invariant interactions of h S , h D and h D involving the Higgs field are L H ¯ hh = − ¯ h S (cid:20) yH † ( h D − ih D ) √ y ∗ H T ( h D + ih D ) √ (cid:21) + h . c . = − ¯ hf ( H ) h , (16) This construction is analogous to that appearing in applications of heavy quark effective theory to processes whereboth a heavy quark and a heavy anti-quark are active degrees of freedom. h = ( h S , h D , h D ) = ( h S , h , h , h , h (cid:48) ). The Higgs coupling matrix is given by f ( H ) = a √ H † + H T i ( H T − H † ) H + H ∗ i ( H − H ∗ ) + a √ − i ( H T − H † ) H T + H † − i ( H − H ∗ ) H + H ∗ , (17)with real parameters a = Re( y ) and a = Im( y ). For comparison, the derivation in Appendix Aobtains (16) at tree level starting from a manifestly relativistic lagrangian. The residual mass matrixis δm = diag ( M S , M D ) − M ref , and we define M S and M D to be real and positive. The gaugecouplings are obtained by trivially extending (13) to include the singlet. This completely specifiesthe heavy-particle lagrangian given in (4).The mass induced by EWSB is accounted for at tree level by including contributions from (17), δM ( v ) = δM + v a a a a . (18)In the following, we use subscripts to denote the electric charge and bracketed superscripts to labelthe mass eigenstate. For neutral states we find the residual mass eigenvalues δ (0)0 = M D − M ref , δ ( ± )0 = M D + M S ± (cid:112) ∆ + ( av ) − M ref , (19)where we define ∆ ≡ M S − M D , a ≡ (cid:113) a + a . (20)By definition a >
0, and regardless of the sign of ∆, the smallest eigenvalue is δ ( − )0 . Let us set thiseigenvalue to zero by appropriately choosing the reference mass M ref . The corresponding normalizedeigenvectors in the ( h S , h , h (cid:48) ) basis of electrically neutral states are then (cid:126) v (0)0 = 1 a a − a , (cid:126) v ( ± )0 = 1 (cid:20)(cid:16) ∆ ± (cid:112) ∆ + ( av ) (cid:17) + ( av ) (cid:21) ∆ ± (cid:112) ∆ + ( av ) a va v , (21)and we may construct the unitary matrix U (on the three-dimensional neutral subspace) to translateto the mass eigenbasis, U = (cid:16) (cid:126) v (0)0 (cid:126) v (+)0 (cid:126) v ( − )0 (cid:17) , h S h h (cid:48) = U h (0)0 h (+)0 h ( − )0 , U † δM ( v ) U = diag (cid:16) δ (0)0 , δ (+)0 , δ ( − )0 (cid:17) . (22) An additional phase redefinition of h ψ , h ψ c could be used to enforce the vanishing of a or a . δ (0) ± = δ (0)0 ,and the corresponding charge eigenstates are given by (cid:32) h h (cid:33) = 1 √ (cid:32) i − i (cid:33) (cid:32) h (0)+ h (0) − (cid:33) . (23)The basis of mass eigenstates is thus given by the column vector h = (cid:16) h (0)0 , h (+)0 , h ( − )0 , h (0)+ , h (0) − (cid:17) ,where h ( − )0 ≡ χ , and the lagrangian is given by (6) with δM ( v ) = diag (cid:16) δ (0)0 , δ (+)0 , δ ( − )0 , δ (0)+ , δ (0) − (cid:17) = av diag (cid:16) t ρ , s − ρ , , t ρ , t ρ (cid:17) ,Q = diag( , , − ,T − s W Q = i | s ρ | i | c ρ | − i | s ρ | − i | c ρ | − s W
00 0 0 0 − + s W ,T + = e − iξ √ − i −| s ρ | −| c ρ | i | s ρ | | c ρ | , T − = e + iξ √ − i
00 0 0 | s ρ |
00 0 0 | c ρ |
00 0 0 0 0 i −| s ρ | −| c ρ | ,f ( φ ) = a | c ρ | φ Z −| s ρ | φ Z | c ρ | φ Z s ρ h c ρ h | c ρ | e + iξ φ − W | c ρ | e − iξ φ + W −| s ρ | φ Z c ρ h − s ρ h −| s ρ | e + iξ φ − W −| s ρ | e − iξ φ + W | c ρ | e − iξ φ + W −| s ρ | e − iξ φ + W | c ρ | e + iξ φ − W −| s ρ | e + iξ φ − W , (24)where we have introducedsin ρ ≡ av (cid:112) ( av ) + ∆ , cos ρ ≡ ∆ (cid:112) ( av ) + ∆ , e ± iξ ≡ ( a ± ia ) a . (25)The shorthand notation c x ≡ cos x , s x ≡ sin x , and t x ≡ tan x is used throughout this paper. Notethat s ρ is positive, and that c ρ can have either sign depending on the hierarchy between M S and M D . It is straightforward to extract Feynman rules from the lagrangian (6) and the matrices (24).For example, the propagator for χ , and its coupling to the physical Higgs boson, h , are (cid:1) kχ χ = iv · k − δ ( − )0 + i , (cid:2) χ χh = ias ρ . (26)8 .2.2 Triplet-doublet admixture The construction for the triplet-doublet case follows closely that for the singlet-doublet case, witha heavy triplet h T in place of the singlet h S . Using τ = ( τ , τ , τ ) and ¯ τ = − ( τ T , τ T , τ T ), thegauge-invariant interactions of h T , h D and h D involving the Higgs field can be written in the form L H ¯ hh = − ¯ hf ( H ) h , where we collect fields in a seven-component column vector h = ( h T , h D , h D ),and the matrix f ( H ) is given by f ( H ) = a √ H † τ − H T ¯ τ i ( − H T ¯ τ − H † τ ) − ¯ τ H ∗ + τ H i ( τ H + ¯ τ H ∗ ) + a √ i ( H T ¯ τ + H † τ ) H † τ − H T ¯ τ i ( − τ H − ¯ τ H ∗ ) − ¯ τ H ∗ + τ H , (27)with real parameters a and a . Upon accounting for mass contributions from EWSB, the basisof mass eigenstates is given by the column vector h = (cid:16) h (0)0 , h (+)0 , h ( − )0 , h (+)+ , h ( − )+ , h (+) − , h ( − ) − (cid:17) , where h ( − )0 ≡ χ , and the lagrangian is given by (6) with δM ( v ) = diag (cid:16) δ (0)0 , δ (+)0 , δ ( − )0 , δ (+)+ , δ ( − )+ , δ (+) − , δ ( − ) − (cid:17) = av diag (cid:16) t ρ , s − ρ , , s − ρ , , s − ρ , (cid:17) ,Q = diag(0 , , , , , − , − ,T = i | s ρ | i | c ρ | − i | s ρ | − i | c ρ | − s ρ − s ρ − s ρ − c ρ − s ρ s ρ s ρ − c ρ ,T + = 1 √ i | s ρ | i | c ρ | c ρ − s ρ − s ρ s ρ − i | s ρ | − − c ρ s ρ − i | c ρ | s ρ − − s ρ , − = 1 √ i | s ρ | i | c ρ | − − c ρ s ρ s ρ − − s ρ − i | s ρ | c ρ − s ρ − i | c ρ | − s ρ s ρ ,f ( φ ) = a | c ρ | φ Z −| s ρ | φ Z − i | c ρ | φ − W i | s ρ | φ − W i | c ρ | φ + W − i | s ρ | φ + W | c ρ | φ Z s ρ h c ρ h φ − W φ + W −| s ρ | φ Z c ρ h − s ρ h − φ − W − φ + W i | c ρ | φ + W − φ + W s ρ h c ρ h − iφ Z − i | s ρ | φ + W φ + W c ρ h + iφ Z − s ρ h − i | c ρ | φ − W − φ − W s ρ h c ρ h + iφ Z i | s ρ | φ − W φ − W c ρ h − iφ Z − s ρ h , (28)where s ρ and c ρ are as defined in (25), with a = (cid:112) a + a and ∆ = ( M T − M D ) /
2. Again, s ρ ispositive and c ρ can have either sign depending on the hierarchy between M T and M D . Appropriate parametric limits can be taken to decouple the pure state constituents of an admixture.This can be used to check the consistency of matching computations in Sec. 6. From the singlet-doublet admixture, we may recover the pure doublet (singlet) case by taking a → | ∆ | → ∞ ,with ∆ > < ρ → ρ → π ). Similarly, to recover the pure doublet (triplet) casefrom the triplet-doublet admixture, we decouple the triplet (doublet) component by taking a → | ∆ | → ∞ , with ∆ > < ρ → ρ → π ). A consistent evaluation of amplitudes beyond tree level demands renormalization of the Higgs-WIMPvertex, h ¯ χχ , that appears for admixtures. We define an extension of the onshell renormalizationscheme for the electroweak SM (e.g., see [39]) by expressing the vertex amplitude in terms of physicalmasses in the SM and DM sectors. We begin by studying the singlet-doublet mixture, and will laterquote the analogous results for the triplet-doublet mixture.To avoid confusion with standard notation for counterterms, in this section (only) we denote aresidual mass by µ , and a residual mass counterterm by δµ . We keep the notation introduced inSec. 4 for the residual mass eigenvalues, δ (0)0 , δ ( ± )0 , etc. Let us write the bare lagrangian as the sum of renormalized and counterterm contributions L = ¯ h bare (cid:104) iv · D − µ bare − f bare ( H bare ) (cid:105) h bare
10 ¯ h (cid:104) iv · D + δZ h iv · D − µ − δµ − f ( H bare ) − δf ( H bare ) (cid:105) h , (29)where the bare quantities are given by µ bare = diag( µ bare S , µ bare D , µ bare D , µ bare D , µ bare D ) ,f bare ( H ) = a bare1 √ H † + H T i ( H T − H † ) H + H ∗ i ( H − H ∗ ) + a bare2 √ − i ( H T − H † ) H T + H † − i ( H − H ∗ ) H + H ∗ ≡ a bare1 f ( H ) + a bare2 f ( H ) , (30)and the expression for f bare ( H ) above is valid for arbitrary H (in particular, for H bare ). The gaugesymmetry preserving counterterms are given by Z h = 1 + δZ h = 1 + diag( δZ S , δZ D ) ,µ + δµ = Z h µ bare Z h = diag( µ S + δµ S , ( µ D + δµ D ) ) ,f ( H bare ) + δf ( H bare ) = Z h f bare ( H bare ) Z h = ( a + δa ) f ( H (cid:48) ) + ( a + δa ) f ( H (cid:48) ) . (31)We have introduced H (cid:48) to absorb the renormalization of v : H bare = Z H (cid:32) φ + W √ ( v − δv + h + iφ Z ) (cid:33) = Z H (cid:18) − δvv (cid:19) H (cid:48) . (32)Note that the renormalization of v introduces a coupling ∼ δvv h ¯ χχ through the a f ( H (cid:48) ) + a f ( H (cid:48) )term in (31). We will fix the counterterms by enforcing renormalization conditions on the residualmass matrix (two point functions). Three point functions involving the Higgs interaction will thenbe determined. Anticipating renormalization conditions that preserve the basis h = (cid:16) h (0)0 , h (+)0 , h ( − )0 , h (0)+ , h (0) − (cid:17) ofmass eigenstates introduced in Sec. 4.2.1, let us express the counterterms in this basis, δµ = δµ D + | c ρ | va ( a δa − a δa ) | s ρ | va ( a δa − a δa ) 0 0 · c ρ ( δ ∆) + s ρ va ( a δa + a δa ) − s ρ ( δ ∆) + c ρ va ( a δa + a δa ) 0 0 · · s ρ ( δ ∆) − s ρ va ( a δa + a δa ) 0 0 · · · · · · · , Z h = δZ D + · c ρ ( δZ S − δZ D ) − s ρ ( δZ S − δZ D ) 0 0 · · s ρ ( δZ S − δZ D ) 0 0 · · · · · · · , (33)where the above matrices are symmetric, and ( δ ∆) = ( δµ S − δµ D ) /
2. Due to the masslessness ofthe photon, the onshell renormalization factor for the electrically charged state, δZ D , is infrared(IR) divergent. To avoid the associated complications, we may turn off δZ D , corresponding to anadditional overall renormalization of the fields with δZ S = δZ D . This overall renormalization willnot impact the determination of physical masses or mass eigenstates. However, we will of course needto include additional wavefunction renormalization factors when computing physical amplitudes. Inthe following, we allow for arbitrary δZ D . (cid:3) W ± , Z (cid:4) h (cid:5) φ ± W , φ Z Figure 1: One-loop corrections to two-point functions. Double lines denote heavy WIMPs, zigzaglines denote gauge bosons, W ± or Z , dashed lines denote the physical Higgs boson, h , and dottedlines denote Goldstone bosons, φ ± W or φ Z .We compute the one-loop corrections to the amputated two-point function, Σ , from virtual Z , W ± , h , φ Z and φ ± W exchange, as illustrated in Fig. 1. In the following results, we set the externalmomentum to zero (i.e., we compute Σ (0)), and the first (second) subscript denotes the final (initial)state, with values (1 , , , ,
5) corresponding to the mass eigenstates ( h (0)0 , h (+)0 , h ( − )0 , h (0)+ , h (0) − ). UsingFeynman-t’Hooft gauge, and expressing results in terms of the basic integral I ( δ, m ) of Appendix B,we find − i [Σ (0)] = − g c W c ρ I ( δ ( − )0 , m Z ) − g c W s ρ I ( δ (+)0 , m Z ) − g I ( δ (0) ± , m W )+ a c ρ I ( δ (+)0 , m Z ) + a s ρ I ( δ ( − )0 , m Z ) , − i [Σ (0)] = − g c W s ρ I ( δ (0)0 , m Z ) − g s ρ I ( δ (0) ± , m W ) + a s ρ I ( δ (+)0 , m h )+ a c ρ I ( δ ( − )0 , m h ) + a c ρ I ( δ (0)0 , m Z ) + 2 a c ρ I ( δ (0) ± , m W ) , − i [Σ (0)] = − i [Σ (0)] = − g c W s ρ I ( δ (0)0 , m Z ) − g s ρ I ( δ (0) ± , m W ) + a s ρ c ρ I ( δ (+)0 , m h ) − a s ρ c ρ I ( δ ( − )0 , m h ) − a s ρ I ( δ (0)0 , m Z ) − a s ρ I ( δ (0) ± , m W ) , − i [Σ (0)] = − g c W c ρ I ( δ (0)0 , m Z ) − g c ρ I ( δ (0) ± , m W ) + a s ρ I ( δ ( − )0 , m h )12 a c ρ I ( δ (+)0 , m h ) + a s ρ I ( δ (0)0 , m Z ) + 2 a s ρ I ( δ (0) ± , m W ) , − i [Σ (0)] = − i [Σ (0)] = − e I ( δ (0) ± , λ ) − g c W (1 − s W ) I ( δ (0) ± , m Z ) − g I ( δ (0)0 , m W ) − g s ρ I ( δ (+)0 , m W ) − g c ρ I ( δ ( − )0 , m W ) + a c ρ I ( δ (+)0 , m W )+ a s ρ I ( δ ( − )0 , m W ) , (34)where λ is a fictitious photon mass, and the self-energy components not displayed above vanish. Wemay evaluate Σ( v · k ) by the substitution I ( δ, m ) → I ( δ − v · k, m ). Let us fix the counterterms δa , δa , δµ S , δµ D and δZ S by enforcing that the physical residual massesof the neutral states are given by the renormalized parameters of the lagrangian,[ δµ ] + Re[Σ ( δ (0) ± )] − δ (0)0 [ δZ h ] = 0 , [ δµ ] + Re[Σ ( δ (+)0 )] − δ (+)0 [ δZ h ] = 0 , [ δµ ] + Re[Σ (0)] = 0 , (35)and that the lightest mass eigenstate is proportional to the vector (0 , , , , δµ ] + Re[Σ (0)] = 0 , [ δµ ] + Re[Σ (0)] = 0 . (36)This scheme defines renormalized values for a and t ρ through the physical mass differences betweenneutral states, M h (+)0 − M h ( − )0 = 2 avs − ρ ,M h (0)0 − M h ( − )0 = avt ρ , (37)where the mass of the neutral mass eigenstate h ( · )0 is denoted M h ( · )0 . Note also that the presence of δZ S (cid:54) = δZ D is required to maintain the orientation of the lightest mass eigenstate under renormal-ization. Solving for the counterterms, we find from [ δµ ] , δa a = δa a = ⇒ a δa + a δa = a δa a . (38)The remaining system of equations involving [ δµ ] , [ δµ ] , [ δµ ] and [ δµ ] then yields av δa a = − [ δµ ] + t − ρ ([ δµ ] − [ δµ ] )= [Σ (0)] + t − ρ (cid:16) [Σ (0)] − [Σ ( δ (0)0 )] + δ (0)0 [ δZ h ] (cid:17) , Z S = δZ D + 1 av (cid:26) t ρ [Σ ( δ (+)0 )] + 2[Σ (0)] + t − ρ [Σ (0)] − s − ρ [Σ ( δ (0)0 )] (cid:27) . (39)We focus here on the counterterms δa , δa , and δZ S which enter in the calculation of amplitudesrelevant for WIMP-nucleon scattering. Explicit expressions for the remaining counterterms δµ S and δµ D may be similarly obtained. We note that the degeneracy between the mass of the h (0)0 state andthe h (0) ± states is lifted by a finite amount, predicted in terms of renormalized parameters as M h (0) ± − M h (0)0 = [Σ ( δ (0) ± )] − [Σ ( δ (0)0 )] , (40)where we have used that [ δµ ] = [ δµ ] , [ δZ h ] = [ δZ h ] and δ (0)0 = δ (0) ± . The extension to the triplet-doublet case is straightforward. The counterterms δa , δa , δµ T , δµ D , δZ T and δZ D are introduced in an analogous manner. In terms of the mass eigenbasis h = (cid:16) h (0)0 , h (+)0 , h ( − )0 , h (+)+ , h ( − )+ , h (+) − , h ( − ) − (cid:17) introduced in Sec. 4.2.2, the counterterms are given bythe 7 × δµ = δµ D + δµ δµ +
00 0 δµ − , δZ h = δZ D + δZ δZ +
00 0 δZ − , (41)where the submatrices for the neutral and charged sectors are specified by the following symmetricmatrices, δµ = | c ρ | va ( a δa − a δa ) | s ρ | va ( a δa − a δa ) · c ρ ( δ ∆) + s ρ va ( a δa + a δa ) − s ρ ( δ ∆) + c ρ va ( a δa + a δa ) · · s ρ ( δ ∆) − s ρ va ( a δa + a δa ) ,δµ ± = c ρ ( δ ∆) + s ρ va ( a δa + a δa ) − s ρ ( δ ∆) + c ρ va ( a δa + a δa ) ± i va ( a δa − a δa ) · s ρ ( δ ∆) − s ρ va ( a δa + a δa ) ,δZ = · c ρ ( δZ T − δZ D ) − s ρ ( δZ T − δZ D ) · · s ρ ( δZ T − δZ D ) ,δZ ± = c ρ ( δZ T − δZ D ) − s ρ ( δZ T − δZ D ) · s ρ ( δZ T − δZ D ) , (42)with ( δ ∆) = ( δµ T − δµ D ) /
2. To fix counterterms, we impose the same renormalization conditionsgiven in (35) and (36). We again require the one-loop corrections to the two-point function, Σ .In the following results, the first (second) subscript denotes the final (initial) state, with values(1 , , , , , ,
7) corresponding to the mass eigenstates (cid:16) h (0)0 , h (+)0 , h ( − )0 , h (+)+ , h ( − )+ , h (+) − , h ( − ) − (cid:17) . Using14eynman-t’Hooft gauge and expressing results in terms of the basic integral I ( δ, m ) of Appendix B,we find − i [Σ (0)] = − g c W s ρ I ( δ (+)0 , m Z ) − g c W c ρ I ( δ ( − )0 , m Z ) − g s ρ I ( δ (+) ± , m W ) − g c ρ I ( δ ( − ) ± , m W ) + a c ρ I ( δ (+)0 , m Z ) + a s ρ I ( δ ( − )0 , m Z )+ 2 a c ρ I ( δ (+) ± , m W ) + 2 a s ρ I ( δ ( − ) ± , m W ) , − i [Σ (0)] = − g c W s ρ I ( δ (0)0 , m Z ) − g (cid:16) c ρ (cid:17) I ( δ (+) ± , m W ) − g s ρ I ( δ ( − ) ± , m W )+ a c ρ I ( δ (0)0 , m Z ) + 2 a I ( δ ( − ) ± , m W ) + a c ρ I ( δ ( − )0 , m h ) + a s ρ I ( δ (+)0 , m h ) , − i [Σ (0)] = − i [Σ (0)] = − g c W s ρ I ( δ (0)0 , m Z ) + g s ρ (cid:16) c ρ (cid:17) I ( δ (+) ± , m W ) + g s ρ (cid:16) s ρ (cid:17) I ( δ ( − ) ± , m W ) − a s ρ I ( δ (0)0 , m Z ) + a c ρ s ρ I ( δ (+)0 , m h ) − a c ρ s ρ I ( δ ( − )0 , m h ) , − i [Σ (0)] = − g c W c ρ I ( δ (0)0 , m Z ) − g (cid:16) s ρ (cid:17) I ( δ ( − ) ± , m W ) − g s ρ I ( δ (+) ± , m W )+ a s ρ I ( δ (0)0 , m Z ) + 2 a I ( δ (+) ± , m W ) + a c ρ I ( δ (+)0 , m h ) + a s ρ I ( δ ( − )0 , m h ) , − i [Σ (0)] = − i [Σ (0)] = − g c W (cid:18) c W − s ρ (cid:19) I ( δ (+) ± , m Z ) − g c W s ρ I ( δ ( − ) ± , m Z ) − g s ρ I ( δ (0)0 , m W ) − e I ( δ (+) ± , λ ) − g (cid:16) c ρ (cid:17) I ( δ (+)0 , m W ) − g s ρ I ( δ ( − )0 , m W ) + a I ( δ ( − ) ± , m Z )+ a I ( δ ( − )0 , m W ) + a c ρ I ( δ (0)0 , m W ) + a s ρ I ( δ (+) ± , m h ) + a c ρ I ( δ ( − ) ± , m h ) , − i [Σ (0)] = − i [Σ (0)] = − i [Σ (0)] = − i [Σ (0)] = g c W s ρ (cid:18) c W − s ρ (cid:19) I ( δ (+) ± , m Z ) + g c W s ρ (cid:18) c W − c ρ (cid:19) I ( δ ( − ) ± , m Z ) − g s ρ I ( δ (0)0 , m W ) + g s ρ (cid:16) c ρ (cid:17) I ( δ (+)0 , m W ) + g s ρ (cid:16) s ρ (cid:17) I ( δ ( − )0 , m W ) − a s ρ I ( δ (0)0 , m W ) + a c ρ s ρ I ( δ (+) ± , m h ) − a c ρ s ρ I ( δ ( − ) ± , m h ) , − i [Σ (0)] = − i [Σ (0)] − g c W s ρ I ( δ (+) ± , m Z ) − g c W (cid:18) c W − c ρ (cid:19) I ( δ ( − ) ± , m Z ) − g c ρ I ( δ (0)0 , m W ) − e I ( δ ( − ) ± , λ ) − g s ρ I ( δ (+)0 , m W ) − g (cid:16) s ρ (cid:17) I ( δ ( − )0 , m W ) + a I ( δ (+) ± , m Z )+ a I ( δ (+)0 , m W ) + a s ρ I ( δ (0)0 , m W ) + a c ρ I ( δ (+) ± , m h ) + a s ρ I ( δ ( − ) ± , m h ) , (43)where λ is a fictitious photon mass, and the self-energy components not displayed above vanish. Theremainder of the renormalization program proceeds as for the singlet-doublet system. In particular,the similarity of the neutral sectors implies relations similar to (39), av δa a = av δa a = [Σ (0)] + t − ρ (cid:16) [Σ (0)] − [Σ ( δ (0)0 )] + δ (0)0 [ δZ h ] (cid:17) ,δZ T = δZ D + 1 av (cid:26) t ρ [Σ ( δ (+)0 )] + 2[Σ (0)] + t − ρ [Σ (0)] − s − ρ [Σ ( δ (0)0 )] (cid:27) , (44)where the self-energy components are those of the triplet-doublet system given in (43). This section describes the matching of the effective theory described by (6) onto the effective theorydescribed by (2), through integrating out weak-scale particles, W ± , Z , h , φ Z , φ ± W , and t . Thecomplete basis of twelve bare matching coefficients, c (0) q , c (2) q , c (0) g , and c (2) g , are determined at leadingorder in perturbation theory.We may write the quark and gluon matching coefficients in terms of contributions from one-bosonexchange (1BE) and two-boson exchange (2BE) diagrams, c (0) q = c (0) q + c (0) q + . . . ,c (0) g = c (0) g + c (0) g + . . . ,c (2) q = c (2) q + . . . ,c (2) g = c (2) g + . . . , (45)where the ellipses denote subleading contributions with more than two bosons exchanged. Note thatspin-2 coefficients do not receive contributions from one-boson exchange amplitudes.In the following analysis, we denote generic up- and down-type quarks by U and D , respectively,and an arbitrary quark flavor by q . We specify the contributions to the matching coefficients in termsof the constants c ( U ) V = 1 − s W , c ( D ) V = − s W , c ( U ) A = − , c ( D ) A = 1 . (46)We systematically neglect subleading corrections involving light quark masses, and use CKM unitarityto simplify sums over quark flavors. Together with | V tb | ≈ | V td | ≈ | V ts | ≈ c ( S ) u = c ( S ) c and c ( S ) d = c ( S ) s for both S = 0 ,
2, reducing the number of independentmatching coefficients to eight. When the interactions are isospin-conserving, e.g., as in the pure tripletcase, we furthermore have c ( S ) u = c ( S ) d and c ( S ) c = c ( S ) s for both S = 0 ,
2, leaving only six independent16oefficients. We use Feynman-t’Hooft gauge for the electroweak sector, and neglect higher-ordercorrections to the tree-level relations between residual masses, δ (0)0 = δ (0) ± for the singlet-doubletsystem, and δ ( ± )0 = δ ( ± ) ± for the triplet-doublet system. In Secs. 6.1 and 6.3, we match to quarkoperators using onshell external quarks, and thus use the equivalence of m q u q ( p ) and p/u q ( p ). (cid:6) + (cid:7) + (cid:8) + (cid:9) + (cid:10) + (cid:11) + (cid:12) + . . . + (cid:13) + . . . = c (0) q (cid:14) Figure 2: Matching condition for one-boson exchange contributions to quark operators. The fulltheory diagrams on the left-hand side illustrate the possible types of contributions to the h ¯ χχ three-point function. Time-reversed diagrams are not shown. Double lines denote heavy WIMPs, zigzaglines denote gauge bosons, W ± or Z , dotted lines denote Goldstone bosons, φ ± W or φ Z , dashed linesdenote the physical Higgs boson, h , and single lines with arrows denote quarks. The solid circledenotes counterterm contributions. The solid square denotes effective theory vertices.The matching condition for one-boson exchange is pictured in Fig. 2. The full-theory amplitudeis given by i M q = i (cid:16) ˆ M tree + ˆ M vertex , + ˆ M vertex , + ˆ M δa + ˆ M δZ + ˆ M δv (cid:17) i − m h − ig m q m W ¯ u q ( p ) u q ( p ) , (47)where the ˆ M i are contributions to the h ¯ χχ three-point function. These come from tree-level Higgsexchange ( ˆ M tree ), one-loop diagrams with Higgs coupling to W ± or Z ( ˆ M vertex , ), one-loop vertexcorrections with Higgs coupling to the heavy particle ( ˆ M vertex , ), the δa counterterm ( ˆ M δa ), wave-function renormalization ( ˆ M δZ ), and the renormalization of the Higgs vacuum expectation value( ˆ M δv ). Having included the counterterms, the sum of these contributions is finite. The one-bosonexchange contribution to the spin-0 quark matching coefficient is thus c (0) q = − g m h m W (cid:16) ˆ M tree + ˆ M vertex , + ˆ M vertex , + ˆ M δa + ˆ M δZ + ˆ M δv (cid:17) . (48)We neglect one-boson exchange contributions containing O ( α ) corrections to the SM h ¯ qq coupling,shown in Fig. 2 within square brackets. This gauge-invariant class of diagrams is loop-suppressedrelative to the tree-level diagram for any value of the h ¯ χχ coupling. On the other hand, the remain-ing loop diagrams (including those in Fig. 4) may compete with, or even dominate, the tree-levelcontribution depending on the size of the h ¯ χχ coupling. Let us proceed to specify the contributions,ˆ M i , for each SM extension in terms of the integrals I ( δ, m ), I ( δ, m ), I ( δ, m ) and I ( δ , δ , m ) ofAppendix B. 17 .1.1 Pure states For pure states the only diagrams are those with Higgs coupling to W ± and Z , and in terms of theconstants C W and C Z specified in (9) the amplitude is given by i ˆ M vertex , = −C Z g c W m Z I (0 , m Z ) − C W g m W I (0 , m W ) . (49)Using (48), we find the contribution to the spin-0 quark matching coefficient, c (0) q = π Γ(1 + (cid:15) ) g (4 π ) − (cid:15) (cid:26) − m − − (cid:15)W x h (cid:18) C W + C Z c W (cid:19) + O ( (cid:15) ) (cid:27) , (50)where x h = m h /m W . The pure triplet (doublet) result is obtained by setting C W = 2 and C Z = 0( C W = 1 / C Z = 1 /
4) above.
For the singlet-doublet case, we have the following contributions to the h ¯ χχ three-point function, i ˆ M tree = ias ρ , i ˆ M δa = ias ρ δa a , i ˆ M δZ = ias ρ δZ χ , i ˆ M δv = ias ρ δvv ,i ˆ M vertex , = − g c W c ρ m Z I ( δ (0)0 , m Z ) + g a c W s ρ I ( δ (0)0 , m Z ) + g a s ρ m h m W I ( δ (0)0 , m Z )+ 3 g a m h m W (cid:104) s ρ I ( δ ( − )0 , m h ) + c ρ I ( δ (+)0 , m h ) (cid:105) − g c ρ m W I ( δ (0)0 , m W )+ g a s ρ I ( δ (0)0 , m W ) + g a s ρ m h m W I ( δ (0)0 , m W ) ,i ˆ M vertex , = − a s ρ I ( δ ( − )0 , δ ( − )0 , m h ) + a s ρ c ρ I ( δ (+)0 , δ (+)0 , m h ) − a s ρ c ρ I ( δ ( − )0 , δ (+)0 , m h ) , (51)where δa is given in (39), the onshell Z factor is given by Z − χ − − δZ χ = [ δZ h ] − ∂∂v · k [Σ ( v · k )] = δZ D − [Σ (cid:48) (0)] + 1 av s ρ (cid:26) − s − ρ [Σ ( δ (0)0 )] + t ρ [Σ ( δ (+)0 )] + 2[Σ (0)] + t − ρ [Σ (0)] (cid:27) , (52)and δv is determined by the SM result [40], δvv = 12 Σ AA (cid:48) (0) − s W c W Σ AZ (0) m Z − c W s W Re[Σ ZZ ( m Z )] m Z + c W − s W s W Re[Σ
W W ( m W )] m W −
12 Re[Σ HH (cid:48) ( m h )] . (53)The two-point functions required in (53) are specified in (121) of Appendix B. The one-bosonexchange quark matching coefficient is obtained by collecting the above amplitudes into (48). Upontaking the pure-case limits described in Sec. 4.3, we recover the results (49) and (50) for a puredoublet. In the pure singlet limit, the one-boson exchange amplitudes vanish. We are here neglecting contributions from states beyond the SM. Renormalization schemes relevant for WIMPs ofmass M ∼ m W are discussed in Refs. [41]. .1.3 Triplet-doublet admixture For the triplet-doublet case we have the following contributions to the h ¯ χχ three-point function, i ˆ M tree = ias ρ , i ˆ M δa = ias ρ δa a , i ˆ M δZ = ias ρ δZ χ , i ˆ M δv = ias ρ δvv ,i ˆ M vertex , = − g c W c ρ m Z I ( δ (0)0 , m Z ) + g a c W s ρ I ( δ (0)0 , m Z ) + g a s ρ m h m W I ( δ (0)0 , m Z )+ 3 g a m h m W (cid:2) s ρ I ( δ ( − )0 , m h ) + c ρ I ( δ (+)0 , m h ) (cid:3) − g s ρ m W I ( δ (+)0 , m W ) − g s ρ ) m W I ( δ ( − )0 , m W ) + g a s ρ I ( δ (+)0 , m W ) + g a m h m W I ( δ (+)0 , m W ) ,i ˆ M vertex , = − g a s ρ I ( δ (+)0 , δ (+)0 , m W ) + g a s ρ ) s ρ c ρ I ( δ ( − )0 , δ (+)0 , m W )+ g a s ρ ) s ρ I ( δ ( − )0 , δ ( − )0 , m W ) + 2 a s ρ I ( δ (+)0 , δ (+)0 , m W )+ a c ρ s ρ I ( δ (+)0 , δ (+)0 , m h ) − a c ρ s ρ I ( δ ( − )0 , δ (+)0 , m h ) − a s ρ I ( δ ( − )0 , δ ( − )0 , m h ) , (54)where δa is specified in (44) and δv in (53). The onshell Z factor takes the same form as in (52),but uses the self-energy components for the triplet-doublet system given in (43). The one-bosonexchange quark matching coefficient is obtained by collecting the above amplitudes into (48). Upontaking the pure case limits described in Sec. 4.3, we recover the results (49) and (50) for both puretriplet and pure doublet. One-boson exchange contributions to gluon matching are pictured in Fig. 3. The two-loop diagramsfactorize into separate one-loop diagrams: the boson loop given by the amplitudes ˆ M i determinedin the previous section, and the fermion loop familiar from, e.g., the top quark contribution to theeffective h ( G Aµν ) vertex (e.g., see [42]). In terms of quark matching coefficients from one-bosonexchange, c (0) q , the leading contribution to the bare gluon matching coefficient is thus c (0) g = − g (4 π ) c (0) q + O ( (cid:15) ) . (55)For the same reason discussed after Eq. (48), we neglect the one-boson exchange contributions con-taining O ( α ) corrections to the effective h ( G Aµν ) coupling, shown within square brackets in Fig. 3.In the above result for c (0) g , the light quark contributions cancel between the full and effectivetheory amplitudes, leaving only contributions from the top quark. Further discussion of effectivetheory contributions can be found in Sec. 6.5. Let us now consider quark matching from two-boson exchange, as displayed in Fig. 4. In covariantgauges, in particular Feynman-t’Hooft gauge employed here, the full theory contributions include19 + (cid:16) + (cid:17) + (cid:18) + (cid:19) + (cid:20) + (cid:21) + (cid:22) + . . . + . . .= c (0) g (cid:23) + c (0) q (cid:24) Figure 3: Matching condition for one-boson exchange contributions to gluon operators. The notationfor the different lines and vertices is as in Fig. 2. All active quark flavors, such as the top quark inthe full theory, are included in the loops. (cid:25) + (cid:26) + (cid:27) + (cid:28) + . . .= c (0) q (cid:29) + c (2) q (cid:30) Figure 4: Matching condition for two-boson exchange contributions to quark operators. The notationfor the different lines and vertices is as in Fig. 2. The full theory diagrams illustrate the possibletypes of two-boson exchange. Crossed diagrams and time-reversed diagrams are not shown.diagrams with exchange of two gauge bosons ( W ± or Z ), two Goldstone bosons ( φ Z or φ ± W ), onegauge and one Goldstone boson ( Z and φ Z , or W ± and φ ± W ), or two Higgs bosons. In terms of thesecontributions the total amplitude is M q = M ZZq + M W Wq + M W φ W q + M Zφ Z q + M φ W φ W q + M φ Z φ Z q + M hhq , (56)where the superscripts denote which bosons are exchanged, and the contributions from crossed dia-grams and time-reversed diagrams are included in each amplitude. Upon expressing the amplitudesin terms of the integrals J ( m V , M, δ ), J µ ( p, m V , M, δ ), J − ( p, m V , M, δ ) and J µ − ( m V , M, δ ) defined inAppendix C, we may write each amplitude in the form M BB (cid:48) q = ¯ u q ( p ) (cid:20) m q c (0) BB (cid:48) q + (cid:18) v/v · p − p/d (cid:19) c (2) BB (cid:48) q (cid:21) u q ( p ) , (57)20here the superscript BB (cid:48) denotes the type of two-boson exchange. The contributions to spin-0 andspin-2 quark matching coefficients can then be read off as c (0) BB (cid:48) q and c (2) BB (cid:48) q , respectively. For pure states the contributions come from diagrams with exchange of W ± or Z bosons. In termsof C W and C Z specified in (9), the amplitudes are i M ZZq = g C Z c W ¯ u q ( p ) (cid:20)(cid:2) c ( q )2 V + c ( q )2 A (cid:3) v/ (cid:2) J/ ( p, m Z , ,
0) + p/J ( m Z , , (cid:3) v/ + m q (cid:2) c ( q )2 V − c ( q )2 A (cid:3) J ( m Z , , (cid:21) u q ( p ) ,i M W WU = g C W u U ( p ) v/ (cid:2) J/ ( p, m W , ,
0) + p/J ( m W , , (cid:3) v/u U ( p ) ,i M W WD = (cid:88) U g C W | V UD | ¯ u D ( p ) v/ (cid:2) J/ ( p, m W , m U ,
0) + p/J ( m W , m U , (cid:3) v/u D ( p ) . (58)Upon writing these amplitudes in the form of (57) and evaluating integrals, we find the contributionsto the matching coefficients, c (0) U = π Γ(1 + (cid:15) ) g (4 π ) − (cid:15) (cid:26) m − − (cid:15)Z C Z c W (cid:2) c ( U )2 V − c ( U )2 A (cid:3) + O ( (cid:15) ) (cid:27) ,c (0) D = π Γ(1 + (cid:15) ) g (4 π ) − (cid:15) (cid:26) m − − (cid:15)Z C Z c W (cid:2) c ( D )2 V − c ( D )2 A (cid:3) + δ Db m − − (cid:15)W C W (cid:20) − x t x t + 1) (cid:21) + O ( (cid:15) ) (cid:27) ,c (2) U = π Γ(1 + (cid:15) ) g (4 π ) − (cid:15) (cid:26)(cid:20) m − − (cid:15)W C W + m − − (cid:15)Z C Z c W (cid:2) c ( U )2 V + c ( U )2 A (cid:3)(cid:21)(cid:20)
13 + (cid:18) −
23 log 2 (cid:19) (cid:15) (cid:21) + O ( (cid:15) ) (cid:27) ,c (2) D = π Γ(1 + (cid:15) ) g (4 π ) − (cid:15) (cid:26)(cid:20) m − − (cid:15)W C W + m − − (cid:15)Z C Z c W (cid:2) c ( D )2 V + c ( D )2 A (cid:3)(cid:21)(cid:20)
13 + (cid:18) −
23 log 2 (cid:19) (cid:15) (cid:21) + δ Db m − − (cid:15)W C W (cid:20) (3 x t + 2)3( x t + 1) −
23 + (cid:18) x t (7 x t − x t − log x t − x t + 2)3( x t + 1) log 2 − x t − x t − x t − ( x t + 1) −
229 + 43 log 2 (cid:19) (cid:15) (cid:21) + O ( (cid:15) ) (cid:27) , (59)where x t = m t /m W , and the Kronecker delta, δ Db , is equal to unity for D = b and vanishes for D = d, s . We obtain the pure triplet (doublet) result upon setting C W = 2 and C Z = 0 ( C W = 1 / C Z = 1 /
4) in (59).
For the singlet-doublet case the amplitudes for the different types of two-boson exchange are i M ZZq = g c W c ρ ¯ u q ( p ) (cid:20)(cid:2) c ( q )2 V + c ( q )2 A (cid:3) v/ (cid:2) J/ ( p, m Z , , δ (0)0 ) + p/J ( m Z , , δ (0)0 ) (cid:3) v/ m q (cid:2) c ( q )2 V − c ( q )2 A (cid:3) J ( m Z , , δ (0)0 ) (cid:21) u q ( p ) ,i M W WU = g c ρ ¯ u U ( p ) v/ (cid:2) J/ ( p, m W , , δ (0)0 ) + p/J ( m W , , δ (0)0 ) (cid:3) v/u U ( p ) ,i M W WD = (cid:88) U g c ρ | V UD | ¯ u D ( p ) v/ (cid:2) J/ ( p, m W , m U , δ (0)0 ) + p/J ( m W , m U , δ (0)0 ) (cid:3) v/u D ( p ) ,i M Zφ Z q = − g a c W s ρ m q m W ¯ u q ( p ) (cid:2) v · J − ( m Z , , δ (0)0 ) (cid:3) u q ( p ) ,i M W φ W U = − g a s ρ m U m W ¯ u U ( p ) (cid:2) v · J − ( m W , , δ (0)0 ) (cid:3) u U ( p ) ,i M W φ W D = (cid:88) U g a s ρ | V UD | ¯ u D ( p ) (cid:20) − m D m W v · J − ( m W , m U , δ (0)0 ) + v/ m U m W J − ( p, m W , m U , δ (0)0 ) (cid:21) u D ( p ) ,i M φ W φ W D = g a s ρ m t m W | V tD | ¯ u D ( p ) (cid:2) − m D J ( m W , m t , δ (0)0 ) + J/ ( p, m W , m t , δ (0)0 ) (cid:3) u D ( p ) ,i M φ W φ W U = 0 , i M φ Z φ Z q = 0 , i M hhq = 0 . (60)The amplitudes M hhq , M φ Z φ Z q , and M φ W φ W U are suppressed by light quark masses. Comparing eachamplitude above with (57), we find the contributions to spin-0 and spin-2 quark matching coefficients, c (0) q ZZ = g c W c ρ (cid:40)(cid:2) c ( q )2 V − c ( q )2 A (cid:3) J ( m Z , , δ (0)0 )+ (cid:2) c ( q )2 V + c ( q )2 A (cid:3) (cid:20) − J ( m Z , , δ (0)0 ) − J ( m Z , , δ (0)0 ) + 1 d ˆ J ( m Z , , δ (0)0 ) (cid:21) (cid:41) ,c (2) q ZZ = g c W c ρ (cid:2) c ( q )2 V + c ( q )2 A (cid:3) ˆ J ( m Z , , δ (0)0 ) ,c (0) U W W = g c ρ (cid:20) − J ( m W , , δ (0)0 ) − J ( m W , , δ (0)0 ) + 1 d ˆ J ( m W , , δ (0)0 ) (cid:21) ,c (2) U W W = g c ρ ˆ J ( m W , , δ (0)0 ) ,c (0) D W W = (cid:88) U g c ρ | V UD | (cid:20) − J ( m W , m U , δ (0)0 ) − J ( m W , m U , δ (0)0 ) + 1 d ˆ J ( m W , m U , δ (0)0 ) (cid:21) ,c (2) D W W = (cid:88) U g c ρ | V UD | ˆ J ( m W , m U , δ (0)0 ) ,c (0) q Zφ Z = − g a c W s ρ m W J − ( m Z , , δ (0)0 ) ,c (2) q Zφ Z = 0 , (0) U W φ W = − g a s ρ m W J − ( m W , , δ (0)0 ) ,c (2) U W φ W = 0 ,c (0) D W φ W = (cid:88) U g a s ρ | V UD | (cid:20) − m W J − ( m W , m U , δ (0)0 ) + 1 d m U m W J − ( m W , m U , δ (0)0 ) (cid:21) ,c (2) D W φ W = (cid:88) U g a s ρ | V UD | m U m W J − ( m W , m U , δ (0)0 ) ,c (0) D φ W φ W = g a s ρ m t m W | V tD | (cid:20) J ( m W , m t , δ (0)0 ) − J ( m W , m t , δ (0)0 ) + 1 d J ( m W , m t , δ (0)0 ) (cid:21) ,c (2) D φ W φ W = g a s ρ m t m W | V tD | J ( m W , m t , δ (0)0 ) , (61)where we have definedˆ J ( m x , m y , δ z ) ≡ J ( m x , m y , δ z ) + 2 J ( m x , m y , δ z ) + 2 J ( m x , m y , δ z ) . (62)The integrals J ( m V , M, δ ), J ( m V , M, δ ), J ( m V , M, δ ), J − ( m V , M, δ ) and J − ( m V , M, δ ) are givenin Appendix C. The matching coefficients c (0) q and c (2) q for a given quark q are obtained bysumming the nonvanishing contributions above, c (0) U = c (0) U ZZ + c (0) U W W + c (0) U Zφ Z + c (0) U W φ W ,c (0) D = c (0) D ZZ + c (0) D W W + c (0) D Zφ Z + c (0) D W φ W + c (0) D φ W φ W ,c (2) U = c (2) U ZZ + c (2) U W W ,c (2) D = c (2) D ZZ + c (2) D W W + c (2) D W φ W + c (2) D φ W φ W . (63)Upon taking the pure-case limits described in Sec. 4.3, we recover the results (58) and (59) for a puredoublet. In the pure singlet limit, the two-boson exchange amplitudes vanish. We may similarly compute the two-boson exchange amplitudes for the triplet-doublet system, andupon comparing with (57), we find the following contributions to spin-0 and spin-2 quark matchingcoefficients, c (0) q ZZ = g c W c ρ (cid:40)(cid:2) c ( q )2 V − c ( q )2 A (cid:3) J ( m Z , , δ (0)0 )+ (cid:2) c ( q )2 V + c ( q )2 A (cid:3) (cid:18) − J ( m Z , , δ (0)0 ) − J ( m Z , , δ (0)0 ) + 1 d ˆ J ( m Z , , δ (0)0 ) (cid:19) (cid:41) ,c (2) q ZZ = g c W c ρ (cid:2) c ( q )2 V + c ( q )2 A (cid:3) ˆ J ( m Z , , δ (0)0 ) , (0) U W W = g (cid:40) (1 + s ρ ) (cid:20) − J ( m W , , δ ( − )0 ) − J ( m W , , δ ( − )0 ) + 1 d ˆ J ( m W , , δ ( − )0 ) (cid:21) + 14 s ρ (cid:20) − J ( m W , , δ (+)0 ) − J ( m W , , δ (+)0 ) + 1 d ˆ J ( m W , , δ (+)0 ) (cid:21) (cid:41) ,c (2) U W W = g (cid:20) (1 + s ρ ) ˆ J ( m W , , δ ( − )0 ) + 14 s ρ ˆ J ( m W , , δ (+)0 ) (cid:21) ,c (0) D W W = (cid:88) U g | V UD | (cid:40) (1 + s ρ ) (cid:20) − J ( m W , m U , δ ( − )0 ) − J ( m W , m U , δ ( − )0 ) + 1 d ˆ J ( m W , m U , δ ( − )0 ) (cid:21) + 14 s ρ (cid:20) − J ( m W , m U , δ (+)0 ) − J ( m W , m U , δ (+)0 ) + 1 d ˆ J ( m W , m U , δ (+)0 ) (cid:21) (cid:41) ,c (2) D W W = (cid:88) U g | V UD | (cid:20) (1 + s ρ ) ˆ J ( m W , m U , δ ( − )0 ) + 14 s ρ ˆ J ( m W , m U , δ (+)0 ) (cid:21) ,c (0) q Zφ Z = − g a c W s ρ m W J − ( m Z , , δ (0)0 ) ,c (2) q Zφ Z = 0 ,c (0) U W φ W = − g a s ρ m W J − ( m W , , δ (+)0 ) ,c (2) U W φ W = 0 ,c (0) D W φ W = (cid:88) U g a s ρ | V UD | (cid:20) − m W J − ( m W , m U , δ (+)0 ) + 1 d m U m W J − ( m W , m U , δ (+)0 ) (cid:21) ,c (2) D W φ W = (cid:88) U g a s ρ | V UD | m U m W J − ( m W , m U , δ (+)0 ) ,c (0) D φ W φ W = g a m t m W | V tD | (cid:20) J ( m W , m t , δ (+)0 ) − J ( m W , m t , δ (+)0 ) + 1 d J ( m W , m t , δ (+)0 ) (cid:21) ,c (2) D φ W φ W = g a m t m W | V tD | J ( m W , m t , δ (+)0 ) , (64)where ˆ J ( m x , m y , δ z ) is given in (62), and the relevant integrals can be found in Appendix C. Thetotal matching coefficients c (0) q and c (2) q are obtained by summing the contributions above asin (63). Upon taking the pure-case limits described in Sec. 4.3, we recover the results (58) and (59)for both pure triplet and pure doublet. The gluon matching condition for two-boson exchange is pictured in Fig. 5. If we consider the externalgluons as a background field [10], we may express the full theory diagrams in terms of electroweak24 + + ! + " + + $ + % + & + ’ + ( + ) + * + . . . = c (0) g + + c (0) q , + c (2) g - + c (2) q . + / Figure 5: Matching condition for two-boson exchange contributions to gluon operators. The notationfor the different lines and vertices is as in Fig. 2. The diagrams with a quark loop are obtainedby closing the external legs of the corresponding diagrams in Fig. 4, and considering the possibleattachments of two external gluons. All active quark flavors, such as the top quark in the full theory,are included in the loops.polarization tensors induced by a loop of quarks. For example, using the Feynman rules for theWIMP- Z coupling from (6), the contributions from exchanging two Z bosons may be written as M ZZ ∼ (cid:90) ( dL ) 1 − v · L − δ + i L − m Z + i v µ v ν i Π µν ( ZZ ) ( L ) , (65)where ( dL ) = d d L/ (2 π ) d (this shorthand notation is used throughout this work), δ is a residualmass depending on the intermediate WIMP state, and Π µν ( ZZ ) ( L ) is the two-gluon part of the Z boson polarization tensor in a background gluon field. The amplitudes with exchange of one gaugeand one Goldstone boson, two Goldstone bosons, or two Higgs bosons, have the same structurebut with vector and scalar electroweak polarization tensors appearing. A similar analysis of gluoncontributions to DM-nucleon scattering in Ref. [3] focused on the spin-0 operator. Here we performa complete matching for both spin-0 and spin-2 gluon operators, and consider new contributionsappearing in the case of mixed states.The background field method presents the following strategy for evaluating the two-loop diagramsof the full theory. First, we determine the two-gluon part of the relevant polarization tensors. Theseamplitudes depend only on SM parameters, and can be used for gluon matching in general DMscenarios; in particular, this part of the computation is independent of whether the heavy-particle25xpansion is employed. Second, we insert the polarization tensors into the boson loop and perform theremaining integrals. We illustrate this second part by identifying a basis of heavy-particle integralsto compute the universal heavy WIMP limit.In our evaluation we neglect subleading corrections of O ( m q /m W ) for light quarks ( q (cid:54) = t ). Thetwo-loop diagrams in the full theory (cf. Fig. 5) are UV finite, and may be evaluated in d = 4.However, we regulate the effective theory with dimensional regularization, and in performing theeffective theory subtractions to determine Wilson coefficients it is convenient to also use dimensionalregularization as IR regulator. Thus we choose to implement dimensional regularization as IR regu-lator also in the full theory. When considering only those terms contributing to the scalar operatorsappearing in (2), the relevant amplitudes do not involve γ or (cid:15) µναβ . In particular, the specificationof γ for d (cid:54) = 4 is unnecessary. Further discussion of effective theory contributions can be found inSec. 6.5. Let us isolate the two-gluon amplitude of the relevant electroweak polarization tensors in a back-ground gluon field. The generalized polarization tensors appearing in two-boson exchange contribu-tions are i Π νµ ( W + W + ) ( L ) = UDµ ν = − (cid:88) U,D g | V UD | (cid:90) d d x e iL · x (cid:104) T { ¯ D ( x ) γ ν (1 − γ ) U ( x ) ¯ U (0) γ µ (1 − γ ) D (0) }(cid:105) ,i Π νµ ( ZZ ) ( L ) = qqµ ν = − (cid:88) q g c W (cid:90) d d x e iL · x (cid:104) T { ¯ q ( x ) γ ν ( c ( q ) V + c ( q ) A γ ) q ( x )¯ q (0) γ µ ( c ( q ) V + c ( q ) A γ ) q (0) }(cid:105) ,i Π µ ( W + φ + W ) ( L ) = UDµ = (cid:88) U,D g | V UD | m W (cid:90) d d x e iL · x (cid:104) T { ¯ D ( x ) (cid:2) − ( m U − m D ) − ( m U + m D ) γ (cid:3) U ( x )¯ U (0) γ µ (1 − γ ) D (0) }(cid:105) ,i Π µ ( Zφ Z ) ( L ) = qqµ = (cid:88) q ig m q c W m W (cid:90) d d x e iL · x (cid:104) T { ¯ q ( x ) c ( q ) A γ q ( x )¯ q (0) γ µ ( c ( q ) V + c ( q ) A γ ) q (0) }(cid:105) ,i Π ( φ + W φ + W ) ( L ) = UD − (cid:88) UD g | V UD | m W (cid:90) d d x e iL · x (cid:104) T { ¯ D ( x ) (cid:2) − ( m U − m D ) − ( m U + m D ) γ (cid:3) U ( x )¯ U (0) (cid:2) − ( m U − m D ) + ( m U + m D ) γ (cid:3) D (0) }(cid:105) ,i Π ( φ Z φ Z ) ( L ) = qq = (cid:88) q g m q m W (cid:90) d d x e iL · x (cid:104) T { ¯ q ( x ) γ q ( x )¯ q (0) γ q (0) }(cid:105) ,i Π ( hh ) ( L ) = qq = − (cid:88) q g m q m W (cid:90) d d x e iL · x (cid:104) T { ¯ q ( x ) q ( x )¯ q (0) q (0) }(cid:105) , (66)where the momentum L is flowing from left to right in the above diagrams. We also require thepolarization tensorsΠ µν ( W − W − ) ( L ) , Π µ ( W − φ − W ) ( L ) , Π µ ( φ ± W W ± ) ( L ) , Π µ ( φ Z Z ) ( L ) , Π ( φ − W φ − W ) ( L ) , (67)which are related to those we have specified in (66) through the identities in (84).Let us now focus on the object, i ˜Π( L ) ≡ (cid:90) d d x e iL · x (cid:104) T { ¯ q (cid:48) ( x )Γ q ( x )¯ q (0)Γ (cid:48) q (cid:48) (0) }(cid:105) , (68)where Γ and Γ (cid:48) denote the possible Dirac structures whose indices we here suppress. The sum overquark mass eigenstates and other prefactors appearing in (66) are included in the final result for thepolarization tensors. Let us write ˜Π in terms of momentum-space propagators in a background field, i ˜Π( L ) = − (cid:90) d d x e iL · x Tr (cid:2) Γ iS ( q ) ( x, (cid:48) iS ( q (cid:48) ) (0 , x ) (cid:3) = (cid:90) ( dp )Tr (cid:2) Γ S ( q ) ( p )Γ (cid:48) ˜ S ( q (cid:48) ) ( p − L ) (cid:3) , (69)where iS ( q ) ( x, y ) = (cid:104) T { q ( x )¯ q ( y ) }(cid:105) (70)and we have used S ( q ) ( p ) ≡ (cid:90) d d x e ip · x S ( q ) ( x, , ˜ S ( q ) ( p ) ≡ (cid:90) d d x e − ip · x S ( q ) (0 , x ) . (71)We may expand the background field propagators at weak coupling, iS ( p ) = ip/ − m + g (cid:90) ( dq ) ip/ − m iA/ ( q ) ip/ − q/ − m + g (cid:90) ( dq )( dq ) ip/ − m iA/ ( q ) ip/ − q/ − m iA/ ( q ) ip/ − q/ − q/ − m + . . . ,i ˜ S ( p ) = ip/ − m + g (cid:90) ( dq ) ip/ + q/ − m iA/ ( q ) ip/ − m g (cid:90) ( dq )( dq ) ip/ + q/ + q/ − m iA/ ( q ) ip/ + q/ − m iA/ ( q ) ip/ − m + . . . , (72)and upon insertion of these expressions into (69), the terms with two gluon fields are readily identified.Furthermore, in Fock-Schwinger gauge the gluon field can be written as A/ ( q ) = t a γ α (cid:90) d d x e iq · x A aα ( x )= t a γ α (cid:20) − i ∂∂q ρ G aρα (0)(2 π ) d δ d ( q ) + . . . (cid:21) , (73)where the ellipsis denotes terms with derivatives acting on G aµν . Thus the amplitudes with gluonemission are given directly in terms of field-strengths, and intermediate steps involving gauge-variantcombinations can be avoided.In isolating the two-gluon amplitude, we may separately consider three cases depending on wherethe gluons are attached. Contributions with both gluons attached to the upper quark line in (66) arereferred to as“ a -type”, those with both gluons attached to the lower quark line in (66) are referredto as “ b -type”, and those with one gluon attached to each of the upper and lower quark lines arereferred to as “ c -type”. We thus have˜Π( L ) = ˜Π a ( L ) + ˜Π b ( L ) + ˜Π c ( L ) , (74)with i ˜Π a ( L ) = − g t a t b ) G aρα (0) G bστ (0) (cid:90) ( dp ) ∂∂q ρ ∂∂q (cid:48) σ Tr (cid:20) Γ 1 p/ − m γ α p/ − q/ − m γ τ p/ − q/ − q/ (cid:48) − m Γ (cid:48) p/ − L/ − m (cid:21) q = q (cid:48) =0 ,i ˜Π b ( L ) = − g t a t b ) G aρα (0) G bστ (0) (cid:90) ( dp ) ∂∂q ρ ∂∂q (cid:48) σ Tr (cid:20) Γ 1 p/ − m Γ (cid:48) p/ − L/ + q/ + q/ (cid:48) − m γ α p/ − L/ + q/ (cid:48) − m γ τ p/ − L/ − m (cid:21) q = q (cid:48) =0 ,i ˜Π c ( L ) = − g t a t b ) G aρα (0) G bστ (0) (cid:90) ( dp ) ∂∂q ρ ∂∂q (cid:48) σ Tr (cid:20) Γ 1 p/ − m γ α p/ − q/ − m Γ (cid:48) p/ − L/ + q/ (cid:48) − m γ τ p/ − L/ − m (cid:21) q = q (cid:48) =0 , (75)where m and m are the masses of the quarks in the upper and lower lines in (66), respectively.To project these onto the spin-0 and spin-2 QCD gluon operators, O (0) g and O (2) g in (3), considerthe four-index tensor T αργδ = G Aαρ G Aγδ with index symmetries T αργδ = T γδαρ = − T ραγδ . We candecompose T into components T = T (0) + T (2) + ∆ T , where T (0) αργδ = 1 d ( d − O (0) g ( g αγ g ρδ − g αδ g ργ ) ,T (2) αργδ = 1 d − (cid:16) − g αγ O (2) g ρδ + g αδ O (2) g ργ − g ρδ O (2) g αγ + g ργ O (2) g αδ (cid:17) , (76)28nd ∆ T , satisfying g αγ g ρδ (∆ T ) αργδ = v α v γ g ρδ (∆ T ) αργδ = 0 , (77)is not needed for the present analysis. The proportionality constants in T (0) and T (2) were obtainedby contraction with g αγ g ρδ or v α v γ g ρδ . Upon applying the above decomposition to the expressionsin (75), we obtain i ˜Π k ( L ) ≡ − g (cid:20) d ( d − O (0) g I (0) k ( L ) + 1 d − O (2) µνg I (2) k µν ( L ) + . . . (cid:21) , (78)where k = a, b, c and the ellipsis denotes irrelevant ∆ T contributions.Let us now determine I (0) k ( L ) and I (2) k µν ( L ) for the different cases of two-boson exchange. Thetrace and derivatives with respect to momenta q and q (cid:48) in (75) are straightforward to evaluate, andthe result is projected onto gluon operators of definite spin using (76). The quark-loop integral overmomentum p is computed using standard methods, leaving an integral over a Feynman parameter, x ,which will be evaluated after performing the boson-loop integral over momentum L . We may expressthe results in the form I ( S ) k ( L ) ≡ i Γ[1 + (cid:15) ](4 π ) − (cid:15) (cid:90) dx u k ( x ) N ( S ) k ( L ) , u a ( x ) = (1 − x ) , u b ( x ) = x , u c ( x ) = x (1 − x ) , (79)where S = 0 , S = 2 the Lorentz indices are suppressed. Let us also introduce the parameters z n ≡ ( − n − n Γ[ n + (cid:15) ]Γ[1 + (cid:15) ] , ∆ ≡ (1 − x ) m + xm − x (1 − x ) L − i , (80)which appear in the expressions for N ( S ) k ( L ) given below.For the operators of interest in (2), the relevant projections of the a - and b -type amplitudes in(75) and (76) are related by CP transformation. This condition can be stated in terms of N ( S ) k ( L ) as N ( S ) b ( L ) = N ( S ) a ( L ) (cid:12)(cid:12)(cid:12)(cid:12) x ↔ − x, m ↔ m . (81)In the case of flavor-diagonal currents ( Z , φ Z , h ) where we set m = m = m q in ˜Π k ( L ), the aboverelation implies I ( S ) b ( L ) = I ( S ) a ( L ). For flavor-changing currents ( W ± , φ ± W ) we set the down-typequark mass m = m D = 0 but keep the up-type quark mass m = m U (cid:54) = 0 to accommodatethe top quark. This asymmetry in treating the masses does not allow us to systematically recover N ( S ) b ( L ) from N ( S ) a ( L ) using the relation (81). Below we provide N ( S ) b ( L ) explicitly for flavor-changingcurrents.To illustrate the explicit implementation of this program, we again focus on the heavy WIMPlimit, retaining the leading order (in 1 /M ) WIMP-SM couplings as in (65). Anticipating the insertionof polarization tensors into the boson loop with leading order heavy-particle Feynman rules, we thuscontract the free Lorentz indices of Γ and Γ (cid:48) in (66), (68) with v µ ’s from the WIMP-vector bosonvertices. It is straightforward to analyze the remaining components of Π µν ( L ) by the same methods.The following results are labelled by the bosons in the corresponding electroweak polarization tensor.For N (0) k ( L ) we find, N (0) a ( W + W + ) = 64(3 − (cid:15) ) m U (cid:26) − (cid:15) ) z ∆ (cid:15) + x (1 − x ) (cid:0) v · L ) − L (cid:1) z ∆ (cid:15) (cid:27) , (0) b ( W + W + ) = 0 ,N (0) c ( W + W + ) = 64(1 − (cid:15) ) (cid:26) − (cid:15) )(3 − (cid:15) ) z ∆ (cid:15) + x (1 − x ) (cid:2) − (cid:15) )( v · L ) + (1 + 2 (cid:15) ) L (cid:3) z ∆ (cid:15) (cid:27) ,N (0) a ( ZZ ) = 32(3 − (cid:15) ) m q (cid:26)(cid:2) c ( q )2 V + c ( q )2 A (cid:3)(cid:20) − (cid:15) ) z ∆ (cid:15) + x (1 − x )(2( v · L ) − L ) z ∆ (cid:15) (cid:21) − (cid:2) c ( q )2 V − c ( q )2 A (cid:3)(cid:20) − (cid:15) ) z ∆ (cid:15) + x L z ∆ (cid:15) (cid:21)(cid:27) ,N (0) c ( ZZ ) = 32 (cid:26)(cid:2) c ( q )2 V + c ( q )2 A (cid:3) (1 − (cid:15) ) (cid:20) − − (cid:15) )(1 + (cid:15) ) z ∆ (cid:15) + x (1 − x ) (cid:2) − (cid:15) )( v · L ) + (1 + 2 (cid:15) ) L (cid:3) z ∆ (cid:15) (cid:21) + (cid:2) c ( q )2 V − c ( q )2 A (cid:3) (cid:15) (3 − (cid:15) ) m q z ∆ (cid:15) (cid:27) ,N (0) a ( W + φ + W ) = − − (cid:15) ) m U v · L (cid:20) (cid:2) − x − (cid:15) (1 − x ) (cid:3) z ∆ (cid:15) + x (1 − x ) L z ∆ (cid:15) (cid:21) ,N (0) b ( W + φ + W ) = 0 ,N (0) c ( W + φ + W ) = − − (cid:15) )(1 − (cid:15) )(1 − x ) m U v · L z ∆ (cid:15) ,N (0) a ( Zφ Z ) = − − (cid:15) ) c ( q )2 A m q v · L (cid:20) (cid:2) − x − (cid:15) (1 − x ) (cid:3) z ∆ (cid:15) + x (cid:2) m q + x (1 − x ) L (cid:3) z ∆ (cid:15) (cid:21) ,N (0) c ( Zφ Z ) = − − (cid:15) )(1 − (cid:15) ) c ( q )2 A m q v · L z ∆ (cid:15) ,N (0) a ( φ + W φ + W ) = 64(3 − (cid:15) ) m U (cid:2) − − (cid:15) ) z ∆ (cid:15) + x (1 − x ) L z ∆ (cid:15) (cid:3) ,N (0) b ( φ + W φ + W ) = 0 ,N (0) c ( φ + W φ + W ) = 64(1 − (cid:15) )(3 − (cid:15) ) m U (cid:2) − − (cid:15) ) z ∆ (cid:15) + x (1 − x ) L z ∆ (cid:15) (cid:3) ,N (0) a ( φ Z φ Z ) = − − (cid:15) ) xm q L z ∆ (cid:15) ,N (0) c ( φ Z φ Z ) = 32(3 − (cid:15) ) (cid:20) − (cid:15) )(2 − (cid:15) ) z ∆ (cid:15) − (cid:2) (2 − (cid:15) ) m q + (1 − (cid:15) ) x (1 − x ) L (cid:3) z ∆ (cid:15) (cid:21) ,N (0) a ( hh ) = 32(3 − (cid:15) ) m q (cid:20) − − (cid:15) ) z ∆ (cid:15) + x (1 − x ) L z ∆ (cid:15) (cid:21) ,N (0) c ( hh ) = 32(3 − (cid:15) ) (cid:20) − − (cid:15) )(2 − (cid:15) ) z ∆ (cid:15) + (cid:2) (1 − (cid:15) ) x (1 − x ) L − (2 − (cid:15) ) m q (cid:3) z ∆ (cid:15) (cid:21) . (82)For N (2) k µν ( L ) the open indices are to be contracted with O (2) µνg , which is symmetric in µ and ν and30atisfies g µν O (2) µνg = 0. The results are N (2) a µν ( W + W + ) = 128(1 − (cid:15) ) (cid:26) − − (cid:15) ) v µ v ν z ∆ (cid:15) + 2 (cid:20) ( m U − x L ) v µ v ν + 2(2 − (cid:15) ) x (1 − x ) v · Lv µ L ν − x (2 − x − (cid:15) ) L µ L ν (cid:21) z ∆ (cid:15) + x (1 − x ) (cid:20) ( m U − x ( v · L ) ) L µ L ν − m U − x L ) v · Lv µ L ν (cid:21) z ∆ (cid:15) (cid:27) ,N (2) b µν ( W + W + ) = 128(1 − (cid:15) ) (cid:26) − − (cid:15) ) v µ v ν z ∆ (cid:15) − − x ) (cid:20) (1 − x ) L v µ v ν − − (cid:15) ) xv · Lv µ L ν + (1 + x − (cid:15) ) L µ L ν (cid:21) z ∆ (cid:15) + 2 x (1 − x ) (cid:20) − ( v · L ) L µ L ν + L v · Lv µ L ν (cid:21) z ∆ (cid:15) (cid:27) ,N (2) c µν ( W + W + ) = 128 (cid:26) − (cid:15) )(1 − (cid:15) ) v µ v ν z ∆ (cid:15) + x (1 − x ) (cid:2) (cid:15)L µ L ν + 2(1 − (cid:15) ) v · Lv µ L ν − (1 − (cid:15) ) L v µ v ν (cid:3) z ∆ (cid:15) (cid:27) ,N (2) a µν ( ZZ ) = 64(1 − (cid:15) ) (cid:26)(cid:2) c ( q )2 V − c ( q )2 A (cid:3) x m q L µ L ν z ∆ (cid:15) + (cid:2) c ( q )2 V + c ( q )2 A (cid:3)(cid:20) − − (cid:15) ) v µ v ν z ∆ (cid:15) + 2 (cid:2) ( m q − x L ) v µ v ν + 2(2 − (cid:15) ) x (1 − x ) v · Lv µ L ν + x (2 − x − (cid:15) ) L µ L ν (cid:3) z ∆ (cid:15) + x (1 − x ) (cid:2) ( m q − x ( v · L ) L µ L ν − m q − x L ) v · Lv µ L ν (cid:3) z ∆ (cid:15) (cid:21)(cid:27) ,N (2) c µν ( ZZ ) = 64 (cid:26) − (cid:2) c ( q )2 V − c ( q )2 A (cid:3) − (cid:15) ) m q v µ v ν z ∆ (cid:15) + (cid:2) c ( q )2 V + c ( q )2 A (cid:3)(cid:20) − (cid:15) )(1 − (cid:15) ) v µ v ν z ∆ (cid:15) + x (1 − x ) (cid:2) (cid:15)L µ L ν + 2(1 − (cid:15) ) v · Lv µ L ν − (1 − (cid:15) ) L v µ v ν (cid:3) z ∆ (cid:15) (cid:21)(cid:27) ,N (2) a µν ( W + φ + W ) = − − (cid:15) ) xm U (cid:26) v µ L ν z ∆ (cid:15) − x (1 − x ) v · LL µ L ν z ∆ (cid:15) (cid:27) ,N (2) b µν ( W + φ + W ) = − − (cid:15) )(1 − x ) m U (cid:26) − (cid:15) ) v µ L ν z ∆ (cid:15) + (1 − x ) (cid:2) L v µ L ν − v · LL µ L ν (cid:3) z ∆ (cid:15) (cid:27) ,N (2) c µν ( W + φ + W ) = − − (cid:15) )(1 − x ) m U v µ L ν z ∆ (cid:15) ,N (2) a µν ( Zφ Z ) = (cid:2) c ( q )2 A (cid:3) − (cid:15) ) xm q (cid:26) − − (cid:15) ) v µ L ν z ∆ (cid:15) + (cid:2) ( m q − x L ) v µ L ν + xv · LL µ L ν (cid:3) z ∆ (cid:15) (cid:27) , (2) c µν ( Zφ Z ) = − (cid:2) c ( q )2 A (cid:3) − (cid:15) ) m q v µ L ν z ∆ (cid:15) ,N (2) a µν ( φ + W φ + W ) = 128(1 − (cid:15) ) xm U L µ L ν (cid:26) − (cid:15) ) z ∆ (cid:15) − (1 − x ) m U z ∆ (cid:15) (cid:27) ,N (2) b µν ( φ + W φ + W ) = 256(1 − (cid:15) )(2 − (cid:15) )(1 − x ) m U L µ L ν z ∆ (cid:15) ,N (2) c µν ( φ + W φ + W ) = 128(1 − (cid:15) ) x (1 − x ) m U L µ L ν z ∆ (cid:15) ,N (2) a µν ( φ Z φ Z ) = 64(1 − (cid:15) ) xL µ L ν (cid:26) − − (cid:15) ) z ∆ (cid:15) + m q z ∆ (cid:15) (cid:27) ,N (2) c µν ( φ Z φ Z ) = − − (cid:15) ) x (1 − x ) L µ L ν z ∆ (cid:15) ,N (2) a µν ( hh ) = 64(1 − (cid:15) ) xL µ L ν (cid:26) − (cid:15) ) z ∆ (cid:15) − (1 − x ) m q z ∆ (cid:15) (cid:27) ,N (2) c µν ( hh ) = 64(1 − (cid:15) ) x (1 − x ) L µ L ν z ∆ (cid:15) . (83)The results for N ( S ) k ( L ) in (82) and (83) specify I ( S ) a ( L ) through (79), and hence ˜Π k ( L ) through(78), and ˜Π( L ) through (74). This completes our determination of the polarization tensors in (66).The polarization tensors in (67) are obtained through the following relationsΠ µν ( W − W − ) ( L ) = Π µν ( W + W + ) ( − L ) , Π µ ( φ Z Z ) ( L ) = Π µ ( Zφ Z ) ( − L ) , Π ( φ − W φ − W ) ( L ) = Π ( φ + W φ + W ) ( − L ) , Π µ ( φ − W W − ) ( L ) = Π µ ( W + φ + W ) ( − L ) , Π µ ( φ + W W + ) ( L ) = Π µ ( W − φ − W ) ( − L ) , Π µ ( W − φ − W ) ( L ) = Π µ ( W + φ + W ) ( − L ) . (84)The identities in the first two lines are consequences of reversing the direction of momentum L inthe diagrams in (66). The last relation follows from Hermitian conjugation and the identification S ( p ) ≡ γ S ( p ) † γ = ˜ S ( p ). We note that polarization tensors with one gauge and one Goldstoneboson are odd in L , while all others are even in L . This property also holds for the corresponding N ( S ) k ( L ), and we use it in the next section to systematically reduce the boson loop integrals into aconvenient basis. Having determined the generalized polarization tensors, we now proceed with the reduction of theremaining boson loop integrals. Upon insertion of the polarization tensors into the boson loop, wefind the required set of basic loop integrals (cid:90) ( dL ) (cid:20) v · L − δ + i − v · L − δ + i (cid:21) L − m V + i N ( S ) k ( L ) ≡ I even ( δ, m V ) N ( S ) k ( L ) , (cid:90) ( dL ) (cid:20) v · L − δ + i − − v · L − δ + i (cid:21) L − m V + i N ( S ) k ( L ) ≡ I odd ( δ, m V ) N ( S ) k ( L ) , (85)32here δ is the residual mass of the intermediate WIMP state, and m V is the mass of the exchangedbosons. We suppress the arguments, ( δ, m V ), of these integral operators when making generic state-ments below. The integral operator I even requires that N ( S ) k ( L ) be even in L as in those for polar-ization tensors with a single type of boson, while the integral operator I odd requires that N ( S ) k ( L )be odd in L as in those for polarization tensors with one gauge and one Goldstone boson. Let usdenote even N ( S ) k ( L ) by N ( S ) k even ( L ) and odd N ( S ) k ( L ) by N ( S ) k odd ( L ). The subscripts even and odd maybe dropped if we mean either type, or if the exchanged bosons are specified.To reduce (85) to a set of basis integrals for evaluation, we begin by replacing factors of L in N ( S ) k ( L ) with L = − ∆ x (1 − x ) + m x + m (1 − x ) , (86)which follows from the definition of ∆ in (80). The N ( S ) k ( L ) of (82) and (83) may then be written interms of ∆ and the vectors v µ and L µ . In N ( S ) k even ( L ) each term must have two or zero v µ ’s, while in N ( S ) k odd ( L ) each term must have one v µ . Organizing the result in powers of ( v · L ), we obtain N (0) k even ( L ) = ( v · L ) (cid:88) n a (1) n ∆ − n − (cid:15) + ( v · L ) (cid:88) n a (2) n ∆ − n − (cid:15) ,N (0) k odd ( L ) = ( v · L ) (cid:88) n a (3) n ∆ − n − (cid:15) ,N (2) µνk even ( L ) = ( v · L ) (cid:88) n (cid:20) v µ v ν a (4) n ∆ − n − (cid:15) + L µ L ν a (5) n ∆ − n − (cid:15) (cid:21) + ( v · L ) (cid:88) n v µ L ν a (6) n ∆ − n − (cid:15) + ( v · L ) (cid:88) n L µ L ν a (7) n ∆ − n − (cid:15) ,N (2) µνk odd ( L ) = ( v · L ) (cid:88) n v µ L ν a (8) n ∆ − n − (cid:15) + ( v · L ) (cid:88) n L µ L ν a (9) n ∆ − n − (cid:15) , (87)where the sums run over n = 1 , , . . . , and the coefficients a ( i ) n are functions of x and (cid:15) . The above N ( S ) k ( L ) structures require the set of integrals H ( n ) = I even ∆ − n − (cid:15) , H µ ( n ) = I odd ∆ − n − (cid:15) L µ , H µν ( n ) = I even ∆ − n − (cid:15) L µ L ν ,F ( n ) = (cid:90) ( dL ) 1( L − m V + i ∆ − n − (cid:15) . (88)The integrals H µ and H µν may be expressed in terms of H ( n ) and F ( n ) through standard reductionmethods and the relation (cid:20) v · L − δ + i ± − v · L − δ + i (cid:21) v · L = δ (cid:20) v · L − δ + i ∓ − v · L − δ + i (cid:21) + 1 ∓ . (89)Furthermore, recursion relations in n may be derived by taking derivatives of parameters. A de-tailed discussion of these relations, as well as the evaluation of the above integrals, can be found inAppendix D. Note that the ( v · L ) term in N (2) µνk even ( L ) also requires the integral (cid:90) ( dL ) 1( L − m V + i ∆ − n − (cid:15) L µ L ν ∼ g µν , (90)33owever this does not contribute since it vanishes upon contraction with the traceless spin-2 gluonoperator, O (2) µνg . Upon feeding the general expressions for N ( S ) k ( L ) in (87) into the integrals in (85),we find the following decomposition in terms of basis integrals, I even N (0) k even ( L ) = (cid:88) n (cid:20) a (1) n H ( n ) + a (2) n (cid:2) δ H ( n ) + 2 δF ( n ) (cid:3)(cid:21) , I odd N (0) k odd ( L ) = (cid:88) n a (3) n (cid:2) δH ( n ) + 2 F ( n ) (cid:3) , I even N (2) µνk even ( L ) = v µ v ν (cid:88) n (cid:20) a (4) n H ( n ) + a (5) n H ( n ) + a (6) n (cid:2) δ H ( n ) + 2 δF ( n ) (cid:3) + a (7) n δ H ( n ) (cid:21) , I odd N (2) µνk odd ( L ) = v µ v ν (cid:88) n (cid:20) a (8) n (cid:2) δH ( n ) + 2 F ( n ) (cid:3) + a (9) n δH ( n ) (cid:21) , (91)where H ( n ) = 13 − (cid:15) (cid:26) (4 − (cid:15) ) (cid:2) δ H ( n ) + 2 δF ( n ) (cid:3) + H ( n − x (1 − x ) − (cid:20) m x + m − x (cid:21) H ( n ) (cid:27) . (92)The above results apply generally to both pure and mixed states. Comparing with the explicitexpressions for N ( S ) k ( L ) in (82) and (83), we find that H ( n ) for n = 1 , , F ( n ) for n = 2 , I odd is irrelevant since the only contributionsare from exchanges of W ± and Z , involving N ( S ) k even ( L ). The vanishing of certain contributions in I even N ( S ) k even ( L ) at δ = 0 can be traced to the identity in (89). Setting δ = 0 in I even N ( S ) k even ( L )above and using the explicit expressions for N ( S ) k ( L ) in (82) and (83), we find pure-state results thatdepend on H ( n ) only, I (0 , m W ) N (0) a ( W + W + ) = 64(1 + (cid:15) )(3 − (cid:15) ) m U (cid:26) (2 + (cid:15) )(1 − x ) m U H (3) − (1 + 2 (cid:15) ) H (2) (cid:27) , I (0 , m W ) N (0) c ( W + W + ) = 32(1 − (cid:15) ) (cid:26) (1 + 2 (cid:15) )(1 − x ) m U H (2) + 2(1 − (cid:15) ) H (1) (cid:27) , I (0 , m Z ) N (0) a ( ZZ ) = 32(1 + (cid:15) )(3 − (cid:15) ) m q − x (cid:26)(cid:2) c ( q )2 V + c ( q )2 A (cid:3) (1 − x ) (cid:2) (2 + (cid:15) ) m q H (3) − (1 + 2 (cid:15) ) H (2) (cid:3) + (cid:2) c ( q )2 V − c ( q )2 A (cid:3)(cid:2) (2 + (cid:15) ) xm q H (3) − (2 − (cid:15) + 2 (cid:15)x ) H (2) (cid:3)(cid:27) , I (0 , m Z ) N (0) c ( ZZ ) = 16(1 + (cid:15) ) (cid:26)(cid:2) c ( q )2 V + c ( q )2 A (cid:3) (1 − (cid:15) ) (cid:2) (1 + 2 (cid:15) ) m q H (2) + 2(1 − (cid:15) ) H (1) (cid:3) + (cid:2) c ( q )2 V − c ( q )2 A (cid:3) (cid:15) (3 − (cid:15) ) m q H (2) (cid:27) , I (0 , m W ) N (2) a µν ( W + W + ) = 128(1 − (cid:15) ) v µ v ν (3 − (cid:15) )(1 − x ) (cid:26) (2 − (cid:15) )(2 − x − (cid:15) + 4 (cid:15)x ) H (1) In particular, this can be used to demonstrate gauge invariance for the electroweak part of the amplitudes since ina general R ξ gauge the ξ -dependent terms carry a factor of ( v · L ).
34 (1 + (cid:15) )(1 − x ) m U (cid:2) (2 + (cid:15) )(1 − x ) m U H (3) + (3 − x − (cid:15) + 2 (cid:15)x ) H (2) (cid:3)(cid:27) , I (0 , m W ) N (2) c µν ( W + W + ) = 64(1 − (cid:15) ) v µ v ν (3 − (cid:15) ) (cid:26) − (3 − (cid:15) − (cid:15) )(1 − x ) m U H (2) + (cid:15) (7 − (cid:15) ) H (1) (cid:27) , I (0 , m Z ) N (2) a µν ( ZZ ) = 64(1 − (cid:15) ) v µ v ν (3 − (cid:15) )(1 − x ) (cid:26)(cid:2) c ( q )2 V + c ( q )2 A (cid:3)(cid:20) (2 − (cid:15) )(2 − x − (cid:15) + 4 (cid:15)x ) H (1)+ m q (1 + (cid:15) ) (cid:2) (2 + (cid:15) )(1 − x ) m q H (3) + (3 − x − (cid:15) + 5 (cid:15)x ) H (2) (cid:3)(cid:21) + (cid:2) c ( q )2 V − c ( q )2 A (cid:3) (1 + (cid:15) )(2 + (cid:15) ) xm q (cid:2) m q H (3) − H (2) (cid:3)(cid:27) , I (0 , m Z ) N (2) c µν ( ZZ ) = 32(1 − (cid:15) ) v µ v ν (3 − (cid:15) ) (cid:26)(cid:2) c ( q )2 V + c ( q )2 A (cid:3)(cid:2) − (1 + (cid:15) )(3 − (cid:15) ) m q H (2) + (cid:15) (7 − (cid:15) ) H (1) (cid:3) − (cid:2) c ( q )2 V − c ( q )2 A (cid:3) (cid:15) )(3 − (cid:15) ) m q H (2) (cid:27) , (93)where the subscript on I even has been suppressed. The reduction for admixtures, where there arenonzero residual masses and the integral I odd is relevant, is also straightforward to obtain.We collect in Appendix D useful results for the remaining task of integrating over Feynmanparameters. The singularity structure and evaluation of integrals can be classified into three casescorresponding to zero, one, or two heavy fermions contributing to the electroweak polarization tensor.The case of zero heavy fermions is for polarization tensors with no top quark in the loop. Withsubleading powers of light quark masses neglected, only polarization tensors of W ± and Z bosonsare relevant in this case. The case of one heavy fermion is for polarization tensors of flavor-changingcurrents with one top quark and one down-type quark. The case of two heavy fermions is forpolarization tensors of flavor-diagonal currents with a top quark loop. Let us now determine the full theory contributions to the matching using the generalized electroweakpolarization tensors and the reduction method for the boson loop integral. For pure states, the totalamplitude receives two-boson exchange contributions from W ± and Z bosons, M = M W W + M ZZ , (94)which may be written in terms of electroweak polarization tensors in a background field as i M W W = ig C W (cid:90) ( dL ) 1 − v · L + i L − m W + i v µ v ν (cid:20) i Π µν ( W + W + ) ( L ) + i Π µν ( W − W − ) ( L ) (cid:21) ,i M ZZ = ig C Z c W (cid:90) ( dL ) 1 − v · L + i L − m Z + i v µ v ν i Π µν ( ZZ ) ( L ) , (95)with C W and C Z given in (9). The parity of the polarization tensors under L → − L and the identitiesin (84) allow us to write the above amplitudes in terms of the integrals defined in (85), i M W W = ig C W I even (0 , m W ) v µ v ν i Π µν ( W + W + ) ( L ) , M ZZ = ig C Z c W I even (0 , m Z ) v µ v ν i Π µν ( ZZ ) ( L ) . (96)Upon inserting the explicit polarization tensors from (66) into the expressions above, we may employthe reduction of integrals given in (93) and write each contribution in terms of the gluon operatorsof definite spin, M BB (cid:48) = M BB (cid:48) (0) O (0) g + M BB (cid:48) (2) v µ v ν O (2) µνg , (97)where the superscript BB (cid:48) denotes the different types of two-boson exchange. From the expressionin (97), we readily identify the contribution of each amplitude to c (0) g and c (2) g as M BB (cid:48) (0) and M BB (cid:48) (2) , respectively. Let us decompose M W W ( S ) , for S = 0 ,
2, into contributions from eachup-type quark flavor, and the a -, b -, and c -type gluon attachments, M W W ( S ) = − [Γ(1 + (cid:15) )] (4 π ) d πg g m (cid:15)W C W (cid:88) U = u,c,t (cid:88) k = a,b,c M W W ( S ) U,k . (98)Similarly, we decompose M ZZ ( S ) into contributions from each quark flavor, and the a -, b -, and c -typegluon attachments, M ZZ ( S ) = − [Γ(1 + (cid:15) )] (4 π ) d πg g m (cid:15)Z C Z c W (cid:88) q = u,c,t,d,s,b (cid:88) k = a,b,c M ZZ ( S ) q,k . (99)The results for W ± exchange are as follows. The amplitudes with one top quark are M W W (0) t,a = 4 x t log x t + 1 x t − x t (6 x t + 9 x t + 2)3( x t + 1) , M W W (0) t,b = 0 , M W W (0) t,c = − x t log x t + 1 x t + 2(6 x t + 9 x t + 2 x t − x t + 1) , M W W (2) t,a = 16(30 x t − x t − x t + 1 x t − x t + 90 x t + 14 x t − x t − x t − x t + 1) , M W W (2) t,b = 16(3 x t + 2)9( x t + 1) (cid:15) + 32 x t (15 x t − x t + 52 x t − x t + 14 x t − x t − log x t − x t − x t + 52 x t − x t + 3 x t − x t − log( x t + 1) − x t − x t + 3 x t + 9 x t − x t − log 2+ 8(180 x t + 90 x t − x t − x t + 285 x t + 111 x t − x t − x t + 71)27( x t − ( x t + 1) , M W W (2) t,c = − x t log x t + 1 x t + 8 x t (6 x t + 9 x t + 2)( x t + 1) , (100)36here x t = m t /m W . The amplitudes with only light quarks are M W W (0)
U,a = M W W (0)
U,b = 0 , M W W (0)
U,c = − , M W W (2)
U,a = M W W (2)
U,b = 329 (cid:15) + 56827 −
649 log 2 , M W W (2)
U,c = 0 , (101)for U = u, c . The results for Z exchange with a top quark loop are M ZZ (0) t,a = M ZZ (0) t,b = (cid:2) c ( t )2 V + c ( t )2 A (cid:3)(cid:20) y t (32 y t − y t + 14 y t − y t − / arctan (cid:0)(cid:112) y t − (cid:1) − πy t
2+ 4 y t ( y t − y t − y t − (cid:21) + (cid:2) c ( t )2 V − c ( t )2 A (cid:3)(cid:20) y t (24 y t − y t + 5)(4 y t − / arctan (cid:0)(cid:112) y t − (cid:1) + 2(144 y t − y t + 9 y t − y t − − πy t (cid:21) , M ZZ (0) t,c = (cid:2) c ( t )2 V + c ( t )2 A (cid:3)(cid:20) − y t (8 y t − y t − y t − / arctan (cid:0)(cid:112) y t − (cid:1) − y t − y t + 1)3(4 y t − + 2 πy t (cid:21) , M ZZ (2) t,a = M ZZ (2) t,b = (cid:2) c ( t )2 V + c ( t )2 A (cid:3)(cid:20) y t − y t + 214 y t − y t + 4)9(4 y t − / arctan (cid:0)(cid:112) y t − (cid:1) + 8(240 y t − y t + 92 y t − y t − − πy t (cid:21) + (cid:2) c ( t )2 V − c ( t )2 A (cid:3)(cid:20) − y t (48 y t − y t + 13)9(4 y t − − y t (16 y t − y t + 4 y t − y t − / arctan (cid:0)(cid:112) y t − (cid:1) + 2 πy t (cid:21) , M ZZ (2) t,c = (cid:8)(cid:2) c ( t )2 V + c ( t )2 A (cid:3) + 2 (cid:2) c ( t )2 V − c ( t )2 A (cid:3)(cid:9)(cid:20) − y t (16 y t − y t + 3)(4 y t − / arctan (cid:0)(cid:112) y t − (cid:1) − y t (8 y t − y t − + 8 πy t (cid:21) , (102)where y t = m t /m Z . The amplitudes for Z exchange with a light quark loop are M ZZ (0) q,a = M ZZ (0) q,b = 0 , M ZZ (0) q,c = (cid:2) c ( q )2 V + c ( q )2 A (cid:3)(cid:20) − (cid:21) , M ZZ (2) q,a = M ZZ (2) q,b = (cid:2) c ( q )2 V + c ( q )2 A (cid:3) (cid:20) (cid:15) + 56827 −
649 log 2 (cid:21) , M ZZ (2) q,c = 0 , (103)where q = u, d, s, c, b . The (cid:15) pieces in the above amplitudes are IR divergences that cancel uponsubtraction of the effective theory contributions, M ( S )EFT , discussed in Sec. 6.5. The bare coefficientsare then given by c ( S ) g = M W W ( S ) + M ZZ ( S ) − M ( S )EFT , (104)where the remaining (cid:15) pieces are UV divergences.37 .4.4 Full theory contributions and matching coefficients for admixtures For admixtures, the total amplitude receives contributions from other types of two-boson exchangebeyond
W W and ZZ , M = M W W + M ZZ + M φ W φ W + M φ Z φ Z + M hh + M Zφ Z + M W φ W . (105)Let us first consider the singlet-doublet case. In terms of the electroweak polarization tensors, wefind integrals involving nonzero residual masses, i M W W = ig c ρ (cid:90) ( dL ) 1 − v · L − δ (0)0 + i L − m W + i v µ v ν (cid:20) i Π µν ( W + W + ) ( L ) + i Π µν ( W − W − ) ( L ) (cid:21) ,i M ZZ = ig c W c ρ (cid:90) ( dL ) 1 − v · L − δ (0)0 + i L − m Z + i v µ v ν i Π µν ( ZZ ) ( L ) ,i M hh = ia (cid:90) ( dL ) (cid:34) s ρ − v · L − δ ( − )0 + i c ρ − v · L − δ (+)0 + i (cid:35) L − m h + i i Π ( hh ) ( L ) ,i M φ Z φ Z = ia s ρ (cid:90) ( dL ) 1 − v · L − δ (0)0 + i L − m Z + i i Π ( φ Z φ Z ) ( L ) ,i M φ W φ W = ia s ρ (cid:90) ( dL ) 1 − v · L − δ (0)0 + i L − m W + i (cid:20) i Π ( φ + W φ + W ) ( L ) + i Π ( φ − W φ − W ) ( L ) (cid:21) ,i M Zφ Z = g a c W s ρ (cid:90) ( dL ) 1 − v · L − δ (0)0 + i L − m Z + i v µ (cid:20) i Π µ ( Zφ Z ) ( L ) − i Π µ ( φ Z Z ) ( L ) (cid:21) ,i M W φ W = ig a s ρ (cid:90) ( dL ) 1 − v · L − δ (0)0 + i L − m W + i v µ (cid:20) i Π µ ( W + φ + W ) ( L ) − i Π µ ( W − φ − W ) ( L )+ i Π µ ( φ + W W + ) ( L ) − i Π µ ( φ − W W − ) ( L ) (cid:21) . (106)Using the behavior of the polarization tensors under L → − L and the identities in (84), we maywrite these amplitudes in terms of the integrals defined in (85), i M W W = ig c ρ I even ( δ (0)0 , m W ) v µ v ν i Π µν ( W + W + ) ( L ) ,i M ZZ = ig c W c ρ I even ( δ (0)0 , m Z ) v µ v ν i Π µν ( ZZ ) ( L ) ,i M hh = ia s ρ I even ( δ ( − )0 , m h ) i Π ( hh ) ( L ) + ia c ρ I even ( δ (+)0 , m h ) i Π ( hh ) ( L ) ,i M φ Z φ Z = ia s ρ I even ( δ (0)0 , m Z ) i Π ( φ Z φ Z ) ( L ) ,i M φ W φ W = ia s ρ I even ( δ (0)0 , m W ) i Π ( φ + W φ + W ) ( L ) ,i M Zφ Z = g a c W s ρ I odd ( δ (0)0 , m Z ) v µ i Π µ ( Zφ Z ) ( L ) , M W φ W = ig a s ρ I odd ( δ (0)0 , m W ) v µ i Π µ ( W + φ + W ) ( L ) . (107)The required polarization tensors are specified in (66), and, in particular, the complete set of functions N ( S ) k ( L ) are explicitly given in (82) and (83). Thus, the general result in (91) for reducing theseintegrals may be applied. Each amplitude may be written in the form of (97), i.e., in terms of itscontributions to the gluon operators of definite spin. The bare coefficients are then given by c ( S ) g = M ( S ) W W + M ( S ) ZZ + M ( S ) hh + M ( S ) φ Z φ Z + M ( S ) φ W φ W + M ( S ) Zφ Z + M ( S ) W φ W − M ( S )EFT , (108)where the remaining (cid:15) pieces are UV divergences. We may again organize each contribution inthe previous equation in terms of the quark flavors in the loop, and the a -, b -, and c -type gluonattachments, as we have done in (98) and (99).For the triplet-doublet case we find, i M W W = ig s ρ I even ( δ (+)0 , m W ) v µ v ν i Π µν ( W + W + ) ( L )+ ig (cid:0) s ρ (cid:1) I even ( δ ( − )0 , m W ) v µ v ν i Π µν ( W + W + ) ( L ) ,i M ZZ = ig c W c ρ I even ( δ (0)0 , m Z ) v µ v ν i Π µν ( ZZ ) ( L ) ,i M hh = ia s ρ I even ( δ ( − )0 , m h ) i Π ( hh ) ( L ) + ia c ρ I even ( δ (+)0 , m h ) i Π ( hh ) ( L ) ,i M φ Z φ Z = ia s ρ I even ( δ (0)0 , m Z ) i Π ( φ Z φ Z ) ( L ) ,i M φ W φ W = ia I even ( δ (+)0 , m W ) i Π ( φ + W φ + W ) ( L ) ,i M Zφ Z = g a c W s ρ I odd ( δ (0)0 , m Z ) v µ i Π µ ( Zφ Z ) ( L ) ,i M W φ W = ig a s ρ I odd ( δ (+)0 , m W ) v µ i Π µ ( W + φ + W ) ( L ) . (109)The rest of the analysis proceeds as above, using the same polarization tensors and integral reductionmethod. We check for both types of admixtures that the expected results are recovered upon takingthe pure-case limits described in Sec. 4.3. In the computation of both pure- and mixed-case amplitudes above, we have neglected subleadingcorrections of O ( m q /m W ) by Taylor expanding integrands about vanishing light quark masses. This requires a regulator to control IR divergences (the full theory diagrams in Figs. 3 and 5 are UVfinite but the projection onto the spin-2 operator O (2) g is IR divergent). For matching onto quark operators, we of course include the leading m q factor appearing in O (0) q and O (2) q . Formatching onto gluon operators we may neglect light quark masses.
39t is technically simplest to compute the full and effective theory amplitudes using dimensionalregularization as IR regulator. Effective theory loop diagrams on the right hand sides of Figs. 3 and 5then result in dimensionfull but scaleless integrals that are required to vanish. Upon subtractingthe effective theory amplitude, remaining 1 /(cid:15) pieces in matching coefficients are identified as UVdivergences.We have obtained identical renormalized matching coefficients by retaining light quark masses, m q (cid:54) = 0, as an alternative IR regulator. In this scheme, the effective theory loop diagrams on theright-hand side of Figs. 3 and 5 yield nonvanishing contributions. The full theory diagrams on theleft-hand side are correspondingly modified so that, upon subtracting the effective theory amplitude,consistent results are obtained. We may now collect the results of the preceding analysis of quark and gluon matching to present thebare coefficients of the effective theory at the weak scale. We have analyzed the Wilson coefficientsof the effective theory described by (2) in terms of contributions from exchanges of one or twoelectroweak bosons, as expressed in (45). The results for one-boson exchange matching to quarkand gluon operators are given by (48) and (55), respectively. The results for two-boson exchangematching to quark and gluon operators are given by summing contributions of the form (57) and(97), respectively.For pure cases, the results for the bare matching coefficients are as follows, c (0) U = π Γ(1 + (cid:15) ) g (4 π ) − (cid:15) (cid:40) − m − − (cid:15)W x h (cid:20) C W + C Z c W (cid:21) + m − − (cid:15)Z C Z c W (cid:2) c ( U )2 V − c ( U )2 A (cid:3) + O ( (cid:15) ) (cid:41) ,c (0) D = π Γ(1 + (cid:15) ) g (4 π ) − (cid:15) (cid:40) − m − − (cid:15)W x h (cid:20) C W + C Z c W (cid:21) + m − − (cid:15)Z C Z c W (cid:2) c ( D )2 V − c ( D )2 A (cid:3) − δ Db m − − (cid:15)W C W x t x t + 1) + O ( (cid:15) ) (cid:41) ,c (0) g = π [Γ(1 + (cid:15) )] g g (4 π ) − (cid:15) (cid:40) m − − (cid:15)W (cid:34) x h (cid:20) C W + C Z c W (cid:21) + C W (cid:20)
13 + 16( x t + 1) (cid:21) (cid:35) + m − − (cid:15)Z C Z c W (cid:34) (cid:2) c ( D )2 V + c ( D )2 A (cid:3) + (cid:2) c ( U )2 V + c ( U )2 A (cid:3)(cid:20)
83 + 32 y t (8 y t − y t − / arctan (cid:0)(cid:112) y t − (cid:1) − πy t + 4(48 y t − y t + 9 y t − y t − (cid:21) + (cid:2) c ( U )2 V − c ( U )2 A (cid:3)(cid:20) πy t − y t − y t + 9 y t − y t − − y t (24 y t − y t + 5)(4 y t − / arctan (cid:0)(cid:112) y t − (cid:1)(cid:21)(cid:35) + O ( (cid:15) ) (cid:41) ,c (2) U = π Γ(1 + (cid:15) ) g (4 π ) − (cid:15) (cid:40)(cid:20) m − − (cid:15)W C W + m − − (cid:15)Z C Z c W (cid:2) c ( U )2 V + c ( U )2 A (cid:3)(cid:21)(cid:20)
13 + (cid:18) −
23 log 2 (cid:19) (cid:15) (cid:21) + O ( (cid:15) ) (cid:41) , (2) D = π Γ(1 + (cid:15) ) g (4 π ) − (cid:15) (cid:40)(cid:20) m − − (cid:15)W C W + m − − (cid:15)Z C Z c W (cid:2) c ( D )2 V + c ( D )2 A (cid:3)(cid:21)(cid:20)
13 + (cid:18) −
23 log 2 (cid:19) (cid:15) (cid:21) + δ Db m − − (cid:15)W C W (cid:20) (3 x t + 2)3( x t + 1) −
23 + (cid:18) x t (7 x t − x t − log x t − x t + 2)3( x t + 1) log 2 − x t − x t − x t − ( x t + 1) −
229 + 43 log 2 (cid:19) (cid:15) (cid:21) + O ( (cid:15) ) (cid:41) ,c (2) g = π [Γ(1 + (cid:15) )] g g (4 π ) − (cid:15) (cid:40) m − − (cid:15)W C W (cid:20) − (cid:15) − − x t + 2)9( x t + 1) (cid:15) + 8(6 x t − x t + 21 x t − x t − x t − log( x t + 1) + 4(3 x t − x t + 3 x t + 9 x t − x t − log 2 − x t − x t + 39 x t + 14 x t − x t − x t − x t − log x t − x t + 72 x t − x t − x t − x t + 47 x t + 9827( x t − ( x t + 1) (cid:21) + m − − (cid:15)Z C Z c W (cid:34)(cid:20) (cid:2) c ( U )2 V + c ( U )2 A (cid:3) + 12 (cid:2) c ( D )2 V + c ( D )2 A (cid:3)(cid:21)(cid:20) − (cid:15) − (cid:21) + (cid:2) c ( U )2 V + c ( U )2 A (cid:3)(cid:20) y t − y t − y t + 5 y t − y t − / arctan (cid:0)(cid:112) y t − (cid:1) − πy t
3+ 16(48 y t + 62 y t − y t + 9)9(4 y t − (cid:21) + (cid:2) c ( U )2 V − c ( U )2 A (cid:3)(cid:20) y t (624 y t − y t + 103)9(4 y t − − πy t
3+ 128 y t (104 y t − y t + 35 y t − y t − / arctan (cid:0)(cid:112) y t − (cid:1)(cid:21)(cid:35) + O ( (cid:15) ) (cid:41) , (110)where, as before, x t = m t /m W and y t = m t /m Z . Above, the Kronecker delta, δ Db , is equal to unityfor D = b , and vanishes for D = d, s . The pure triplet (doublet) results are given by setting C W = 2and C Z = 0 ( C W = 1 / C Z = 1 / c (2) g given above and the renormalized coefficient c (2) g ( µ ) involves a nontrivial subtractionrequiring the O ( (cid:15) ) part of c (2) q which we have retained.The results for admixtures are similarly obtained by collecting contributions to the coefficientsspecified in (45). For example, the amplitudes in (51) for a singlet-doublet admixture, combined withthe integrals defined in Appendix B, specify c (0) q through (48), and c (0) g through (55). Thecoefficients c ( S ) q are specified in (63) in terms of the results in (61), which require the integrals inAppendix C. Finally, c ( S ) g is specified in (108) in terms of the amplitudes in (107) which requirethe polarization tensors in (66), the basis reduction in (91), and the integrals in Appendix D.The matching coefficients for admixtures are functions of the mass splitting ∆ and the coupling a , as defined in (20) for the singlet-doublet mixture. We illustrate numerical values in Fig. 6 forboth the singlet-doublet and triplet-doublet mixtures. Numerical inputs are collected in Table 1 of41 ure doubletpure singlet (-1)(-1) a (cid:61) g (cid:45) (cid:45) (cid:45) pure doubletpure singlet a (cid:61) g
100 (-1) (-1) (cid:45) (cid:45) (cid:45) ∆ /m W ∆ /m W ± c π α m W ± c π α m W pure triplet pure doublet(-1)(-1) a (cid:61) g (cid:45) (cid:45) pure doubletpure triplet (-1)(-1) a (cid:61) g (cid:45) (cid:45) ∆ /m W ∆ /m W ± c π α m W ± c π α m W Figure 6:
Renormalized coefficients (with πα /m W extracted) for the singlet-doublet (upper panels)and triplet-doublet (lower panels) mixtures as a function of the respective mass splittings ∆ =( M S − M D ) / M T − M D ) /
2, in units of m W . The panels on the left (right) use a = g / a = g / c (0) q and c (2) g are presented with opposite sign, as indicatedby ( − − c (0) U = u,c , − c (0) D = d,s , and − c (0) b . Thedashed red, green, and blue lines are respectively for c (2) U = u,c , c (2) D = d,s , and c (2) b . Some quark matchingcoefficients appear degenerate. The orange band with solid borders is c (0) g , and the orange bandwith dashed borders is − c (2) g . The band thickness represents renormalization scale variation, taking m W / < µ t < m t [6]. We indicate the pure-case limits at large | ∆ | .Appendix E. Depending on the value of a , the O ( α ) tree-level Higgs exchange contribution to thespin-0 coefficients may dominate near ∆ = 0. When m W / ∆ suppression is significant, the O ( α ) loopcontributions dominate. The curves approach the correct pure-case values upon taking the limitsdescribed in Sec. 4.3. In particular, the coefficients vanish in the pure singlet limit.The contributions of these coefficients to scattering cross sections depend on the detailed mappingonto the low-energy n f = 3 flavor theory through renormalization group running and heavy quarkthreshold matching, and on the evaluation of nucleon matrix elements at a low scale, µ ∼ α s counting reflected inthe relative magnitudes of the high-scale coefficients. One example is the enhancement of the spin-0gluon contribution due both to a large anomalous dimension in the RG running, and to the largenucleon matrix element of the scalar gluon operator [43]. Another example is the enhanced impactof numerically subleading contributions due to a partial cancellation at leading order. The relativesigns between high-scale coefficients in Fig. 6, combined with details of the mapping onto low-energycoefficients and evaluation of matrix elements, lead to a cancellation between the spin-0 and spin-242mplitude contributions [5, 6]. Therefore, a robust determination of DM-nucleon scattering crosssections demands a careful analysis of the complete set of leading operators in (3).The coefficient c (2) g has been omitted in previous works [3, 5]. Due to a cancellation betweenspin-0 and spin-2 amplitude contributions to cross sections, the effect of neglecting c (2) g ranges froma factor of a few to an order of magnitude difference in cross sections. For the pure-doublet andpure-triplet states, neglecting c (2) g leads to an O (10 − O (70%). For comparison, neglecting c (2) q for q = b, c, s, d, u shifts the spin-2 amplitude by O (1%), O (10%), O (10%), O (30%), and O (50%), respectively. The present analysis focused on weak-scale matching conditions necessary for robustly computingWIMP-nucleon interactions, both in specified UV completions involving electroweak-charged DM,and in the model-independent heavy WIMP limit. Careful computation of competing Standard Modelcontributions is necessary to estimate the correct order of magnitude of scattering cross sections inmany simple and motivated models of DM. For example, a simple dimensional estimate of the crosssection for spin-independent, low-velocity scattering of a pure-state WIMP on a nucleon yields σ SI ∼ α m N m W (cid:18) m W , m h (cid:19) ∼ − cm . (111)Cross sections of this order of magnitude are currently being probed by direct detection searches(e.g., see Refs. [22] for detection prospects computed using tree-level cross sections). However, acancellation between spin-0 and spin-2 amplitude contributions leads to much smaller cross sectionvalues for motivated candidates such as the pure wino ( σ SI ∼ − cm ) and the pure higgsino( σ SI (cid:46) − cm ) of supersymmetric SM extensions. This cancellation demands a careful analysis ofperturbative contributions from weak-scale matching amplitudes presented here, e.g., the inclusionof the spin-2 gluon contribution, and of remaining theoretical and input uncertainties, which will bediscussed in a companion paper [34]. Robust predictions for the cross sections of the pure triplet,pure doublet, singlet-doublet admixture, and triplet-doublet admixture can be found in Refs. [6].Given the matching coefficients in (110), the cross sections for pure states with arbitrary electroweakquantum numbers can also be computed.Although we find that cancellations are generic, their severity depends on SM parameters andon properties of DM such as its electroweak quantum numbers. The presence of additional low-lyingstates could also have impact, and the formalism for weak-scale matching presented here can bereadily extended to investigate such scenarios. For example, including a second Higgs doublet inthe pure-state analysis simply requires modification of the vertices in the amplitudes computed inFigs. 2 and 3. An extra Higgs boson modifies the spin-0 amplitude, and could potentially weakenthe cancellation between spin-0 and spin-2 amplitudes. The case where the second Higgs-like doubletitself plays the role of DM (e.g., “inert Higgs DM” [7]) is related to the pure-doublet case in theheavy WIMP limit by heavy particle universality.While we have focused here on the case of a heavy, self-conjugate WIMP, deriving from one or twoelectroweak multiplets, much of the formalism applies more generally. The construction of the heavy Cross sections of this magnitude were obtained in previous estimates that missed the cancellation between spin-0and spin-2 amplitude contributions (and ignored gluon contributions) [2]. n f = 3 or n f = 4 flavor QCD plus interactions with DM)where hadronic matrix elements are evaluated. Acknowledgements
We acknowledge useful discussions with C.E.M. Wagner. M.S. also thanks T. Cohen and J. Kearneyfor useful discussions. This work was supported by the United States Department of Energy underGrant No. DE-FG02-13ER41958. M.S. is supported by a Bloomenthal Fellowship.
A The singlet-doublet mixture
The heavy-particle lagrangians in Sec. 4 may be obtained from a manifestly relativistic lagrangianby performing field redefinitions at tree level. Consider the case of a singlet-doublet mixture (seealso [19]), L = L SM + 12¯ b ( i∂/ − M ) b + ¯ ψ ( iD/ − M ) ψ − ( y ¯ bP L H † ψ + y (cid:48) ¯ bP L H T ψ c + h . c . ) , (112)where b is a gauge singlet (Majorana) fermion represented as a Dirac spinor with b c = b , and ψ is a Dirac fermion in the ( , /
2) representation of SU (2) W × U (1) Y . In the above equation, P R , L = (1 ± γ ) /
2, and we have included all renormalizable gauge-invariant interactions involving theSM Higgs field. Expressing the result in terms of Majorana combinations, λ = 1 √ ψ + ψ c ) , λ = i √ ψ − ψ c ) , (113)and collecting the fermions in the column vector λ = ( b, λ , λ ), we may write the interactions withthe Higgs field as L H ¯ λλ = − √ b − γ (cid:104) ( yH † + y (cid:48) H T ) λ − i ( yH † − y (cid:48) H T ) λ (cid:105) + h . c . ≡ −
12 ¯ λ (cid:20) f ( H ) + iγ g ( H ) (cid:21) λ , (114)with f ( H ) = a √ H † + H T i ( H T − H † ) H + H ∗ i ( H − H ∗ ) + a √ − i ( H T − H † ) H T + H † − i ( H − H ∗ ) H + H ∗ , ( H ) = b √ − i ( H T − H † ) H T + H † − i ( H − H ∗ ) H + H ∗ + b √ H † + H T i ( H T − H † ) H + H ∗ i ( H − H ∗ ) . (115)The real parameters a i and b i are given by a = 12 Re( y + y (cid:48) ) , a = 12 Im( y − y (cid:48) ) , b = 12 Re( y − y (cid:48) ) , b = −
12 Im( y + y (cid:48) ) . (116)We employ phase redefinitions of b , ψ L and ψ R to ensure that M and M are real and positive. The gauge generators will be those given in (13), extended trivially to include the singlet. Uponperforming the tree-level field redefinition λ = √ e − i ( M − δM ) v · x ( h v + H v ) , (117)where the fields h v and H v obey v/ h v = h v and v/ H v = − H v , we obtain the heavy-particle lagrangianin (4). It follows from λ c = λ that the resulting lagrangian is invariant under the simultaneoustransformations in (1). Note that f ( H ) is the only term surviving the projection from the condition v/ h v = h v . The remaining analysis follows that of Sec. 4.2.1. B Self energy integrals and Standard Model two-point functions
Here and in the following sections we use the notation[ c (cid:15) ] = i Γ(1 + (cid:15) )(4 π ) − (cid:15) , ( dL ) = d d L (2 π ) d . (118)The self-energies in Sec. 5 and the h ¯ χχ three-point functions in Sec. 6.1 require the following integrals, I ( δ, m ) = (cid:90) ( dL ) 1 v · L − δ + i L − m + i = ∂∂m I ( δ, m )= [ c (cid:15) ] m − (cid:15) (cid:26) √ m − δ − i (cid:20) arctan (cid:18) δ √ m − δ − i (cid:19) − π (cid:21) + O ( (cid:15) ) (cid:27) ,I ( δ, m ) = (cid:90) ( dL ) v · L v · L − δ + i L − m + i = δI ( δ, m ) + i (4 π ) B (0 , m, m )= [ c (cid:15) ] m − (cid:15) (cid:26) (cid:15) + 2 δ √ m − δ − i (cid:20) arctan (cid:18) δ √ m − δ − i (cid:19) − π (cid:21) + O ( (cid:15) ) (cid:27) ,I ( δ, m ) = (cid:90) ( dL ) 1 v · L − δ + i L − m + i An additional phase redefinition could be used to eliminate a , a , b or b .
45 [ c (cid:15) ] m − (cid:15) (cid:26) − δ(cid:15) + 4 (cid:112) m − δ − i (cid:20) arctan (cid:18) δ √ m − δ − i (cid:19) − π (cid:21) − δ + O ( (cid:15) ) (cid:27) ,I ( δ , δ , m ) = (cid:90) ( dL ) 1 v · L − δ + i v · L − δ + i L − m + i . (119)For I ( δ , δ , m ), let us specialize to δ = 0 or δ = δ , I ( δ, , m ) = 1 δ (cid:2) I ( δ, m ) − I (0 , m ) (cid:3) = [ c (cid:15) ] m − (cid:15) (cid:26) − (cid:15) + 4 √ m − δ − i δ (cid:20) arctan (cid:18) δ √ m − δ − i (cid:19) − π (cid:21) − πmδ + O ( (cid:15) ) (cid:27) ,I ( δ, δ, m ) = ∂∂δ I ( δ, m )= [ c (cid:15) ] m − (cid:15) (cid:26) − (cid:15) − δ √ m − δ − i (cid:20) arctan (cid:18) δ √ m − δ − i (cid:19) − π (cid:21) + O ( (cid:15) ) (cid:27) . (120)The two-point functions for the electroweak SM bosons appearing in (53) are obtained by summingthe fermionic and bosonic contributions given below. Following Denner [40], we haveΣ AA (cid:48) (0) = − α π (cid:26) B (0 , m W , m W ) + 4 m W B (cid:48) (0 , m W , m W ) − (cid:88) f,i (cid:2) N fc Q f B (0 , m f,i , m f,i ) (cid:3)(cid:27) , Σ AZ (0) m Z = − α π (cid:26) − c W s W B (0 , m W , m W ) (cid:27) , Σ ZZ ( m Z ) fermion m Z = − α π (cid:26) (cid:20) − B ( m Z , ,
0) + 13 (cid:21) (cid:88) f,i N fc [( g + f ) + ( g − f ) ]+ 23 N tc (cid:20) [( g + t ) + ( g − t ) ] (cid:20) − (cid:18) m t m Z (cid:19) B ( m Z , m t , m t ) + B ( m Z , , m t m Z B (0 , m t , m t ) (cid:21) + 34 s W c W m t m Z B ( m Z , m t , m t ) (cid:21)(cid:27) , Σ ZZ ( m Z ) boson m Z = − α π s W c W (cid:26)
112 (4 c W − c W + 20 c W + 1) B ( m Z , m W , m W ) − c W (12 c W − c W + 1) B (0 , m W , m W ) − B (0 , m Z , m Z ) − (cid:18) m h m Z − m h m Z + 12 (cid:19) B ( m Z , m Z , m h ) − m h m Z B (0 , m h , m h )46 112 (cid:18) − m h m Z (cid:19) B (0 , m Z , m h ) −
19 (1 − c W ) (cid:27) , Σ W W ( m W ) fermion m W = − α π s W (cid:26) (cid:20) − B ( m W , , (cid:21) (cid:88) f,i N fc
2+ 23 N tc (cid:20) (cid:18) m t m W + m t m W − (cid:19) B ( m W , m t ,
0) + B ( m W , , m t m W B (0 , m t , m t ) − m t m W B (0 , m t , (cid:21)(cid:27) , Σ W W ( m W ) boson m W = − α π (cid:26) B ( m W , m W , λ ) − B (0 , m W , m W ) + 23 B (0 , m W , λ ) + 29+ 112 s W (cid:20) c W (4 c W − c W + 20 c W + 1) B ( m W , m W , m Z ) − c W + 1) B (0 , m W , m W ) − c W (8 c W + 1) B (0 , m Z , m Z )+ s W c W (8 c W + 1) B (0 , m W , m Z ) −
23 (1 − c W ) (cid:21) + 112 s W (cid:20) − (cid:18) m h m W − m h m W + 12 (cid:19) B ( m W , m W , m h ) − B (0 , m W , m W ) − m h m W B (0 , m h , m h ) + (cid:18) − m h m W (cid:19) B (0 , m W , m h ) − (cid:21)(cid:27) , Σ HH (cid:48) ( m h ) fermion = − α π m t s W m W (cid:20) (4 m t − m h ) B (cid:48) ( m h , m t , m t ) − B ( m h , m t , m t ) (cid:21) , Σ HH (cid:48) ( m h ) boson = − α π (cid:26) − s W (cid:20) (cid:18) m W − m h + m h m W (cid:19) B (cid:48) ( m h , m W , m W ) − B ( m h , m W , m W ) (cid:21) − s W c W (cid:20) (cid:18) m Z − m h + m h m Z (cid:19) B (cid:48) ( m h , m Z , m Z ) − B ( m h , m Z , m Z ) (cid:21) − m h s W m W B (cid:48) ( m h , m h , m h ) (cid:27) , (121)where the sums over indices f and i are for SM fermion flavors and generations, respectively. Above, N fc and Q f respectively denote the number of colors and the electric charge of fermion f . We havealso used α = g s W π , g + f = 18 s W c W (cid:2) c ( f )2 V + c ( f )2 A (cid:3) , g − f = 18 s W c W (cid:2) c ( f )2 V − c ( f )2 A (cid:3) , (122)where c ( (cid:96) ) V = − s W , c ( (cid:96) ) A = 1 , c ( ν ) V = 1 , c ( ν ) A = − , (123)47ith (cid:96) and ν denoting charged lepton and neutrino, respectively. The coefficients c ( f ) V and c ( f ) A forquarks can be found in (46). The basic integral appearing above is i (4 π ) B ( M, m , m ) = (cid:90) ( dL ) 1 L − m + i L + p ) − m + i
0= [ c (cid:15) ] (cid:20) (cid:15) + 2 − log( m m ) + m − m M log m m − m m M (cid:18) r − r (cid:19) log r + O ( (cid:15) ) (cid:21) , (124)where p = M and r = X + (cid:112) X − , r = X − (cid:112) X − , X = m + m − M − i m m . (125)We find the following limits, B (0 , m, m ) = (4 π ) (cid:15) Γ(1 + (cid:15) ) (cid:20) (cid:15) − m + O ( (cid:15) ) (cid:21) ,B (0 , m,
0) = (4 π ) (cid:15) Γ(1 + (cid:15) ) (cid:20) (cid:15) − m + 1 + O ( (cid:15) ) (cid:21) ,B (0 , m , m ) = (4 π ) (cid:15) Γ(1 + (cid:15) ) (cid:20) (cid:15) − m m − m log m + m m − m log m + 1 + O ( (cid:15) ) (cid:21) ,B ( M, m,
0) = (4 π ) (cid:15) Γ(1 + (cid:15) ) (cid:20) (cid:15) + 2 − m M log m + m − M M log( m − M − i
0) + O ( (cid:15) ) (cid:21) ,B ( M, ,
0) = (4 π ) (cid:15) Γ(1 + (cid:15) ) (cid:20) (cid:15) + 2 − log( − M − i
0) + O ( (cid:15) ) (cid:21) , lim λ → B ( m, m, λ ) = (4 π ) (cid:15) Γ(1 + (cid:15) ) (cid:20) (cid:15) + 2 − log m + O ( (cid:15) ) (cid:21) . (126)In the present application, only the real parts of the integrals are relevant. For the derivative of theintegral we have, B (cid:48) ( M, m, m ) ≡ ∂∂p B ( M, m, m )= (4 π ) (cid:15) Γ(1 + (cid:15) ) (cid:20) m M (cid:18) r − r (cid:19) log r − M (cid:18) r + 1 r − r (cid:19) + O ( (cid:15) ) (cid:21) , (127)which has the following limits, B (cid:48) (0 , m, m ) = (4 π ) (cid:15) Γ(1 + (cid:15) ) (cid:20) m + O ( (cid:15) ) (cid:21) ,B (cid:48) ( M, ,
0) = (4 π ) (cid:15) Γ(1 + (cid:15) ) (cid:20) − M + O ( (cid:15) ) (cid:21) . (128)48 Box integrals
The integrals required for the two-boson exchange amplitudes in Sec. 6.3 may be written in terms ofthe integral operators I even and I odd defined in (85) as J ( m V , M, δ ) = I even ( δ, m V ) 1 L − M + i ,J µ ( p, m V , M, δ ) = I even ( δ, m V ) 1 L + 2 L · p − M + i L µ = v · pv µ J ( m V , M, δ ) + p µ J ( m V , M, δ ) + O ( p ) ,J − ( p, m V , M, δ ) = −I odd ( δ, m V ) 1 L + 2 L · p − M + i v · pJ − ( m V , M, δ ) + O ( p ) ,J µ − ( m V , M, δ ) = −I odd ( δ, m V ) 1 L − M + i L µ = v µ J − ( m V , M, δ ) . (129)Note that J µ ( p, m V , M, δ ) and J − ( p, m V , M, δ ) vanish when p µ vanishes since the integrands are thenodd in L µ . By standard manipulations, we may express the integrals J , J , J , and J − , as J ( m V , M, δ ) = − c (cid:15) ](1 + (cid:15) ) ∂∂m V (cid:90) ∞ dρ (cid:90) dxρ (1 − x ) (cid:2) xm V + (1 − x ) M + ρ + 2 ρδ − i (cid:3) − − (cid:15) ,J ( m V , M, δ ) = 4[ c (cid:15) ] ∂∂m V (cid:90) ∞ dρ (cid:90) dx (1 − x ) (cid:2) xm V + (1 − x ) M + ρ + 2 ρδ − i (cid:3) − − (cid:15) ,J ( m V , M, δ ) = − c (cid:15) ] ∂∂m V (cid:90) ∞ dρ (cid:90) dx (cid:2) xm V + (1 − x ) M + ρ + 2 ρδ − i (cid:3) − − (cid:15) ,J − ( m V , M, δ ) = 4[ c (cid:15) ] ∂∂δ ∂∂m V (cid:90) ∞ dρ (cid:90) dx (1 − x ) (cid:2) xm V + (1 − x ) M + ρ + 2 ρδ − i (cid:3) − − (cid:15) . (130)Let us introduce the integralˆ J ( m V , M, δ ) = [ c (cid:15) ] (cid:90) ∞ dρ (cid:90) dx (1 − x ) (cid:2) xm V + (1 − x ) M + ρ + 2 ρδ − i (cid:3) − − (cid:15) , (131)and write the above integrals in terms of ˆ J ( m V , M, δ ) as J ( m V , M, δ ) = 4 ∂∂m V ˆ J ( m V , M, δ ) ,J − ( m V , M, δ ) = 4 ∂∂δ ∂∂m V ˆ J ( m V , M, δ ) ,J ( m V , M, δ ) = − ∂∂m V (cid:2) ˆ J ( m V , M, δ ) + ˆ J ( M, m V , δ ) (cid:3) ,J ( m V , M, δ ) = 4 ∂∂m V (cid:20) − ˆ J ( m V , M, δ ) + ∂∂A ˆ J ( m V , M, δ/A ) (cid:12)(cid:12) A =1 (cid:21) . (132)49or J − , we may use the identity (89) to write J − ( m V , M, δ ) = − (cid:90) ( dL ) 1( L − m V + i L − M + i − δJ ( m V , M, δ )= 2[ c (cid:15) ] m − − (cid:15)V (cid:15) (1 − (cid:15) ) (cid:18) − M m V (cid:19) − (cid:20) (cid:15) + M m V (cid:18) − (cid:15) − m (cid:15)V M (cid:15) (cid:19) (cid:21) − δ J ( m V , M, δ ) . (133)Having determined the above integrals in terms of ˆ J ( m V , M, δ ), it remains to compute this function.Let us writeˆ J ( m V , M, δ ) = − [ c (cid:15) ] (cid:15) ∂∂M (cid:90) ∞ dρ (cid:90) dx (cid:2) xm V + (1 − x ) M + ρ + 2 ρδ − i (cid:3) − (cid:15) = − [ c (cid:15) ] (cid:15) ∂∂M (cid:90) ∞ dρ m V − M − (cid:15) (cid:26) [ m V + ρ + 2 ρδ − i − (cid:15) − [ M + ρ + 2 ρδ − i − (cid:15) (cid:27) = − [ c (cid:15) ] (cid:15) ∂∂M m V − M − (cid:15) (cid:26) m − (cid:15)V f ( δ/m V , − (cid:15) ) − M − (cid:15) f ( δ/M, − (cid:15) ) (cid:27) , (134)where f ( δ, a ) = (cid:90) ∞ dρ (1 + ρ + 2 ρδ − i a = (1 − δ − i a + √ π − a − )Γ( − a ) − δ a +1 (cid:90) dx (cid:2) δ − − x − i (cid:3) a . (135)Although for the present application we require only δ >
0, the expression is for general sign of δ .We presently need f ( δ, a ) for a = 1 − (cid:15) , and hence consider √ π − + (cid:15) )Γ( − (cid:15) ) = − π (cid:15) + 2 π − (cid:15) + O ( (cid:15) ) , (cid:90) dx (cid:2) δ − − x − i (cid:3) − (cid:15) = B + 13 + (cid:15) (cid:26)
29 + 43 B − B arccot B − (cid:18) B + 13 (cid:19) log( B + 1) (cid:27) + (cid:15) (cid:26)
427 + 209 B + 49 B (6 log 2 B −
5) arccot B + 43 B i (cid:20) Li (cid:18) iB − iB (cid:19) − arccot B + π (cid:21) + 12 (cid:18) B + 13 (cid:19) log ( B + 1) − (cid:18) B + 29 (cid:19) log( B + 1) (cid:27) + O ( (cid:15) ) , (136)where B = 1 /δ − − i
0. For B >
0, the bracket involving dilogarithm may be written i (cid:20) Li (cid:18) iB − iB (cid:19) − arccot B + π (cid:21) = − Im Li (cid:18) iB − iB (cid:19) = − Cl (cid:20) arccos (cid:18) − B B (cid:19)(cid:21) , (137)where Cl is the Clausen function of order two. The general expression is required for continuing toarbitrary mass parameters. Having determined f ( δ, − (cid:15) ), we may proceed to compute ˆ J ( m V , M, δ )50sing (134), and then J ( m V , M, δ ), J ( m V , M, δ ), J − ( m V , M, δ ) and J ( m V , M, δ ) using (132), and J − ( m V , M, δ ) using (133).For M = 0, the expressions in (130), the expressions for J ( m V , M, δ ), J ( m V , M, δ ), and J − ( m V , M, δ ) in (132), and the expression for J − ( m V , M, δ ) in (133), remain valid. The integral J ( m V , , δ ) is now given by J ( m V , , δ ) = − c (cid:15) ] ∂∂m V (cid:26) − (cid:15) m − − (cid:15)V (cid:20) f ( δ/m V , − (cid:15) ) − f ( δ/m V , − (cid:15) ) (cid:21)(cid:27) , (138)and the integral ˆ J ( m V , , δ ) byˆ J ( m V , , δ ) = [ c (cid:15) ] m − V (cid:15) (cid:90) ∞ dρ (cid:26) ( ρ + 2 ρδ − i − (cid:15) − m − V − (cid:15) (cid:20) ( m V + ρ + 2 ρδ − i − (cid:15) − ( ρ + 2 ρδ − i − (cid:15) (cid:21)(cid:27) = [ c (cid:15) ] m − − (cid:15)V (cid:15) (cid:26) f ( δ/m V , − (cid:15) ) − − (cid:15) ) (cid:2) f ( δ/m V , − (cid:15) ) − f ( δ/m V , − (cid:15) ) (cid:3)(cid:27) , (139)where f ( δ, a ) is given by (135) and f ( δ, a ) = (cid:90) ∞ dρ ( ρ + 2 ρδ − i a = δ a Γ(1 + a )Γ (cid:0) − a − (cid:1) √ π . (140)We also need f ( δ/m V , a ) for a = − (cid:15) , which we may write as f ( δ/m V , − (cid:15) ) = 11 − (cid:15) m − (cid:15)V ∂∂m V (cid:20) m − (cid:15)V f ( δ/m V , − (cid:15) ) (cid:21) . (141)At vanishing residual mass, δ = 0, only the integrals J ( m V , M, J ( m V , M,
0) and J ( m V , M, J ( m V , M,
0) = [ c (cid:15) ] 2 √ π (1 − (cid:15) ) Γ( + (cid:15) )Γ(1 + (cid:15) ) m − (cid:15)V ( M − m V ) (cid:20) (cid:15) − (cid:18) Mm V (cid:19) − (cid:15) + (1 − (cid:15) ) (cid:18) Mm V (cid:19) (cid:21) ,J ( m V , M,
0) = − J ( m V , M,
0) = [ c (cid:15) ] 4 √ π (3 − (cid:15) )(1 − (cid:15) ) Γ( + (cid:15) )Γ(1 + (cid:15) ) m − (cid:15)V ( M − m V ) (cid:20) (cid:15) − (3 − (cid:15) ) (cid:18) Mm V (cid:19) − (cid:15) + (3 − (cid:15) ) (cid:18) Mm V (cid:19) − (1 + 2 (cid:15) ) (cid:18) Mm V (cid:19) − (cid:15) (cid:21) . (142)The result J ( m V , M,
0) = − J ( m V , M,
0) follows from the observation that when δ = 0 the identityin (89) implies v µ J µ ( p, m V , M,
0) = 0. The case δ = M = 0 is simply obtained by substitution in(142). D Heavy particle integrals with electroweak polarization tensor in-sertion
The two-boson exchange amplitudes for gluon matching require the integrals H ( n ), F ( n ), H µν ( n ),and H µ ( n ) defined in (88). Let us parameterize the last two as H µν ( n ) = H ( n ) v µ v ν + H ( n ) g µν , H µ ( n ) = H ( n ) v µ . (143)51pon contracting the above expressions with v µ and g µν , we may solve for the relations H ( n ) = 13 − (cid:15) (cid:2) (4 − (cid:15) ) v µ v ν H µν ( n ) − H µµ ( n ) (cid:3) ,H ( n ) = 13 − (cid:15) (cid:2) H µµ ( n ) − v µ v ν H µν ( n ) (cid:3) ,H ( n ) = v µ H µ ( n ) . (144)Using the identities in (86) and (89), we further obtain v µ H µ ( n ) = δH ( n ) + 2 F ( n ) ,v µ v ν H µν ( n ) = δ H ( n ) + 2 δF ( n ) ,H µµ ( n ) = (cid:20) m x + m (1 − x ) (cid:21) H ( n ) − H ( n − x (1 − x ) , (145)and hence the boson loops are completely specified by H ( n ) and F ( n ). In evaluating these functionsit may be advantageous to relate to more basic integrals by means of derivatives. Let us write, H ( n ) = 2 ∂∂m V (cid:90) ( dL ) 1 v · L − δ + i L − m V + i − n − (cid:15) ,F ( n ) = ∂∂m V (cid:90) ( dL ) 1 L − m V + i − n − (cid:15) , (146)with ∆ as defined in (80). The singularity structure and evaluation of the above integrals can beclassified into three cases, corresponding to zero, one, or two heavy fermions contributing to theelectroweak polarization tensor. For pure states we obtain analytic expressions for all integrals, whilefor mixed states we encounter several integrals that require numerical evaluation of one Feynmanparameter integral. D.1 Case of zero heavy fermions
Upon setting m = m = 0 in ∆ and performing the integration in d = 4 − (cid:15) dimensions, we obtain F ( n ) = [ c (cid:15) ] Γ(2 − n − (cid:15) )Γ( n + 2 (cid:15) )Γ(2 − (cid:15) )Γ(1 + (cid:15) ) [ x (1 − x )] − n − (cid:15) m − n − (cid:15)V ,H ( n ) = [ c (cid:15) ] 4Γ( n + 2 (cid:15) )Γ( n + (cid:15) )Γ(1 + (cid:15) ) [ x (1 − x )] − n − (cid:15) ∂∂m V I ( n ) , (147)where I ( n ) = (cid:90) dy (1 − y ) n − (cid:15) (cid:90) ∞ dρ ( ρ + 2 ρδ + ym V − i − n − (cid:15) . (148)We may reduce to the case of I (1) by noticing that I ( n + 1) = − m − V n + 2 (cid:15) (cid:90) dy (1 − y ) n + (cid:15) ddy (cid:90) ∞ dρ ( ρ + 2 ρδ + ym V − i − n − (cid:15) m − V n + 2 (cid:15) (cid:20) (cid:90) ∞ dρ ( ρ + 2 ρδ − i − n − (cid:15) + ( n + (cid:15) ) I ( n ) (cid:21) = m − V n + 2 (cid:15) (cid:20) δ − n − (cid:15) Γ(1 − n − (cid:15) )Γ (cid:0) n − + 2 (cid:15) (cid:1) √ π + ( n + (cid:15) ) I ( n ) (cid:21) . (149)Finally, for I (1) we require I (1) = δ − − (cid:15) (cid:90) dy (1 + (cid:15) log(1 − y ) + . . . ) (cid:90) ∞ dρ ( ρ + α ) − (cid:0) − (cid:15) log( ρ + α ) + . . . (cid:1) , (150)where α = (cid:0) ym V /δ − − i (cid:1) . The relevant integrals are (cid:90) ∞ dρ ρ + α = 1 α arctan α , (cid:90) ∞ dρ log( ρ + α ) ρ + α = 1 α (cid:20) α ) arctan α − i (cid:18) Li (cid:18) − iα iα (cid:19) − Li (cid:18) iα − iα (cid:19) (cid:19)(cid:21) . (151)We perform the remaining integral over Feynman parameter y numerically. D.2 Case of one heavy fermion
Let us set m = M (not to be confused with heavy WIMP mass M used elsewhere in the paper) and m = 0 in ∆, and consider separately the finite integrals for a - and c -type contributions, and the IRdivergent integrals for b -type contributions. D.2.1 Finite integrals for a - and c -type contributions For the finite a - and c -type contributions we may take d = 4. Let us evaluate the required integrals F (2) and H (1), and obtain the remaining integrals by differentiating with respect to M . We find F (2) = i (4 π ) ∂∂m V (cid:40)(cid:34) x (1 − x ) m V (cid:18) − M xm V (cid:19) (cid:35) − (cid:34) − log M xm V + M xm V − (cid:35)(cid:41) ,H (1) = i (4 π ) ∂∂m V (cid:40) (cid:34) x (1 − x ) m V (cid:18) − M xm V (cid:19) (cid:35) − (cid:34)(cid:113) m V − δ arctan (cid:32)(cid:114) m V δ − (cid:33) − (cid:114) M x − δ arctan (cid:32)(cid:114) M xδ − (cid:33) − δ xm V M (cid:35)(cid:41) . (152)The integrals have been obtained by breaking an integration region into pieces, e.g., (cid:90) ∞ δ dρ (cid:20) log( ρ + m V − δ ) − log (cid:18) ρ + M x − δ (cid:19)(cid:21) = lim ε → (cid:90) ∞ δ dρ (cid:20) log( ρ + m V − δ − iε ) − log (cid:18) ρ + M x − δ − iε (cid:19)(cid:21) = δ lim ε → (cid:90) ∞ dρ (cid:20) log (cid:18) ρ + m V δ − − iε (cid:19) − log (cid:18) ρ + M xδ − − iε (cid:19)(cid:21) δ lim ε → (cid:26) (cid:90) ∞ dρ (cid:20) log (cid:18) ρ + m V δ − − iε (cid:19) − log (cid:18) ρ + M xδ − − iε (cid:19)(cid:21) − (cid:90) dρ (cid:20) log (cid:18) ρ + m V δ − − iε (cid:19) − log (cid:18) ρ + M xδ − − iε (cid:19)(cid:21) (cid:27) . (153)Since the original integral is independent of ε , either choice of sgn( ε ) is correct provided it is usedconsistently in both terms. The continuation away from δ → δ → δ + iε everywhere. For the evaluation of integrals over x involving H (1), let us write H (1) ≡ ∂∂m V K (1) ≡ ∂∂m V (cid:40) M xm V − M k (1) (cid:41) . (154)We then have x n K (1) = (cid:18) M m V (cid:19) n K (1) + (cid:16) M m V (cid:17) n − x nM m V − x M m V k (1) , (155)so that all powers x n K (1) can be reduced to the case n = 0, in addition to the remaining straight-forward integral involving a polynomial in x times k (1), which in practice is evaluated numerically.The remaining integrals involving F (2) are similarly straightforward to evaluate. D.2.2 Infrared divergent integrals for b -type contributions Let us now turn to the integrals for b -type contributions, where we work in d = 4 − (cid:15) spacetimedimensions to account for singular behavior at the endpoints of the x integration. We find, F (1) = [ c (cid:15) ][ x (1 − x )] − − (cid:15) Γ(1 + 2 (cid:15) )[Γ(1 + (cid:15) )] (cid:40) m − − (cid:15)V (cid:20) (cid:0) r − (cid:1) − (cid:18) r log r − r + 1 (cid:19) + (cid:15) (cid:0) r − (cid:1) − (cid:18) r log r − r log r − r + 1 + r Li (cid:0) − r (cid:1) (cid:19)(cid:21) + m − V (cid:34) (cid:18) r x − (cid:19) − (cid:18) r x log r x − r x + 1 (cid:19) − (cid:0) r − (cid:1) − (cid:18) r log r − r + 1 (cid:19)(cid:35)(cid:41) , (156)where r ≡ M/m V . The first term in curly braces is obtained by taking x = 1 inside the (cid:82) dy integral,and the second term is the remainder having no singularity in the final (cid:82) dx integral at x = 1.Similarly we find, H (1) = [ c (cid:15) ][ x (1 − x )] − − (cid:15) (cid:15) )[Γ(1 + (cid:15) )] ∂∂m V (cid:40) δ − − (cid:15) (cid:34) Y (1) + (cid:15) (cid:18) Y + Y (cid:19)(cid:35) + δ − (cid:34) Y ( x ) − Y (1) (cid:35)(cid:41) , (157)where Y ( x ) = 2 r V − r M x (cid:26)(cid:113) r V − (cid:18)(cid:113) r V − (cid:19) − (cid:114) r M x − (cid:32)(cid:114) r M x − (cid:33) −
12 log xm V M (cid:27) , (158)54ith r V ≡ m V /δ and r M ≡ M/δ . As in the discussion after (153), continuation away from δ = 0 isgiven by taking δ → δ + iε with arbitrary choice of sgn( ε ). The remaining terms Y and Y are givenby Y = (cid:90) dy (cid:90) ∞ dβ (cid:0) r M − r V (cid:1) − ddy log (cid:104) β + 2 β + yr V + (1 − y ) r M (cid:105) = (cid:0) r M − r V (cid:1) − (cid:26) − π (cid:113) r V − (cid:104) − log (cid:16) (cid:113) r V − (cid:17)(cid:105) + 4 π (cid:113) r M − (cid:104) − log (cid:16) (cid:113) r M − (cid:17)(cid:105) − y (cid:16)(cid:113) r V − (cid:17) + y (cid:16)(cid:113) r M − (cid:17)(cid:27) ,Y = (cid:90) dy log(1 − y ) (cid:16) yr V + (1 − y ) r M − (cid:17) − arctan (cid:18)(cid:113) yr V + (1 − y ) r M − (cid:19) , (159)where y ( A ) ≡ (cid:90) dx log ( x + A ) . (160)For Y , we evaluate the remaining integral over Feynman parameter y numerically. D.3 Case of two heavy fermions
Let us set m = m = M (not to be confused with heavy WIMP mass M used elsewhere in thepaper) in ∆, and work in d = 4 dimensions. Naming x (1 − x ) ≡ z , we find, F (1) = i (4 π ) (cid:34) zm V (cid:18) − M zm V (cid:19) (cid:35) − (cid:34) M zm V log M zm V − M zm V + 1 (cid:35) ,H (1) = i (4 π ) ∂∂m V (cid:40) (cid:34) zm V (cid:18) − M zm V (cid:19) (cid:35) − (cid:34)(cid:113) m V − δ arctan (cid:32)(cid:114) m V δ − (cid:33) − (cid:114) M z − δ arctan (cid:32)(cid:114) M zδ − (cid:33) − δ zm V M (cid:35)(cid:41) . (161)The remaining integrals can be obtained by differentiating the above results with respect to M . Inpractice, we evaluate the remaining integral over Feynman parameter x (or z ) numerically. E Numerical inputs
We use the inputs of Table 1 in the numerical analysis of coefficients appearing in Fig. 6. Lightfermion masses enter the analysis indirectly via the onshell renormalization scheme. The matchingin (53) requires a limit of the photon two-point function which receives contributions from momentumregions of light ( u , d and s ) quark loops that are outside the domain of validity of QCD perturbationtheory. A complete nonperturbative treatment of this function is not numerically relevant to thepresent analysis; for definiteness, we model these contributions using MS light quark masses (cf.Table 1) in the one-loop evaluation of the two-point function. Varying these mass inputs by an orderof magnitude in either direction does not appreciably change the numerical matching coefficients ofFig. 6. 55arameter Value Reference | V td | , | V ts | ∼ | V tb | ∼ m e .
511 MeV [44] m µ
106 MeV [44] m τ .
78 GeV [44] m h
126 GeV [28, 29] m W . m Z .
188 GeV [44] Parameter Value Reference m t
172 GeV [45] m b .
75 GeV [45] m c . m s . m d .
70 MeV [44] m u .
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