Shining LUX on Isospin-Violating Dark Matter Beyond Leading Order
Vincenzo Cirigliano, Michael L. Graesser, Grigory Ovanesyan, Ian M. Shoemaker
AACFI-T13-05CP3-Origins-2013-047 DNRF90LA-UR-13-27904
Shining LUX on Isospin-Violating Dark Matter Beyond Leading Order
Vincenzo Cirigliano a , Michael L. Graesser a , Grigory Ovanesyan a,b , Ian M. Shoemaker a,c a Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA b Physics Department, University of Massachusetts Amherst, Amherst, MA 01003, USA c CP -Origins and the Danish Institute for Advanced Study, University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark Abstract
Isospin-violating dark matter (IVDM) has been proposed as a viable scenario to reconcile conflicting positive and nullresults from direct detection dark matter experiments. We show that the lowest-order dark matter-nucleus scatteringrate can receive large and nucleus-dependent corrections at next-to-leading order (NLO) in the chiral expansion. Thesize of these corrections depends on the specific couplings of dark matter to quark flavors and gluons. In general thefull NLO dark-matter-nucleus cross-section is not adequately described by just the zero-energy proton and neutroncouplings. These statements are concretely illustrated in a scenario where the dark matter couples to quarks throughscalar operators. We find the canonical IVDM scenario can reconcile the null XENON and LUX results and therecent CDMS-Si findings provided its couplings to second and third generation quarks either lie on a special line orare suppressed. Equally good fits with new values of the neutron-to-proton coupling ratio are found in the presence ofnonzero heavy quark couplings. CDMS-Si remains in tension with LUX and XENON10 /
100 but is not excluded.
1. Introduction
To date, the dominant component of the matter in the Milky Way has only been detected through its gravitationalinteractions. However, a number of experiments around the world are currently seeking to directly detect this DarkMatter (DM). The aim is detect the recoil energy deposited by an incident DM particle as it scatters on a nucleartarget, producing a characteristic spectrum [1].At present, the field of DM direct detection is in an uncertain and exciting state with a number of experimentsfinding evidence of such a signal [2, 3], and others seeming to exclude these same signals with null observations[4, 5, 6, 7]. An apparent reconciliation however may be achieved by allowing the coupling of the DM to protons, f p ,to di ff er from its coupling to neutrons, f n . While such isospin-violating Dark Matter (IVDM) has been studied bymany authors [8, 9, 10], it has become especially intriguing given the latest results from CDMS-Si [11], which arena¨ıvley at odds with the limits from XENON100 [6] and LUX [7]. For example, the authors of [12] surveyed manydi ff erent possible astrophysical and microphysical possibilities for DM and concluded that only IVDM or inelasticdown-scattering significantly reduce the tension between CDMS-Si and XENON100. After LUX, similar conclusionsare found in Refs. [13, 14], with “Xenophobic” WIMP couplings still providing a reconciliation of existing results,albeit under increasing pressure.In this paper we study the phenomenological implications of chiral NLO corrections to IVDM in light of the recentresults by LUX [7]. The chiral corrections to WIMP-nucleus cross section have been studied in Refs. [15, 16] assum-ing scalar WIMP-quark interactions (for axial interactions see [17]). In contrast to the one-nucleon-level e ff ectivefield theory (EFT) developed in Ref. [18], the chiral EFT approach includes two-body e ff ects and is particularly wellsuited to connect the phenomenological bounds on WIMP-nucleus cross sections to the WIMP-quark short-distancecouplings, controlling other aspects of WIMP phenomenology (indirect detection, production at colliders). In [16] itwas found that for generic isospin-conserving WIMP-quark couplings the magnitude of the NLO e ff ects is of the sizeexpected from chiral power counting ∼ m π / (1GeV) ∼ Preprint submitted to Elsevier October 4, 2018 a r X i v : . [ h e p - ph ] N ov anonical IVDM point r ≡ f n / f p (cid:39) − .
7, where the signal for Xe is suppressed at LO by several orders of magnitude,it was found that the chiral corrections wash out the LO cancelation generically, and move the “Xenophobic” point toother regions in the parameter space of WIMP-quark couplings. In this letter we explore in detail these points.The remainder of this paper is organized as follows. In Sec. 2 we review and update our results on scalar-mediatedDM-quark interactions, including now the momentum dependence in the two-body amplitude. In Sec. 3 we studythe degradation in sensitivity experienced by a Xenon target at NLO and compare the e ff ect of chiral corrections forXenon, Silicon and Germanium targets. In Sec. 4 we discuss parameter degeneracies and the role of hadronic andnuclear uncertainties. Then in Sec. 5 we compute the best-fit and excluded regions from the CDMS-Si, XENON, andLUX experiments respectively. There we find that the well-known r = − . pro-vided either that the strange and heavy quark couplings in the e ff ective low-energy theory are su ffi ciently suppressed,or that these couplings lie on a line corresponding to an approximate degeneracy in the total recoil rate. In addition,we also find new regions of partial compatibility for which f n / f p is significantly di ff erent from − .
7. Finally in Sec. 6we discuss the implications of these findings for future DM data, including direct detection and collider searches.
2. Setup
Below the scale of the heavy quarks, the scalar interaction of WIMPs (denoted by X ) with light quarks is given bythe e ff ective Lagrangian [16] L e ff = (cid:88) q = u , d , s λ q v Λ XX m q qq + λ θ v Λ XX θ µµ , (1)where Λ is a generic new physics scale, v = ( √ G F ) − / is the electroweak scale and θ µµ is the trace of the energy-momentum tensor. The e ff ect of WIMP couplings to heavy quarks is encoded in the coe ffi cient λ θ = (2 / (cid:80) Q ˜ λ Q − (8 / λ G , and also in the couplings of the light quarks through the relation λ q = ˜ λ q − λ θ . Here ˜ λ q , Q and ˜ λ G are theshort-distance couplings of dark matter to light quark, heavy quarks, and the gluon field strength.At leading order (LO) in chiral EFT, the four quark-level couplings λ u , d , s ,θ collapse into two independent combi-nations, i.e. the zero momentum transfer matrix elements of L e ff in the proton and neutron, f p , n , f p , n = v Λ (cid:20) σ π N ( λ + ± λ − ξ ) + λ s σ s + λ θ m p (cid:21) , λ ± = ( λ u m u ± λ d m d ) / ( m u + m d ) , (2)where σ π N = (( m u + m d ) / (cid:104) p | ¯ uu + ¯ dd | p (cid:105) , ξ = (cid:104) p | ¯ uu − ¯ dd | p (cid:105) / (cid:104) p | ¯ uu + ¯ dd | p (cid:105) , σ s = (cid:104) p | m s ¯ ss | p (cid:105) , and the upper (lower)sign refers to p ( n ) . These relations are valid up to small isospin-breaking e ff ects of order ( m u − m d ) / Λ QCD . Workingto LO in chiral EFT, it is convenient to trade f n , p for σ p ≡ k X µ f p /π and r ≡ f n / f p ( µ is the WIMP-proton reducedmass and k X = / ff erential rate is then given by: dRdE R LO = σ p ρ µ m X (cid:12)(cid:12)(cid:12)(cid:12)(cid:16) Z + ( A − Z ) r (cid:17) F ( E R ) (cid:12)(cid:12)(cid:12)(cid:12) × η ( E R , m X , m A ) , (3)where m X and m A are the WIMP and target nucleus masses, F ( E R ) is the one-body nuclear form factor, ρ is thelocal DM mass density, and η ( E R , m X , m A ) is the flux factor involving an integral over the local WIMP velocitydistribution [21, 22, 23, 24]. This is the familiar result used in phenomenological applications. Note that any value of σ p and r can be obtained by an appropriate choice of the quark couplings λ i / Λ . However, in the limit ξ → r = λ i , as seen from Eq. (2).As discussed in Ref. [16], at next-to-leading order (NLO) one needs all four λ u , d , s ,θ parameters to describe thescattering rate. The λ u , d , s ,θ couplings appear in the recoil energy dependence of neutron and proton matrix elements,as well as a new two-body contribution to the amplitude ( A ( E R )). In order to make contact with the existing phe-nomenology we choose as independent parameters the “standard” quantities σ p and r , as well as the rescaled strangeand gluonic (heavy quark) couplings λ s ,θ ≡ λ s ,θ /λ u . With this choice, the NLO WIMP-nucleus di ff erential rate reads dRdE R NLO = σ p ρ µ m X (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:20) Z (cid:16) + s p E R (cid:17) + (cid:16) A − Z (cid:17) (cid:16) r + s n E R (cid:17)(cid:21) F ( E R ) + A ( E R ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) × η ( E R , m X , m A ) , (4) For the nucleon sigma-terms we use the lattice QCD ranges σ π N = σ s = ξ can be related to y ≡ (cid:104) p | ¯ ss | p (cid:105) / (cid:104) p | ¯ uu + ¯ dd | p (cid:105) through an analysis of baryon masses in the S U (3) limit [20], leading to ξ = (1 − y ) 0 . = . up s dp s un s dn s sp , n t u t d t s - 0.116 -0.192 -0.096 -0.232 -0.472 -0.63 MeV -1.27 MeV 0.070 MeV Table 1:
Numerical values of the coe ffi cients entering the NLO amplitude. The uncertainty in the combination of low-energyconstants F / ( F + D ) ∈ [0 . , . a ff ects s u , dN at the 5% level and s sN at the 20% level [16]. The dimensionful two-body coe ffi cientst u , d , s have been estimated through a nuclear shell model calculation in Ref. [16], and are in principle subject to larger uncertainties. where s p = f u [ r , λ s ,θ ] (cid:16) s up + f d [ r , λ s ,θ ] s dp + s sp λ s (cid:17) · A (5) s n = f u [ r , λ s ,θ ] (cid:16) s un + f d [ r , λ s ,θ ] s dn + s sn λ s (cid:17) · A (6) A ( E R ) = f u [ r , λ s ,θ ] (cid:20)(cid:16) t u + f d [ r , λ s ,θ ] t d (cid:17) F ππ ( E R ) + t s λ s F ηη ( E R ) (cid:21) · A , (7)and the common factor of A arises in s p , n from q = m A E ∝ A . The quantities f u , d [ r , λ s ,θ ] arise in the change ofvariables from λ u , d to f p and r . f d is the ratio λ d /λ u expressed in terms of the independent variables r , λ s ,θ . Similarly, f u represents the ratio λ u / ( v Λ f p ) expressed in terms of r , λ s ,θ . The explicit form of f u , d depends not only on r , λ s ,θ butalso on the hadronic matrix elements appearing in (2): f u = + ξ − r (1 − ξ )2 ξ (cid:104) δ f + m u m u + m d σ π N (cid:105) , δ f = λ s σ s + λ θ m p , (8) f u f d = ( r − δ f − m u m u + m d σ π N (cid:2) − ξ − r (1 + ξ ) (cid:3) ξ (cid:104) δ f + m u m u + m d σ π N (cid:105) m d m u + m d σ π N . (9)Note that there is an apparent singularity in the above expressions when the denominators vanish. This correspondsto the limit f p →
0. In that case the fractional correction diverges, but that’s simply because we are factoring out f p .The coe ffi cients appearing in s p , n and A ( E R ) are known from the NLO EFT analysis of Ref. [16] and are reported inTable 1 . Extending the work in Ref. [16], within the shell model we include here the recoil energy dependence ofthe two-body amplitude: F ππ ( E R ) = F exp (cid:104)(cid:16) . − . A − / + . A − / (cid:17) ·| q | (cid:105) , | q | = (cid:112) m A E R (10) F ηη ( E R ) = F Bessel (cid:104)(cid:16) . + . A − / − . A − / (cid:17) ·| q | (cid:105) . (11)In the above expressions we have F exp ( q ) = exp (cid:16) − q R / (cid:17) with R ≡ (cid:104) . + .
91 ( m A / GeV) / (cid:105) fm, and F Bessel ( q ) = (cid:16) (sin( qr n ) − qr n cos( qr n )) / ( qr n ) (cid:17) × e − ( qs ) / with r n ≡ A / fm , s = A = , , , , A -dependent rescalingof the argument.
3. Degradation factors beyond leading order
Scalar-mediated interactions induce coherent WIMP-nucleus scattering, which for f p ∼ f n implies the well-knownoverall factor of A in the cross-section. In general, for f p (cid:44) f n interference e ff ects can suppress the cross-sectionrelative to the case f p = f n , and a useful measure of this suppression is provided by the so-called degradation factor [25, Note that the numerical values of t u , t d , t s depend on the nuclear matrix elements N ππ (0) and N ηη (0) [16]. In Ref. [16] these were computedwithin the shell model, using an unconventional cut on relative nucleon distance of d c = / · . N ππ (0) = − . A and N ηη (0) = . A . Here we use the more conventional cut d c = . N ππ (0) = − . A and N ηη (0) = . A . (cid:45) (cid:45) (cid:45) (cid:45) r D (cid:72) N (cid:76) L O Xe, m X (cid:61)
10 GeV; Λ s (cid:8) (cid:61) LONLO: Λ Θ (cid:61) Λ Θ (cid:61) Λ Θ (cid:61)(cid:45) Λ Θ (cid:61)(cid:45) AEB CD F (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) Λ Θ r m i n Xe, m X (cid:61)
10 GeV; position of the minimum of D NLO (cid:45) (cid:45) (cid:45) (cid:45) Λ Θ D N L O (cid:72) r m i n (cid:76) Xe, m X (cid:61)
10 GeV; value at the minimum of D NLO
Figure 1:
Left panel : Xenon degradation factors. Solid lines represent D NLO ( r , λ s , λ θ ) (Eq. (13)) with λ θ = λ s = λ θ = . λ θ = − . λ θ = − .
025 (purple line). The dashed blue line represents D LO ( r ). D LO ( r ) and D NLO ( r , ,
0) are nearly degenerate,as explained in the text. Note that for other values of λ s and λ θ the degradation factor at NLO has a sizable shift. Middle panel : Dependence ofthe position of the minimum of D NLO , denoted by r min , on λ θ , with λ s =
0. Benchmarks discussed further in the text are also shown.
Right panel :Dependence of the value of D NLO ( r min ) on λ θ with λ s =
0. Note that at r min the values of the degradation factor are nearly independent of λ θ .
26, 27, 28]. The original references worked to LO in ChPT and their definition can be cast in terms of the integratedrates ¯ R as D LO ( r ) = R LO (cid:16) r , σ p (cid:17) R LO (cid:16) , σ p (cid:17) , R ≡ (cid:90) E max R E min R dE R dRdE R , (12)with experiment-dependent integration limits E min / max R . Note that for a given isotope D LO ∝ [ Z + ( A − Z ) r ] and onecan use either the integrated or the di ff erential rate, as the energy-dependence cancels in the ratio. This is not trueanymore to NLO, so we generalize the definition of degradation factor as follows D NLO ( r , λ s , λ θ ) = R NLO (cid:16) r , σ p , λ s , λ θ (cid:17) R LO (cid:16) , σ p (cid:17) , (13)and note that while the dependence on σ p drops in the ratio, D NLO depends not only on r , but also on λ s ,θ .Inspection of Eqs. (4) through (9) shows that D NLO is still a quadratic form in r . However, as illustrated below,for a given target the location of the minimum and the value at the minimum are a ff ected in a non-trivial way by thechiral corrections.In Fig. 1 we illustrate the impact of chiral corrections on the degradation factor, using as a benchmark the Xenontarget (summing over isotopes). In the left panel we show both D LO (dashed line) and D NLO versus r for λ s = λ θ = , ± .
1. A few salient features emerge: first, in the absence of 2 nd and 3 rd generation couplings (in the low-energy theory) the NLO corrections are %-level and do not significantly a ff ect the degradation factors . However, asone “turns on” the WIMP coupling to strange and θ µµ , even at a level of 10% of the light quark couplings, the resultschange dramatically, with an O (1) shift in the value of r for which the degradation factor has a dip (compared to thewell-known LO case r (cid:39) − . A in Eq. (4), as one canverify using Eqs. (4) through (9) and typical recoil energies of O (10) keV. That the NLO corrections depend on λ θ may at first seem strange, since they do not have any such explicit dependence. Such a dependence is induced throughour choice of independent parameters (namely λ d ≡ λ d /λ u depends not only on r , but also on λ s and λ θ ).Varying λ s while keeping λ θ = ff ectof λ s ,θ is degenerate, as they appear in the linear combination δ f = σ s λ s + m p λ θ . Finally, we note that sizable shifts This can be understood as follows: in the region r ∼ − f d ∼ − m u / m d ∼ − /
2, which combined with the numerical values in Table 1simultaneously suppresses both the slopes s p , n and A , i.e. the entire NLO corrections. In the region r (cid:44) − f u , that gets suppressed by a factor of ξ ∼ .
18 compared to its value at r ∼ − Λ s , Λ Θ , Σ Π N (cid:144) MeV (cid:77) (cid:61) (cid:72)
0, 0, 45 (cid:76) , (cid:72) (cid:45) (cid:76) , (cid:72)
1, 1, 60 (cid:76)(cid:72) (cid:45) (cid:76) , (cid:72)
0, 0.1, 45 (cid:76) , (cid:45) (cid:45) (cid:45) (cid:45) f n (cid:144) f p R X e (cid:144) R S i m X (cid:61)
10 GeV (cid:45) (cid:45) (cid:45) (cid:45) f n (cid:144) f p R X e (cid:144) R G e m X (cid:61)
10 GeV
Figure 2: Double ratio of total rates ¯ R NLO ( Xe ) / ¯ R NLO ( S i ) (left panel) and ¯ R NLO ( Xe ) / ¯ R NLO ( Ge ) (right panel) versus r , for λ s = σ π N =
45 MeV,and λ θ = , − . , ± .
1. Also shown is the double ratio for σ π N =
60 MeV and λ s ,θ = σ π N =
45 MeV and λ θ = .
1. Such degeneracies are described in further detail in Sec. 4. in the minimum location arises when varying the nucleon sigma term σ π N . We will discuss in greater detail thesedegeneracies and hadronic uncertainties in Sec. 4.Given the sensitivity to the strange quark and θ µµ couplings demonstrated above, it is interesting to track the locationand depth of the “dip” in the degradation factor as a function of λ s ,θ . We illustrate this variation in the middle andright panels of Fig. 1. One can see that at NLO the dip can occur at virtually any value of r (even positive values!)provided we adjust λ θ accordingly. In the middle panel we indicate six benchmark scenarios A , B , C , D , E , F in the r , λ θ plane. We chose them in such a way that A is the canonical IVDM scenario ( r = − . λ s = λ θ =
0) while B and C are perturbations around it. The last three benchmarks correspond to plausible reconciliation of XENON / LUXwith CDMS-Si. Values of r are r = − . A , B and C , r = + .
15 for D , r = − .
45 for E , and r = − F . Wewill come back to these benchmark scenarios in the Sec 5. The right panel of Fig. 1 shows that for most values of r min the degradation is close to 10 − , suggesting that indeed there is a manifold of “Xenophobic” couplings in whichXENON / LUX exclusion regions might be consistent with signals claimed in experiments using Ge or Si targets.To make the latter point plausible, however, one needs to check that Xenon degradation at NLO is not accompaniedby excessive degradation in other targets. To this end, we plot in Fig. 2 the ratio of integrated rates ¯ R NLO ( Xe ) / ¯ R NLO ( S i )(left panel) and ¯ R NLO ( Xe ) / ¯ R NLO ( Ge ) (right panel) versus r , for λ s = λ θ = , − . , ± .
1. In obtaining theseplots we use experiment-specific energy windows, corresponding to LUX ([3 ,
30] keV), CDMS-Si ([7 , , ff erent energy thresholds, Fig. 2 stronglysupports the existence of a manifold of “Xenophobic” couplings consistent with current data.Based on these results, we expect two qualitative changes in the phenomenology of IVDM: (1) turning on sizablenonzero strange quark and / or θ µµ couplings with r = − . / LUX such that these couplings are excluded, and (2) new regions of compatibility arise in which r (cid:44) − . r = − . r , by turning on specific couplings of the WIMP to heavy quarks or gluons. Thismakes the IVDM scenario far richer, but of course more model-dependent. In Sec. 5 we investigate these possibilitiesfurther with a more detailed examination of the CDMS-Si, CDMS-Ge, XENON10 /
4. Parameter degeneracies, hadronic uncertainties, and higher order corrections
In this section we provide an analytic description of how the dominant NLO chiral corrections a ff ect direct-detection phenomenology. This explains the observed degeneracies in the λ s ,θ parameter space and allows us to assessthe impact of hadronic and nuclear uncertainties, and higher order corrections. While in the numerical studies we usethe full NLO corrections, in this Section we obtain an approximate analytic solution by keeping only the dominant5LO e ff ects. This means we: neglect (i) all slope terms compared to the two-body corrections ( s qN E R (cid:28) t u , d , s ); (ii)ignore the strange contribution to 2-body amplitude ( t s (cid:28) t u , d ); (iii) and drop terms of O ( ξ ) compared to terms of O (1). With these assumptions we find that the NLO corrections are controlled by the quantity ∆ : dRdE R NLO ∝ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:20) ZF ( E R ) + A ∆ F ππ ( E R ) (cid:21) + r (cid:20) ( A − Z ) F ( E R ) − A ∆ F ππ ( E R ) (cid:21)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (14) ∆ = ξ (cid:104) δ f σ π N + m u m u + m d (cid:105) · (cid:34) t u σ π N − t d σ π N m u + m d m d (cid:32) δ f σ π N + m u m u + m d (cid:33)(cid:35) . (15)Setting F ( E R ) = F ππ ( E R ) = r min = − ¯ Z − ¯ Z · + ∆ ¯ Z − ∆ − ¯ Z ¯ Z = Z / A , (16)where the first factor is the LO result and the second factor represents the NLO shift. After appropriate averaging overmultiple isotopes, the above expressions explain quite accurately the corrections we observe in our parameter scan.In particular, the above expressions explain very peculiar degeneracies observed when one scans in both theWIMP-quark couplings λ θ, s and in the hadronic and nuclear matrix elements σ π N , t u , d (see Figs. 2,3). All the degen-eracies derive from the relation ∆ [ λ s ,θ , σ π N , σ s , t u , d ] = constant . (17)For fixed hadronic matrix elements, this constraint describes a sub-surface in the space of couplings, independent of r . Allowing for hadronic uncertainties pu ff s the surface out into a sub-volume. For example, keeping λ s = σ s , t u , d fixed to their central values, we obtain very similar results for the three following choices: (1) σ π N =
45 MeV, λ θ = − .
15 ; (2) σ π N =
30 MeV, λ θ = + . σ π N =
60 MeV, λ θ = − .
1. They correspond to very close values of ∆ = . , . , . r min (where ¯ R NLO ( Xe ) / ¯ R NLO ( S i ) is minimized for fixed couplings λ θ and λ s ). Here one finds a range of values for r min . The right panel shows contours of ¯ R NLO ( Xe ) / ¯ R NLO ( S i ) evaluatedat r min . Here one finds the double ratio to have only O (1) variation across the plane, demonstrating the existenceof other values of r , λ s and λ θ having equally good suppression of the relative rate as compared to the canonicalIVDM scenario. In comparing the two panels note the approximate analytic and full numerical expressions have goodagreement for contours of r min , whereas for the double ratio ¯ R NLO ( Xe ) / ¯ R NLO ( S i )[ r min ] there is also good agreementover much of the panel, except in the region where r min becomes large. These two seemingly contrasting features canbe easily understood. The point is that the numerator of the double ratio is a quadratic form in r and ∆ , with slightlydi ff erent coe ffi cients between the exact and approximate expressions. Since the value of the quadratic form at theminimum is suppressed (with only one isotope it would be zero) through a cancellation between terms that are eachlarge, small di ff erences in the coe ffi cients between the full and approximate expressions lead to larger variation in thevalue of the minimum, especially as r min becomes large .An approximate degeneracy also passes through the canonical IVDM point having r = − . λ s = λ θ =
0. Thispoint has ∆ =
0, which selects δ f (cid:39) − .
118 MeV. One finds almost perfect degeneracy in the degradation variablealong this line, provided λ s < O (1). Values of couplings along this line will provide as good a fit to the direct detectiondata as the canonical point. For larger values of λ s the slope terms become important and the degeneracy weakens.This analysis illustrates an important point: hadronic uncertainties a ff ect the extraction of quark-WIMP couplingsfrom phenomenologically interesting regions in the σ p − r plane. In turn, this a ff ects other aspects of WIMP phe-nomenology such as indirect detection or collider searches. This can be understood in more detail. Indeed consider a quadratic function V ( r ) = ar + br + c . The position of the minimum and valueat the minimum are: r min = − b / (2 a ) and V ( r min ) = c − b / (4 a ). If we know the coe ffi cients a , b , c only approximately: a = a (1 + (cid:15) ) , b = b (1 + (cid:15) ) , c = c (1 + (cid:15) ), then the approximate formulas for r apprmin = r min (1 + (cid:15) − (cid:15) ) and V ( r min ) appr = V ( r min )(1 + (cid:15) ) + ar ( (cid:15) − (cid:15) + (cid:15) ) andthus if ar >> V ( r min ) the value at the minimum cannot be resolved by an approximate formula. In reality in right panel of Fig. 3 the ratio of twoquadratic equations is minimized, but the conclusions from our toy model apply. (cid:45) (cid:45) (cid:45) (cid:43)(cid:165)(cid:45)(cid:165)(cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) Λ Θ Λ s m X (cid:61)
10 GeV; r min contours of R (cid:72) Xe (cid:76)(cid:144) R (cid:72) Si (cid:76) (cid:45) (cid:45) (cid:45) (cid:45) Λ Θ Λ s m X (cid:61)
10 GeV; contours of R (cid:72) Xe (cid:76)(cid:144) R (cid:72) Si (cid:76) x10 at r min Figure 3: These two panels show the behavior of ¯ R NLO ( Xe ) / ¯ R NLO ( S i ) as a function of λ θ and λ s . The left panel shows contour lines of constant r min , where r min is the location of the minimum of ¯ R NLO ( Xe ) / ¯ R NLO ( S i ) for fixed couplings λ θ and λ s . The right panel shows contour lines ofconstant ¯ R NLO ( Xe ) / ¯ R NLO ( S i ) evaluated at the minimum r = r min . In both panels the solid lines correspond to full expressions. The dashed linescorrespond to the approximate expression of Eq. (14) properly averaged over isotopes. In the right panel the red colors give the value of the fullexpression along that contour, whereas the black colors give the value of the approximate expression. Note that good agreement between theanalytic and full expression for r min in the left panel. In contrast, in the right panel the di ff erence in the double ratio between the full and analyticexpression becomes O (1) as r min gets large; see Sect. 4 for more details. Central values of the hadronic matrix elements are assumed. Finally, the above expressions also show how the e ff ect of chiral corrections on the location of the minimum isamplified. For example a typical chiral correction ∆ ∼ .
15 implies that for both Ge and Xe the second factor inEq. (16) is about 1.8 and nearly the same for both elements because they each have ¯ Z (cid:39) .
4. The amplification arisesfrom the factors of ¯ Z , − ¯ Z and from the fact that the corrections to numerator and denominator have the oppositesign.Generalizing the current NLO analysis, one can show that to all orders in the chiral expansion the rate (Eq. 4)takes the form of Eq. (14), with the replacement ∆ · F ππ ( E R ) → ∆ χ ( E R ; λ s ,θ ) , and ∆ χ depending non-trivially on E R and λ s ,θ (keeping O ( ξ ) terms results in two di ff erent functions ∆ (1) , (2) χ in the two terms of Eq. (14)). Now, as long as ∆ χ has a well behaved expansion (i.e. there are no dynamical enhancements on the nuclear side, which we do not expectfor scalar operators), then the corrections to the rate and key quantities such as r min , R ( r min ) are well behaved. Wetherefore conclude that our analysis is robust against higher order corrections in the chiral expansion.
5. CDMS-Si vs XENON and LUX at NLO
Throughout, we will assume the Standard Halo Model (SHM), which posits ρ = . / cm and a Maxwell-Boltzmann velocity distribution with variance v =
220 km / s, earth-dark matter relative velocity v e =
220 km / s, andescape velocity v esc =
544 km / s. In this letter, we will not consider the sizeable uncertainty in the details of thelocal DM halo. The interested reader can consult previous direct detection studies which have examined in detail theastrophysical uncertainties a ffl icting direct detection experiments [29, 30, 31, 32, 24, 33, 34, 35, 36, 37, 12, 38, 14].We summarize below the key features of our fitting procedure: CDMS Si:
We use the acceptance from [11] and a total exposure of 140.2 kg-days. We consider an energyinterval [7,100] keV and bin the data in 2 keV intervals. The 3 candidate events appear in the first 3 bins. Following[12], we take the normalized background distributions from [39] and rescale them so that neutrons contribute 0.13events, Pb recoils 0.08 events, and the surface event background 0.41 surface events. To find best-fit regions we obtainthe likelihood function and simply plot constant values of the likelihood that would correspond to 68% and 90% CLregion under the assumption that the likelihood distribution is Gaussian.
CDMS Ge:
The CDMS collaboration performed a dedicated analysis of their detector at low threshold energy[40]. The experiment has a signal region from 2 keV to 100 keV. Following [41] and [35], we set limits using only7
DMS-Si L UX X E NON X E NON C D M S - G e (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) m X (cid:72) GeV (cid:76) Σ p (cid:72) c m (cid:76) Benchmark A : f n (cid:144) f p (cid:61)(cid:45) Λ s (cid:61) Λ Θ (cid:61) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) m X (cid:72) GeV (cid:76) Σ p (cid:72) c m (cid:76) Benchmark B : f n (cid:144) f p (cid:61)(cid:45) Λ s (cid:61) Λ Θ (cid:61)(cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) m X (cid:72) GeV (cid:76) Σ p (cid:72) c m (cid:76) Benchmark C : f n (cid:144) f p (cid:61)(cid:45) Λ s (cid:61) Λ Θ (cid:61) Figure 4: Best-fit CDMS-Si (contours at 68% and 90% CL) and XENON / CDMS-Ge / LUX exclusions (at 90 % CL) under di ff ering assumptionslabelled on the top of each panel. In all cases, we have set λ s = r = − . λ θ turned on. Note that for both points the region allowed in the left panel is now excluded. one of their Ge detectors - T1Z5 - that apparently has the best quality data. We use the e ffi ciencies and total exposureprovided by the supplemental information to [40]. The total exposure of this detector was 35 kg–days. To account forthe finite energy resolution of the detector, the energy of the nuclear recoil is smeared according to [42] with an energyresolution ∆ E = . √ E / keV keV [35]. This experiment saw 36 events in their signal region whose origin remainsundescribed. To set a conservative upper limit we attribute all of these events to signal - following the experimentalcollaboration and other theory papers [41, 35]. Using Poisson statistics a 90% C.L. signal upper limit of 44 events isobtained.For the Xenon10, Xenon100 and LUX experiments we follow [43] and convolve the energy-rate dR / dE with aPoisson distribution in the number of photoelectrons or electrons detected. The mean number of electrons expected ν ( E ) is specific to each experiment, depending on energy-dependent light or electron yields, and on scintillatione ffi ciencies. LUX:
The first data release from LUX [7] has an exposure of 10,065 kg–days. An upper limit of 2.4 signal eventsfor m DM <
10 GeV is reported [44], with up to 5.3 events allowed for larger masses. We conservatively apply a limitof 2.4 signal events to the whole mass range m DM ∈ (5 ,
30) GeV. We use the acceptance provided by [7]. We use theenergy-dependent light-yield L y presented in [44], including a sharp cuto ff at 3 keV. We use the scintillation e ffi ciency L e f f provided by [45]. After convolving, we then sum over the S1 signal region (2,30), finding good agreement withthe LUX limits [7]. Smearing the number of photoelectrons produced with a gaussian to model the response of thedetector, as in [43], with a variance of 0.5 PE (photoelectrons), does not appreciably a ff ect our limits. Xenon10:
While the values of the electron yield Q y ( E ) at low energies are controversial, here we simply adopt thecollaboration’s parameterization from Fig.1 of [5], assuming a sharp cuto ff to zero at 1.4 keV nuclear recoil energy.Their signal region is from 5 electrons to ≈
35 electrons, corresponding to nuclear recoils of ≈ ff ective exposure of 6.25 kg–days. A limit is obtained using Poisson statistics with 23 events expected and 23detected, allowing 9.2 events. Xenon100:
We use the mean ν ( E ) characterized by [43]. For the scintillation e ffi ciency L e f f we use the e ffi ciencyused in Xenon100’s 225-live-day analysis [6], that can be found in Fig.1 of ref. [46] and includes a linear extrapolationto 0 for E below 3 keV. The response of the detector is modeled by a Gaussian smearing with a mean n and variance √ n σ PMT with σ PMT = . ff erential rate over the signal region - whichfor the analysis in [6] corresponds to S ∈ (3 ,
30) PE - and use a total exposure of 225 ×
34 kg-days [6]. We then usePoisson statistics to obtain a 90% C.L. upper limit where 1 background event is expected and 3 observed.In general we find our exclusions and best-fit region of LO analysis for r = r = − . λ s = λ θ = , ± .
1. Our fit for r = − . λ s = λ θ = A ) agrees8 (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) m X (cid:72) GeV (cid:76) Σ p (cid:72) c m (cid:76) Benchmark D : f n (cid:144) f p (cid:61) Λ s (cid:61) Λ Θ (cid:61)(cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) m X (cid:72) GeV (cid:76) Σ p (cid:72) c m (cid:76) Benchmark E : f n (cid:144) f p (cid:61)(cid:45) Λ s (cid:61) Λ Θ (cid:61)(cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) m X (cid:72) GeV (cid:76) Σ p (cid:72) c m (cid:76) Benchmark F : f n (cid:144) f p (cid:61)(cid:45) Λ s (cid:61) Λ Θ (cid:61) Σ Π N (cid:61)
60 MeV (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) m X (cid:72) GeV (cid:76) Σ p (cid:72) c m (cid:76) Benchmark G : f n (cid:144) f p (cid:61)(cid:45) Λ s (cid:61) Λ Θ (cid:61) Figure 5: Same assumptions as in Fig. 3 with unconventional choices of r that are excluded at LO. Note especially the panel on the bottomright-side which compared to the other panels has a di ff erent choice of σ π N =
60 MeV. The allowed and excluded regions are practically identicalto the panel on the bottom left-side having the same value of r . The similarity of these two panels illustrates the interplay of allowed or excludedregions and uncertainties in the hadronic parameters. well with the LO fits in the literature (see e.g. [12, 25], and recently, [13]). The r = − . λ θ = λ s = r , they lead to qualitatively di ff erentfits as expected, with a valid region in the parameter space consistent with CDMS-Si signal and LUX bound onlyfor λ s = λ θ = λ θ = ± . λ u , results in a completely excludedregion with r = − .
7. Thus for r = − . δ f (cid:39) r = − . ff erent valuesof r arise. This indeed seems plausible given the results of Sec. 3. Inspecting the left panel of Fig. 1 we see threechoices of parameters that may result in an improved compatibility between LUX and CDMS-Si: (1) Benchmark D : λ θ = − . λ u with r = + .
15, (2) Benchmark E : λ θ = − . λ u with r = − .
45, and (3) Benchmark F : r = − λ θ = + . λ u . This observation motivates the choice of Benchmarks D , E and F whose fits are shown in Fig. 5. Wesee that these very di ff erent choices of − . (cid:46) f n / f p (cid:46) .
15 can result in a comparable reduction in tension betweenthe Xenon based experiments and CDMS-Si. In the absence of NLO corrections, these benchmarks would be stronglyexcluded.Lastly, we choose Benchmark G ( λ θ = λ s = r = −
1) to illustrate one of the degeneracies discussed in Sect.4. The fit with this set of parameters is illustrated in the bottom right panel of Fig. 5. This final benchmark is chosenwith σ π N =
60 MeV, such that it is roughly degenerate with Benchmark F . Upon inspection of the fits resulting fromthe two benchmarks, we see that indeed all the experiments have nearly identical sensitivities. This final benchmarkrequires σ π N to be high in order to remain consistent with the constraints from LUX, and is completely excluded at90% CL with σ π N at its central value of 45 MeV. 9 . Conclusions The CDMS-Si data remain intriguing and may point to a DM candidate with couplings to quarks that are isospin-violating. For a representative case of scalar-mediated DM-quark interactions, we have studied the e ff ect of long-distance QCD corrections for IVDM models. We use chiral EFT and connect the short-distance coe ffi cients directlyto the DM-nucleus cross section.At leading order in chiral power counting it is well-known only two short-distance parameters appear, r and σ p .At next-to-leading order, however, for a scalar operator two additional parameters appear. We choose for conveniencethe following independent parameters r , σ p , λ s , λ θ , that can all take arbitrary values. In the limit of light DM particles,the chiral corrections are dominated by the two-nucleon amplitude, for which more work beyond the nuclear shellmodel would be highly desirable. We find that for a broad set of values of extra parameters λ s and λ θ qualitativechanges for IVDM phenomenology occur. These can be divided into two categories.In the first category, the standard r = − . ff ect of theLO tuning of Xenon signal. It should be noted however, that for special scenarios, when in the low-energy theoryeither the DM only has couplings to the first generation quarks or has couplings lying on the δ f ≈ degeneracy (seeSec. 4), we find that NLO corrections are small, which can be seen from the left panel in Figure 1. This situation isquite special, as can be seen from the same figure: by turning up λ θ by only 10% of the value for λ u , the value for r shifts by a number of the order of 1. Consistently, in Figure 4 we see that holding r = − . | λ θ | from 0 to 0 . r (cid:44) − .
7, that are excluded by the leading order analysis, canat NLO partially reconcile the LUX and CDMS experiments, though strong tension remains. We find that values aslow as r = − . r = −∞ to r = ∞ isallowed for given (tuned) values of the extra parameters λ s and λ θ . So there is a manifold of “Xenophobic” couplingsthat extends beyond the canonical point r = − . , λ s ,θ = A ). This makes the IVDM scenario richer,but more model-dependent. A case in point is provided by the comparison of benchmarks points A and G . Whileleading to very similar direct detection phenomenology, they have quite distinct short-distance couplings. Benchmark A , having r = − .
7, corresponds to λ d /λ u = ˜ λ d / ˜ λ u (cid:39) − . λ s ,θ =
0. On the other hand benchmark G , having r = −
1, has a much larger relative coupling to the heavy quarks or gluons ( λ s = λ θ =
1) and an even larger relativee ff ective coupling to the down quark: λ d (cid:39) −
34. In terms of ratios of short-distance couplings, ˜ λ s / ˜ λ u = λ θ / ˜ λ u = . λ d / ˜ λ u (cid:39) − . ff erent values of the λ d , λ s and λ θ couplings can lead to similar direct-detection phenomenology, one expectsthe constraints and signatures arising from colliders will be important to further distinguish viable scenarios.
7. Acknowledgements
We would like to thank Peter Sorensen for invaluable information on the XENON10 and LUX experiments. MGand VC would like to acknowledge support from the Department of Energy O ffi ces of High Energy and NuclearPhysics, and the Los Alamos LDRD program o ffi ce. MG would also like to thank the GGI for hospitality and partialsupport were part of this work was completed. IMS would like to thank Josef Pradler for discussions on the analysisof direct detection data. The CP -Origins centre is partially funded by the Danish National Research Foundation,grant number DNRF90. References [1] M. W. Goodman and E. Witten, Phys.Rev.
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