The µ d capture rate in effective field theory

Transcription

1 USCNT)-Report-1-9 The µ d capture rate in effective field theory S. Ando, T.-S. Park, K. Kubodera,F.Myhrer Department of Physics and Astronomy, University of South Carolina, Columbia, SC 98, U.S.A. Muon capture on the deuteron is studied in heavy baryon chiral perturbation theory HBχPT). It is found that by far the dominant contribution to µd capture comes from a region of the final three-body phase-space in which the energy of the two neutrons is sufficiently small for HBχPT to be applicable. The single unknown low-energy constant having been fixed from the tritium beta decay rate, our calculation contains no free parameter. Our estimate of the µd capture rate is consistent with the existing data. The relation between µd capture and the νd reactions, which are important for the SNO experiments, is briefly discussed. PACS : 1.39.Fe, 3.4.-s address : sando@nuc3.psc.sc.edu address : tspark@nuc3.psc.sc.edu address : kubodera@sc.edu address : myhrer@sc.edu 1

2 1 Introduction Electroweak processes in the two-nucleon systems invite detailed studies for multiple reasons. From the nuclear physics point of view, these processes offer a valuable testing ground of the basic inputs of nuclear physics. In astrophysics, the precise knowledge of the pp fusion cross section is of crucial importance for building a reliable model for stellar evolution [1]. Furthermore, experiments at the Sudbury Neutrino Observatory SNO) [, 3] to observe solar neutrinos with a heavy-water Cerenkov counter have made it extremely important to estimate the νd reaction cross sections with high precision. In this note we study µd capture: µ + d ν µ + n + n, using effective field theory EFT). Our work is connected to the above-mentioned urgent need of accurate estimates of the νd cross sections, σ νd. To expound this connection, we will first explain the standard nuclear physics approach SNPA), see e.g. Ref.[4]. This is a highly successful method for describing nuclear responses to electroweak probes. In this approach we consider one-body 1B) impulse approximation terms and two-body B) exchange-current terms acting on non-relativistic nuclear wave functions, with the exchange currents derived from a oneboson exchange model. The vertices in the relevant Feynman diagrams are obtained from a Lagrangian constructed to satisfy the low-energy theorems and current algebra [5], while the nuclear wave functions are generated by solving the A-body Schrödinger equation, H Ψ A = E Ψ A, where the Hamiltonian H contains realistic phenomenological nuclear potentials. The most elaborate study of µd capture based on SNPA was carried out by Tatara et al. TKK) [6] and by Adam et al. [7]. Now, the best available estimation of σ νd based on SNPA is due to Nakamura et al. NSGK) [8], while that based on EFT is due to Butler, Chen, and Kong BCK) [9]. Since EFT is a general framework [1], it can give model-independent results, provided all the low-energy coefficients LEC) in the effective Lagrangian, L eff, are predetermined. L eff considered by BCK, however, does contain one unknown parameter L 1A ), which they adjusted to reproduce σ νd obtained by NSGK. After this adjustment, the results of BCK are found to be in perfect agreement with those of NSGK. The fact that an ab initio calculation modulo one free parameter) based on EFT reproduces σ νd of NSGK extremely well offers strong support to the calculation based on SNPA. At the same time, it stresses the importance of carrying out an EFT calculation free from an adjustable parameter. An interesting possibility is to use µd capture data as input to control the unknown LEC. An immediate question, however, is whether this process is gentle enough to be amenable to EFT. The substantial energy transfer accompanying the disappearance of a muon can lead to a region of the final three-particle phase space in which the intrinsic state of the two neutrons receives such a large momentum that the applicability of EFT becomes a delicate issue. Let this unfavorable kinematical region be called the dangerous region.

3 The problem of the dangerous region is reminiscent of the difficulty one encounters in applying EFT to threshold pion production in N + N N + N + π [11, 1, 13, 14]. It will turn out see below), however, that, unlike the pion production case, µd capture receives only a tiny fraction of contribution from the dangerous region, and therefore the theoretical uncertainty caused by the dangerous region is practically negligible. The calculation in [9] used the power divergence scheme PDS) [15]. We employ here another form of EFT, the Cut-off Weinberg scheme CWS). In CWS, the transition operators are derived from irreducible diagrams in heavy-baryon chiral perturbation theory HBχPT), and the nuclear matrix elements are obtained by sandwiching them between initial and final nuclear wave functions obtained in SNPA. As discussed in [16], a nextto-next-to-next-to-leading order N 3 LO) calculation based on CWS contains one unknown LEC, denoted by ˆd R.LikeL 1A discussed by BCK, the parameter ˆd R controls the strength of a short-range exchange-current term and, once ˆd R is fixed from data, we can make a definite EFT prediction for σ νd. We therefore investigate here the relation between ˆd R and the µd capture rate, Γ µd. Our study is essentially of exploratory nature, given the present limited accuracy see below) of the experimental value of Γ µd. Another important point concerning ˆd R is that, as emphasized in [16], the strength of ˆd R can be reliably related to the tritium β-decay rate, Γ t β. Thus, using the experimental value of Γ t β, which is known with high precision, one can determine ˆd R and then proceed to make predictions on various two-nucleon weak-interaction processes, including the µd capture rate, pp fusion rate, and νd cross sections. We will present here the first estimate of the µd capture rate obtained in this approach; the pp fusion rate has already been discussed in [16], and the νd cross sections will be reported elsewhere [17]. The capture rate Although µd capture can in principle occur from the two µd hyperfine states S µd =1/ and S µd =3/), the capture is known to take place practically uniquely from the hyperfine doublet state. Therefore, concentrating on this dominant capture, we refer to hyperfinedoublet µd capture simply as µd capture and denote the hyperfine-doublet µd capture rate by Γ µd. The measured value of Γ µd is Γ exp µd = 49 ± 4 s 1 [18] and Γ exp µd = 47 ± 9 s 1 [19]. We remark that a high-precision measurement of Γ µd is being contemplated at PSI []. Denoting by L the orbital angular momentum of the two-neutron relative motion in the final state, we can write Γ µd = Γ L µd, 1) L=,1,... where Γ L µd istherateofµd capture leading to the L state. Here we shall be primarily concerned with Γ L= µd since it is this quantity that contains information about ˆd R. The 3

4 contributions of Γ L µd L 1) are significant,#1 but their calculation is not expected to involve any major EFT-related issues. In general, due to the centrifugal force, the L 1 contributions cannot be too sensitive to short-range physics, which implies that chiral expansion for them should converge rapidly. Specifically, the L = 1 contributions are dominated by the axial-charge AC) and E1 transitions, whose one-body operators which are NLO in chiral counting) are well-known. The lowest order meson-exchange corrections MEC) to the one-body operators come from soft one-pion-exchange OPE), which is N LO in chiral counting. These soft-ope terms, dictated by chiral symmetry, are well known, and they are model-independent. For L states, within the accuracy of our evaluation only one-body contributions [6] have to be included. In our exploratory study, therefore, we concentrate on a detailed evaluation of Γ L= µd,andforγl µd L 1) we simply use the results obtained by Ref.[6]. Muon capture by the deuteron is effectively described by the current-current hamiltonian of weak interactions H W = G V d 3 xl α x)j α x)+h.c., ) where the leptonic and the hadronic charged currents are L α x) = ψ ν x)γ α 1 γ 5 )ψ µ x) and J α x) = V α A α ) a=1 x) iv α A α ) a= x), 3) respectively, and G V = GeV [1]; α a) is the Lorentz isospin) index. In the center-of-mass system of the initial µ-d atom from which capture occurs, we can safely assume p µ = p d =. Consequently, the four-momentum transfer to the leptonic system, q α p ν p µ ) α,readsq, q) =E ν m µ, p ν ). The µ-d capture amplitude is then given by f H W i = G V Ψ µ d ) l α Ψ nn q, p; s 1 s ) j α q) Ψ d s d ), 4) where the Fourier transformed currents are j α q) d 3 x e i q x J α x), 5) l α e i x q νp ν,s ν ) L α x) µ p µ,s µ ) =ū ν p ν,s ν )γ α 1 γ 5 )u µ p µ,s µ ). 6) In Eq.4), Ψ µ d ) = 1/ πa 3 is the 1S wave function of the µ-d atom, where a m µ + m d )/m µ m d α,withα 1/ the fine structure constant. # Ψ d s d ) in Eq.4) represents #1 For example, Γ L=1 µd 1 3 Γ µd according to Ref.[6]. # The small correction due to the finite size of the deuteron is taken into account in the actual calculation. 4

5 the deuteron wave function with the z-component of its spin s d ;Ψ nn q, p; s 1 s )represents the final nn wave function, with total nn momentum q, relativenn momentum p = p 1 p )/, and the z-components of the neutron spins, s 1 and s. It is easy to obtain Γ µd = G V Ψ µ ) d 3 p d 3 p ν 4J µd +1) π) 3 π) 3 πδ E) J µd Ψ nn j α q) Ψ d l α 1 S µd = J µd s 1 s s µs d,s µ;1,s d J µd,s µd, 7) where J µd = 1, E is the energy difference between the final and initial states, Ψ d Ψ d s d ) and Ψ nn Ψ nn q, p; s 1 s ). 3 HBχPT Lagrangian and the hadronic currents To calculate the capture rate we adopt Weinberg s power counting rule. In HBχPT the leading order LO) Lagrangian is given by L = B [iv D +ig A S ] B 1 C A BΓA B ) + f π Tr i µ i µ )+ f π 4 Trχ +)8) A [ ] { } with D µ µ + 1 ξ, µ ξ i ξ R µ ξ i ξl µξ, µ = 1 ξ, µ ξ + i ξ R µ ξ i ξl µξ and ) ) χ + = ξ χξ + ξχ ξ,wherer µ = τ a Vµ a + Aa ν and L µ = τ a Vµ a Aa ν denote the external gauge fields. In the absence of the external scalar and pseudo-scalar fields χ = m π,andwe define the pion field as ξ =exp i τ π f π ). It is convenient to choose the four-velocity v µ and ) the spin operator S µ as v µ =1, ) and S µ =, σ. The next-to-leading-order NLO) Lagrangian including the 1/m N terms) in the one-nucleon sector is given in [] while that in the two-nucleon sector is given in [1] #3. Combining them, we can write the NLO Lagrangian relevant to our case as ) L 1 = B v µ v ν g µν D µ D ν + c 1 Trχ + + 4c g A v i ) +4c 3 i i + m N m N + c ) [S µ,s ν ][i µ,i ν ] i 1+c ) 6 [S µ,s ν ] f µν + B m N m N 4id 1 BS B BB +id ɛ abc ɛ µνλδ v µ ν,a BS λ τ b B BS δ τ c B +, 9) where ɛ 13 =1, µ = τ a a µ,andf µν + = ξ µl ν ν L µ i [L µ, L ν ])ξ + ξ µ R ν ν R µ i [R µ, R ν ])ξ. We find it convenient to use the dimensionless low-energy constants ĉ s and ˆd s defined by c 1,,3,4 = 1 ĉ 1,,3,4, and d 1, = g A ˆd m N m N fπ 1,. 1) #3 Our definition of the pion field differs from that used in Ref. [1] by a minus sign. 5

6 The values of these low energy constants, ĉ 1,,3,4, are taken from Ref.[]: ĉ 1 =.6 ±.13, ĉ =1.67 ±.9, ĉ 3 = 3.66 ±.8, ĉ 4 =.11 ±.8 11) and c 6 = κ V =3.7. These values were determined at tree level or NLO) in the one-nucleon sector, which correspond to N 3 LO in our two-nucleon calculation. 3.1 The one-body currents The one-body currents can be obtained either by an explicit HBχPT calculation or by the Foldy Wouthuysen FW) reduction of the well-known relativistic expressions. The former method requires one-loop diagrams which consist of vertices from L for N LO contributions) and those which contain one vertex from L 1 for N 3 LO contributions); also needed are the corresponding counter-terms from L and L 3. We adopt here the FW reduction method for convenience. Since the range of t q = m µ m µ E ν )forµd capture is small m µ < t m µ ), the t-dependences in the standard form factors, F 1, V t)andg At), give only less than % effects, which can be reliably taken into account by expansion in t, F1 V t) =1+t 6 r V + Ot ), F V t) =κ V + Ot) andg A t) =g A 1+ t 6 r A + Ot ) ). Keeping the terms linear in t is consistent with HBχPT to the order we calculate. Some caution is required for the G P t) term, which contains the pion-pole contributions, G P t) m N βt) m NG A t) m, 1) π t where βt) is a slowly varying function. Comparison with the explicit HBχPT calculation up to N LO [3] leads to βt) = f πg πnn g A m N 1 ) Q 3 6 r A m π + O Λ 3 =1+[. 1.5) ±.3] %. 13) χ The authors of Ref. [6] found that Λ µd is reduced only by %whenβ increases by 1 %. Thus, limiting ourselves to the β = 1 case entails at most.4 % error. #4 The resulting one-body vector 1B) current components are V, 1B q) = i V 1B q) = i τ i e i q r i τ i e i q r i [ 1+ t 6 r V q 8m N +1+κ V ) i q σ i p i [ p i + i 1 + κ V ) q σ i m N +1+κ V ) 4m N ] q κ V 4m, N i σ i p i 1 q ) ω 4m N #4 If the deviation of β from 1 were important, we would have to include the one-body pseudoscalar term, ˆP 1B in Eq.15), as is necessary for the two-body ˆP B = ˆP in Eq.18). ] 14) 6

7 where τ i 1 τ x i iτ y i )and p i = p i + p i)/. The one-body axial-vector current components can be written for convenience as which defines Â1B, where Â, 1B q) = i ˆ A 1B q) = i g A τ i e i q r i g A τ i e i q r i A α 1B = Âα 1B + [ σi p i ω σ ] i q, m N qα m π tq βâβ 1B, 15) 8m N [ σ i 1+ t ) 6 r A + p i σ i p i σ i p i ) 1 q σ ] i q + i q p i 4m 16), N The above equations correspond to N LO in HBχPT [4]. Apart from the mentioned t- dependence of the form factors, the N LO contribution is found to be negligible,.1 %, indicating a rapid convergence. We therefore limit ourselves to N LO for the one-body currents although, to be completely consistent, we should in principle include N 3 LO. 3. Two-body exchange currents Since the evaluation of the two-body exchange current is the focus of this work we discuss the various parts of this exchange current in detail. We write the two-body vector and axial-vector currents as V µ B k 1, k )anda µ B k 1, k ), where k i = p i p i is the momentum transferred to the i-th nucleon. For the two-body vector-charge current we know VB k 1, k )=OQ 4 ) [5], which therefore can be ignored. The spatial components of the vector current have a one-pion-exchange contribution of order Q [6]: [ V B k 1, k ) = iτ 1 τ ) g A σ1 σ k ) 4fπ m π σ σ 1 k 1 ) k m π k 1 σ1 + k ) 1 σ k ) ] m π k1 m π k k k 1 ) + OQ 4 ), 17) where f π 93 MeV is the pion decay constant, and we make use of the notation τ 1 τ ) τ 1 τ ) x iτ 1 τ ) y. Analogously to Eq.15), we write the axial-vector current as [7, 16], A α B = Âα B + qα m q β Â β B π t + ˆP ), 18) with ) Â B k 1, k ) = i τ 1 τ ) ga σ 1 k ) 1 4fπ m π k1 g ) A m N fπ ĉ +ĉ 3 g A τ k1 σ 1 k m π k1 +1 ), 19) 7

8 ˆ A B k 1, k ) = g A m N fπ 1+c6 + 4 [ + ˆP k 1, k ) = g Am π m N f π {[ i τ 1 τ ) p 1 +4ĉ 3 τ k +ĉ ) τ 1 τ ) σ 1 k ) ) ] τ 1 τ ) σ σ 1 q) k m π k ] } ˆd 1 τ1 σ 1 + τ σ )+ ˆd τ 1 τ ) σ 1 σ ) +1 ), ) { 8ĉ 1 τ σ k ) } m π k +1 ). 1) Only one combination of the LEC, ˆd 1 and ˆd, is relevant for the µd capture process, ˆd R ˆd 1 +ˆd + 1 3ĉ3 + 3ĉ ) Exactly the same combination of LEC s appears in triton β-decay, pp-fusion and the solar hep process[16]. Adopting the same strategy as in Ref. [16], we fix ˆd R from Γ t β exp), the experimental value of the tritium β-decay rate. To facilitate the calculations, we perform a Fourier transformation FT) of the above two-body currents. To control short-range physics in performing FT, we introduce a Gaussian cut-off regulator S Λ k )=exp k ) Λ. 3) where Λ is a cut-off parameter. It is to be emphasized that, although our calculation without regularization involves no infinities, we still need a regulator since EFT, by definition, is valid only up to a certain momentum scale. The regulated delta and Yukawa functions read δ 3) d 3 Λ r) q π) 3 S Λ q )e i q r = Λ3 exp Λ r ), 4π) 3 4 d 3 q y Λ m, r) π) 3 S Λ q )e i q r 1 q + m, 4) We remark that this is exactly the same regularization method as used in Ref. [16]. In performing FT, we need to specify the time components of the momentum transferred to the nucleons. Energy conservation imposes the constraint: k1 + k = q = m µ E ν. In our calculation we will adopt the so-called fixed-kinematics assumption FKA) [11], where the energy transfer is assumed to be shared equally between the two nucleons, i.e. k1 = k = mµ Eν, which naturally brings in the quantity m π m π m µ E ν ) /4. The uncertainty related to FKA becomes large as q grows. The contribution from the large q region, however, will turn out to be so tiny that the assumptions related to ki cause little uncertainty in our calculation. 8

9 4 The capture rate for the transition to the 1 S nn state The deuteron and the 1 S wave function may be written as ψ d r; s d )= 1 [ 4πr u d r)+ S 1ˆr) 8 ] w d r) χ 1,sd ξ,, ψ r) = 1 u r)χ, ξ 1, 1 5) 4πr with dr [ u d r)+w d r)] = 1 and lim r u r) = sin δ p [cos pr +cotδ sin pr]. Here S 1 ˆr) =3 σ 1 ˆr σ ˆr σ 1 σ, χ ξ) is the Pauli spinor isospinor), and δ is the nn 1 S phase shift. To facilitate numerical work, we approximate E as E = E ν + m + p + E ν 4 M µd + O pν p) ) 4m 3, 6) where m m n = MeV is the neutron ) mass, M µd m µ + m d = MeV. In our calculation we will neglect the O pν p) 4m term since, as we shall show, the major 3 contributions comes from the low p p region. Choosing the z-axis along p ν, we write q = p ν = E ν ẑ. This simplifies the structure of the transition amplitudes as ψ j q) Ψ d s d ) = δ sd,m t, ψ ê λ j q) Ψ d s d ) = δ sd,λm λ, 7) where ê ± = ˆx ± iŷ)/, ê =ẑ, andλ = ±1,. We decompose the matrix elements into vector and axial vector current contributions, M t,λ = M t,λ [V ] M t,λ [A], and arrive at Γ L= µd = G V Ψ µ ) p max π dp p Eν 1 E ) ν M 1 + M M t. 8) M µd 3 Note that M 1 = M +1 [V ]+M +1 [A]), M = M [A] and, to the order under consideration, M t = M t [A]. The matrix elements of the vector current are M λ [V ] = λ dr {q u u d j j ) w d µv ωu 1) u d + w ) } d µ V 1 j 1 m N 4m N + λ 4 ) ) g 1 [ A q dr u dx j x u d jx w d )y L ) 3 yl 1 xqrj x 1 8f π u d + w ) d y L j x u d j ) ) ] x + jx w d y1 L, 9) ) where jn x j n qrx) are the spherical Bessel functions, yn L y nλ m π + 1 4x 4 q,r and y 1Λ m, r) r r y Λm, r) andy Λ m, r) 1 m r r r y Λm, r). Using Eq.18), we obtain for the axial current { } { Mt [A] Mt = [Â] } { 1 ω M λ [A] M λ [Â] + m π t δ λ, q 9 1 r } ωm t [Â] q M[Â]+M[ ˆP ) ], 3)

10 M t [Â] = { 1 g A dr u 1) m u d w d )j 1 qω N 8m u u d j + } w d j ) N [ ) ] ga fπ 1 ĉ +ĉ 3 g A mµ E ν dr u u d y 1Λ w d )j 1 8 m N r, 31) M λ [Â] = ] t6 p q g A dr {[1+ ra 3m δ λ, N 8m u u d j w ) d j λ N where 1 6m N u ) g A 4 ) m N fπ m π 3 y Λ + m π 3 y Λ [ u d w ) d j λ ] } w d j dr [ y1 O kin 1+c 6 r 4 ) u u d j w d j λ ĉ 3 +ĉ ) ĉ 3 ĉ 4 1 ) u wd j u d w ) d )j λ 4 1 δ λ, ) q u u d + w d )j 1 ) ] + ˆd R δ Λ r)u u d, 3) M[ ˆP ] = 4 g [ A ) m N fπ dr ĉ 1 m y 1Λ π r u u d ] w d )j 1, 33) O kin q = δ λ, 8 u u d w d )j [ 1 j + j λ ) u u d w d) u u d ] w d ) 1 4 j j λ )u w d, 34) j λ 1 3δ λ,)j, j n = j n 1 qr), u1) r) =u r) u r) r and u ) r) =u r) 3u r) r +3 u r). r In the above expressions the curly brackets denote 1B contributions, and for clarity we have suppressed the dependence on r in some equations. 5 Results Table 1 shows Γ L= µd as a function of the cut-off parameter, Λ. As discussed, the shortrange exchange current contribution depends on the single low-energy constant ˆd R,seeEqs. 3), ), and ˆd R determined from Γ t β exp) is a function of Λ see Ref.[16]). We observe that the variation of Γ L= µd over the range of Λ under consideration is less than.7 s 1. The ˆd R -dependence in the table indicates the importance of the contribution of the shortdistance exchange current. Without the ˆd R term, Γ L= µd would change as much as 16 s 1 for Λ = 5-8 MeV. Thus, renormalizing the ˆd R -term using Γ t β exp) reduces the variation of Γ µd with respect to Λ by a factor, leading to the practically Λ-independent behavior 1

11 Λ MeV) ˆdR Γ L= µd [s 1 ] 5 1. ± ˆd R ˆd R ) = 45. ± ± ˆd R +.13 ˆd R ) = 45.3 ± ± ˆd R +.7 ˆd R ) = 45.7 ±.6 Table 1: L = capture rate in s 1 ) calculated as a function of the cutoff Λ. Also listed are the corresponding values of ˆd R determined from Γ t β exp) [16]. of Γ µd. Considering this stability we will hereafter only discuss the case corresponding to Λ = 6 MeV and ˆd R =1.78. The capture rate contains several interference terms, which are listed in Table in a cumulative manner. We note that the axial charge AC) plays only a minor role; its Γ µd [s 1 ] L = L 1 Total GT GT+AC GT+AC+M1 1B B+B Table : Cumulative contributions to Γ µd calculated for Λ = 6 MeV and ˆd R =1.78). The row labeled 1B corresponds to the case that contains one-body contributions only, while the row labeled 1B+B to the case that includes both the one-body and MEC contributions. The three columns labeled L= show contributions from the L = channel, with the contributions of the different transition operators displayed in a cumulative manner. The fifth column gives contribution from the L 1 channels, as evaluated in TKK, Ref.[6], and the last column shows the sum of the L =andl 1 contributions. destructive interference with GT decreases the capture rate by 1 s 1. Meanwhile, the M1 contribution interferes constructively with GT, increasing Γ µd by 59 s 1. Furthermore, the two-body MEC in the L = channel increases the capture rate by 13 s 1. Our final result for Γ L= µd = 45 s 1 in Table should be compared with TKK s result, Γ L= µd TKK) = 59 s 1. #5 By adding the 1 L 5 contribution, Γ L 1 µd = 141 s 1, calculated by TKK, we arrive at the total capture rate Γ µd = 386 s 1, 35) #5 We have re-run the code of TKK using g A =1.67. TKK s original result corresponding to g A =1.6 was Γ L= µd TKK)= 57 s 1. 11

12 to be compared with TKK s result Γ µd TKK) = 397 4) s 1. As mentioned earlier, a primary question is whether µd capture process is gentle enough for applying HBχPT with reasonable confidence. As noted the dangerous region for HBχPT occurs when the two neutrons carry most of the final energy. To address this issue, it is useful to consider the differential capture rate, dγ µd /de nn,wheree nn m n + p m n ) is the energy of the final two-neutron relative motion. An equally informative quantity is the cumulative capture rate Γ µd E nn ) Enn dγ µd /de nn ) de nn. 36) From these quantities we can assess to what extent µd capture is free from the dangerous kinematic region. We show in Table 3 the matrix elements, M +1 [A], M +1 [V ], M [A] and E nn M +1 [A] M +1 [V ] M [A] M t [A] Γ L= µd [s 1 ] Enn max Table 3: Matrix elements calculated for representative values of E nn MeV) and the cumulative L = capture rate for the case: Λ = 6 MeV and ˆd R =1.78. In each entry for the matrix element, the first number preceding a + or sign) gives the one-body contribution, while the second number gives the two-body contribution. M t [A], calculated for representative values of E nn, and for Λ = 6 MeV and ˆd R =1.78. Table 3 also gives Γ L= µd E nn). The graphical representation of Γ µd E nn ) can be found in Fig. 1. We learn from Table 3 that the matrix elements decrease quite fast as E nn increases, a feature that can be easily understood as follows. The 1 S nn radial wave function is proportional to sin δ p = ± [ p cot δ ) + p ] 1. Since the nn scattering length is very large, p cot δ diminishes rapidly when the nn relative momentum p gets small. The examination of Table 3 also reveals that the one-body amplitudes decrease more quickly than the twobody amplitudes. This is a consequence of the softness of the deuteron wave function, which cannot supply high momentum transfers needed for producing large values of p. As a result, the contributions from high E nn where the applicability of EFT is questionable is negligible. For instance, the contribution to Γ L= µd from E nn > 3 MeV is just 1.1 s 1, and that from E nn > 5 MeV is less than.1 s 1. We now can make a rough estimate of the theoretical error associated with this calculation. Uncertainty related to the G P term, Eq.1) or β)is 1 s 1, while uncertainty 1

13 4.ffl ffi. 1 3 E nn MeV) Figure 1: Cumulative µ-d capture rate in s 1 ) calculated for Λ = 6 MeV and ˆd R =1.78. The dashed line gives the L = contribution, Γ L= µd E nn), while the solid line shows the total contribution, Γ µd E nn ) Γ L= µd E nn) +Γ L 1 µd E nn). The empty and solid circles for the values at E nn = Enn max 1 MeV, for L =andl, respectively. reflecting the Λ-dependence is less than 1 s 1 ; uncertainty in Γ t β exp) or that in ˆd R for a given Λ) can affect Γ µd at the level of 1 s 1. If we assign a rather conservative error, s 1, to the L 1 contributions obtained in Ref. [6], the overall uncertainty in our estimate becomes 4 s 1 or 1 % in the total capture rate. As mentioned, there is a serious disagreement between the two measured values of Γ µd. Our theoretical result is consistent with Γ µd exp) in Ref.[18]. In the present exploratory study we have not considered radiative corrections [8], which are expected to be smaller than the existing uncertainty in Γ µd exp). When the planned precision measurement of the Γ µd at PSI [] is realized, the issue of radiative corrections should certainly be addressed. The EFT approach as described here will provide a useful tool for this purpose as well. Once the accuracy in Γ µd exp) is significantly improved, we will be able to use µd capture to determine the low energy constant ˆd R, a quantity critically important for the accurate evaluation of the νd cross sections used in the analysis of the SNO experiments. At present the tritium β-decay is a much more accurate source of information on ˆd R than µd capture, but it is hoped that in the near future Γ µd will provide an independent constraint on ˆd R. We consider this redundancy extremely important. We thank P. Kammel and J.-W. Chen for useful communications. This work is 13

14 supported in part by NSF Grant No. PHY and No. INT References [1] J. N. Bahcall and R. May, Astrophys. J ) 51. [] The SNO Collaboration, Phys. Lett. B ) 31, and references therein. [3] Q. R. Ahmad et al., Phys. Rev. Lett. 87 1) [4] J. Carlson and R. Schiavilla, Rev. Mod. Phys ) 743. [5] M. Chemtob and M. Rho, Nucl. Phys. A ) 1; E. Ivanov and E. Truhlík, Nucl. Phys. A ) 451, 437. [6] N. Tatara, Y. Kohyama, and K. Kubodera, Phys. Rev. C4199) [7] J. Adam, E. Truhlík, S. Ciechanowicz, and K.-M. Schmitt, Nucl. Phys. A ) 675. [8] S. Nakamura, T. Sato, V. Gudkov, and K. Kubodera, Phys. Rev. C631) [9] M. Butler, J.-W. Chen, and X. Kong, Phys. Rev. C631) [1] For reviews, see U. van Kolck, Prog. Part. Nucl. Phys ) 49; S. R. Beane et al., in At the Frontier of Particle Physics Handbook of QCD. ed. M. Shifman, vol.1 World Scientific, Singapore, 1) p. 133; G. E. Brown and M. Rho, hep-ph/131, to appear in Phys. Repts. [11] B.-Y. Park, F. Myhrer, J. R. Morones, T. Meissner, and K. Kubodera, Phys. Rev. C ) 661. [1] T. D. Cohen, J. L. Friar, G. A. Miller, and U. van Kolck, Phys. Rev. C531996) 661. [13] T. Sato, T.-S. H. Lee, F. Myhrer, and K. Kubodera, Phys. Rev. C561997) 146. [14] S. Ando, T.-S. Park, and D.-P. Min, Phys. Lett. B 59 1) 53. [15] D. B. Kaplan, M. J. Savage, and M. B. Wise, Phys. Lett. B ) 39; Nucl. Phys. B ) 39. [16] T.-S. Park, L. E. Marcucci, R. Schiavilla, M. Viviani, A. Kievsky, S. Rosati, K. Kubodera, D.-P. Min, and M. Rho, nucl-th/166; nucl-th/171. [17] S. Ando et al., in progress. 14

15 [18] M. Cargelli et al., Proceedings of the XXIII Yamada Conference on Nuclear Weak Processes and Nuclear Structure, Osaka, 1989, edited by M. Morita, H. Ejiri, H. Ohtsubo, and T. Sato, World Scientific, Singapore, 1989), P [19] G. Bardin et al., Nucl. Phys. A ) 591. [] P. Kammel, private communication. [1] J. C. Hardy, I. S. Towner, V. T. Koslowsky, E. Hagberg, and H. Schmeing, Nucl. Phys. A ) 49. [] V. Bernard, N. Kaiser, and U.-G. Meißner, Nucl. Phys. B ) 147. [3] V. Bernard, H. W. Fearing, T. R. Hemmert, and U.-G. Meißner, Nucl. Phys. A ) 11. [4] S. Ando, F. Myhrer, and K. Kubodera, Phys. Rev. C63 1) 153. [5] T.-S. Park, K. Kubodera, D.-P. Min, and M. Rho, Phys. Lett. B 47 ) 3. [6] T.-S. Park, D.-P. Min, and M. Rho, Phys. Rev. Lett ) 4153; Nucl. Phys. A ) 515. [7] T.-S. Park, D.-P. Min, and M. Rho, Phys. Rept ) 341; T.-S. Park, I. S. Towner, and K. Kubodera, Nucl. Phys. A ) 381; T.-S. Park, H. Jung, and D.-P. Min, Phys. Lett. B ) 6; T.-S. Park, K. Kubodera, D.-P. Min, and M. Rho, Astrophys. J ) 443. [8] J. F. Beacom and S. J. Parke, hep-ph/