Table Of ContentFixing Two-Nucleon Weak-Axial Coupling L From µ−d Capture
1,A
Jiunn-Wei Chen,1 Takashi Inoue,1,2 Xiangdong Ji,3 and Yingchuan Li3
1Department of Physics, National Taiwan University, Taipei, Taiwan 10617
2Department of Physics, Shophia University, Chiyoda-ku, Tokyo 102-8554, Japan
3Department of Physics, University of Maryland, College Park, Maryland 20742
(Dated: February 9, 2008)
We calculate the muon capture rate on the deuteron to next-to-next-to-leading order in the
pionless effective field theory. The result can be used to constrain the two-nucleon isovector axial
coupling L1,A to ±2 fm3 if the muon capture rate is measured to 2% level. From this, one can
determine theneutrino-deuteron break up reactions and the pp-fusion cross section in thesun to a
same level of accuracy.
6
The strong evidence of neutrino oscillations observed fusionratewaspredictedwithasmallerror. Whencom-
0
atthe Sudbury Neutrino Observatory(SNO)[1]is based paredwith the EFT(/π) calculation[7],their resultyields
0
2 on detecting the 8B solar neutrino flux through the fol- L1,A =4.2 2.5 fm3.
±
lowing three reactions: In this paper we aim to make a high-precision deter-
n
mination of L from the µ−d capture process
a ν +d p+p+e− (CC), 1,A
J e →
1 νx+d→p+n+νx (NC), µ−+d→νµ+n+n , (2)
1 ν +e− ν +e− (ES). (1)
x x by calculating the rate to next-to-next-to leading order
→
2 Thechargedcurrent(CC)reactioninvolvesonlytheelec- (N2LO)inEFT(/π). Theµ−dcaptureratehasbeenmea-
v suredpreviouslybydifferentgroupswithratherdifferent
tron neutrinos, while the neutral current (NC) reaction
1 resultsΓexp =470 29s−1 [12]andΓexp =409 40s−1
0 and elastic scattering (ES) involve all the active neutri- ± ±
[13]. A measurement of this rate with 1% precision
0 nos (x=e,µ,τ). The νe and νx fluxes are found to be
is under investigation at PSI [14]. An earlier poten-
6 significantly different[1]. Further detailed measurements
0 of the fluxes could sharpen the constraints to neutrino tial model calculation [15] gave Γ = 397 ∼ 400 s−1.
5 More recently, the hybrid approach mentioned above
oscillation parameters and provide precision tests to the
0 gave Γ=386 5 s−1 [16].
h/ ssteactnidoanrdisskonloawrnmtoodehlig[h2].accHuorwaceyv,ert,hewChiCleatnhde NECS ccrroossss A concern i±n applying EFT(/π) to the µ−d capture is
t that the energy transfer into the hadronic system might
- sections have hadronic uncertainties. As shown by But-
l be too large to apply EFT(/π). However, as shown in
c ler, Chen and Kong [3], the dominant uncertainties in
u lowenergyCCandNCcrosssectionscomesfromthecou- Ref.[16] and also in this calculation, the contribution to
n the total rate from high-energy neutrons is small, and
plingofatwo-bodyisovectoraxialcurrent,L ,inpion-
: 1,A it is possible to impose a neutron energy cut to isolate
v less effective field theory(EFT(/π)). The potentialmodel
Xi resultsofRefs. [4]and[5]canbereproducedbydifferent the low-energy(≤10−20 MeV) neutron events without
significantly increasing the statistical errors [8, 14].
choices of L , indicating that the 5% difference be-
r 1,A ∼ Effective field theory is useful when low and high en-
a tween the models comes from the different assumptions
ergyscalesintheproblemarewidelyseparated. Forlow-
abouttheshort-distancenuclearphysics. Thereareother
energyprocesses,short-distancephysicscanbetakeninto
interesting weak reactions involving the same two-body
account by local operators in an effective lagrangian in-
current, for example, the pp and pep fusion processes
volvingonly low-energydegreesoffreedom. Forν(ν¯) d
(pp → de+νe, ppe− → dνe) which power the sun [6, 7]. scattering with neutrino energy below 20 MeV and µ−−d
It is one of the great current interests to measure these
capture with small final-state neutron energy, the pion
neutrino fluxes to further test the standard solar model.
andothermesonexchangesarenotdynamicaldegreesof
Recently much effort has been going into determin-
freedom,andtheirphysicscanbecapturedbycontactin-
ing the effective two-body axial current interaction [8].
teractions involving nucleons and the external currents.
Butler, Chen, and Vogel attempted to fix L from re-
1,A
To make predictions with controlled precision, calcula-
actor antineutrino-deuteron breakupreactions, and they
found L = 3.6 5.5 fm3 [8]. Chen, Heeger, and tions are done with the perturbative expansion param-
RCCobearntdso1Nn,ACobdtaaitnae,dc±Lal1i,bAra=te4d.0b±y6t.h3efmE3SbeyveunstisngofSNSNOO’s teoterheQavy≡s(c1a/leas(n.1nS0T)h,γe,lpig)/hΛt s,cawlehsicihncilsutdheethraetiionvoefrsleighSt-
and Super-Kamiokande(SK) [9]. Schiavilla et al.’s idea waveneutron-neutronscatteringlength1/a(1S0) = 10.6
nn
−
[10] of using the tritium β decay rate to control the MeV in the 1S channel, the deuteron binding momen-
0
strength of the two-body current was adopted by Park tum γ = 45.7 MeV in the 3S channel, and typical nu-
1
et al. in their hybrid EFT calculation [11], and the pp cleon momentum p in the system. The heavy scale Λ is
2
set by the pion mass m . This EFT(/π) (see e.g. [17]) capture process are
π
and its dibaryon version [18, 19, 20] have been applied
Ldb
successfully to many processes involving the deuteron, Vk,a = 1 ǫ t† s +h.c. ,
including electro-magnetic processes such as Compton (2) MN r(3S1)rn(1nS0) kij i∇j a
scattering γd γd [21, 22], np dγ relevant to the
→ → qLdb
big-bang nucleosynthesis [23, 24], weak processes such Ak,a = 1,A t†s (4)
apsroνcdessre[a6c,t7io],nasnfdorpSaNritOy vpihoylasitcisn[g3]o,btsheervsaoblalersp[2p5f]u.sFioonr (2) MN r(3S1)rn(1nS0) (cid:16) k a
reviews on three-body systems, see [26]. + qGp t† s +h.c.,
4M2g j∇j∇k a
The effective Lagrangian for the CC weak interaction N A (cid:19)
is given by LCC = −√ωiGFVudl+µJµ−/√2+h.c., where where ti and sa are dibaryon fields for the two-nucleon
GF =1.166 10−5GeV−2 istheFermicouplingconstant 3S1 and 1S0 states, respectively. The second term in
and ωi =1.0×24 takes into account the inner electroweak Ak(2,a) is induced by the Gp term in the one-nucleon cur-
radiative correction [27]. l+µ = νµγµ(1−γ5)µ is the lep- rent. r(3S1) = 1.764 fm and rn(1nS0) = 2.8 fm are the
tonic current. The quark current J− = V− A− =
µ µ − µ effective ranges in triplet and two-neutron singlet chan-
(Vµ1−A1µ)−i(Vµ2−A2µ) contains both vector and axial- nels, respectively. Ld1b and Ld1,bA are coupling constants
vectorinteractions,wherethesuperscripts1and2arethe
in dibaryon formalism. The vector current is N2LO and
isospin indices. At the scale relevant to nuclear physics,
its coupling Ldb = 4.08fm has been determined by the
thequarkcurrentneedbematchedtoahadroniccurrent 1 −
rate of n+p d+γ near threshold. The axial cur-
whichingeneralcontainsone-nucleon,two-nucleon,etc., →
rentis NLO,andits coupling Ldb is proportionalto the
operators. 1,A
renormalization-scale-µ-independent L˜ in Ref. [3] as
1,A
Uptotheorderofourinterest,theone-nucleonisovec- Ld1,bA = M2πNL˜1,A, through which Ld1,bA is related to the µ-
tor vector and axial vector currents are dependentL (µ)inRef. [3]. Thenumericalrelationbe-
1,A
tweenLdb andL (µ)isLdb = 13.8+0.28L (M ),
1,A 1,A 1,A − 1,A π
V(01,)a = N†τ2a 1+ 16hr2iIc=1∇2 N w∼h6erfemL3.1,A(Mπ) is in units of fm3 and has a naturalsize
(cid:18) (cid:19) The µ−d atom has a ground state with a hyperfine
i τa µ(1) τa
Vk,a = N†←→ N ǫ N†σ N , structure,correspondingtothetotalangularmomentum
(1) 2MN ∇k 2 − MN kij i∇j 2 F = 3/2 and F = 1/2. The µ−d capture process is
A0,a = igA N†~σ ←→τaN (3) known to take place almost uniquely from the doublet
(1) 2MN · ∇ 2 F = 1/2 state. The differential capture rate for muon
1 N†G 2 ∂ ~σ τaN , and deuteron in their specific polarization states can be
−4M2 p ∇ 0 ·∇ 2 written in terms of leptonic tensor lµν andhadronic ten-
N
Ak,a = g N†σ τa 1(cid:16)+ 1(cid:17)r2 I=1 2 N sor Wµν as
(1) +A 1 kN2†G(cid:18) 26h iA~σ ∇τ(cid:19)aN , ddΓE(Sd,Ωξˆ) = ω3iG2π2F2|MVud(|21E+|ψM(0µ))|2lµν(S)Wµν(ξˆ), (5)
4M2 p ∇ ∇k ·∇ 2 µ Md
N
(cid:16) (cid:17)
3
where the ψ(0)2 = 1 α MµMd is the 1S-state
| | π emMµ+Md
where =←−+−→, ∂ =←∂−+−∂→, and ←→=←− −→. The wave-function-at-origin-s(cid:16)quared, E is(cid:17)the outgoing neu-
0 0 0
∇ ∇ ∇ ∇ ∇−∇
superscript a is the isospin index, µ(1) = (µp µn)/2 = trino energy, and Md is the mass of the deuteron.
−
2.353,istheisovectormagneticmoment. IsovectorDirac Thecaptureratedependsonthepolarizationvectorof
cthhearigsoevreacdtoiursahxri2ail-Icc=h1ar=gehrra2idpciu−s hrr22iInc==1 0.08.7435ffmm22,.aWnde tlehpetmonuiocnteSnµsoarnidstghiveedneubtyerlµoνn(Spo)l=ari4z(aktµiokn′νv+eckto′µrkξˆν. Thke
haveneglectedtermsoforderp2/Mh N2iAorev≃enµ(1)p2/MN2. k′gµν +iǫµνρσkρkσ′)−4Mµ(Sµk′ν +k′µSν −k′·Sgµ−ν +·
The pseudoscalar form factor is, to a good approxima- iǫµνρσS k′), where k = (M ,~0) and k′ = (E,k~′), with
ρ σ µ
qti2ond,epdeonmdiennacteedwbilylnthotebpeioenx-ppaonledGedp(bqe2c)a=use4MMtπ2hN2−egqAm2 owmhoense- Eνµ,=re|ks~p′|e,catriveetlhy.eTfohuer-hmaodmroennitcateonfsionritiisal muon and final
tum transfer q is of order muon mass M with low en-
ergy final sta|te|neutrons. The Gp contrµibution to the Wµν(ξˆ)= π1Im d4xeiqxThd|Jµ+(x)Jν−(0)|di , (6)
axial current is counted of order Q. (cid:20)Z (cid:21)
Thelowestdimensionaltwo-nucleonisovectorcurrents, where qµ = kµ−kµ′, and |di ≡ |d(P,ξˆ)i is the deuteron
in the dibaryon version of EFT(/π), relevant to the µ−d state with momentum P =(M ,~0) and polarization ξˆ.
d
3
The diagrams contributing to W (ξˆ) up to N2LO are
µν
shown in Fig. 1. A straightforwardcalculation finds
(a) (b) (c)
dΓ = G2F|Vud|2E2|ψ(0)|2 1 F F2 = +
dE 2π(1+ Mµ) 1 εγr(3S1) 1− 2
Md − (cid:20)(cid:18) (cid:19)
(cid:18)3G2V −2(GV −GA)2−ε4gA3((1M−π2g−A)qM2)µE lFinIGes. i1n:siDdeiagtrhaemlsoocponatrreibuthtiengnutcoleWonµνp(ξrˆo)puapgattoorNs,2LaOnd. Tthhee
8(1 g )E F 2 thin and thick lines outside represent the triplet and singlet
+εκ(1) 3−MA + F1− 62 + 3F4a 9G2A dibaryonfields,respectively. Thesmallopencirclerepresents
N (cid:19) (cid:18) (cid:19) the insertion of hadronic one-body current. In diagram (c),
(cid:0)
6g2M E g2E4 the large gray circle indicates the two possible hadronic cur-
−εMAπ2−µq2 +ε2(Mπ2A−q2)2! rdeenntoitnessetrhtieontwsosh-boowdnyincutrhreensetcaosnsdocrioawtedwhweirtehtLhe1,dAaarkndsqLu1a.re
ε2κ(1)2E2
+(10F F +8F )
1− 2 4b 3M2 175
N dΓ
+ F F2 + 2F 12εg κ(1) E −dEnn 150
(cid:18) 1− 6 3 4c(cid:19) A MN (s·M1eV)125
8ε2g κ(1)E3
A
(F +F )
− 1 4c 3M (M2 q2) 100
N π −
14 16 75
+F ε2E g2 + g +2 (7)
5 3 A 3 A
(cid:18) (cid:19)(cid:21) 50
where GV = 1+ε261hr2iIc=1q2 is the Dirac form factor, 25
and G =g (1+ε21 r2 I=1q2) is the axial form factor.
A A 6h ic
The ε is formally introduced to keep track of the Q ex- 2 4 6 8 10 12 14
pansion. After expanded in ε and truncated at ε2, the Enn(MeV)
N2LO result is obtained by setting ε=1. The functions FIG.2: (Color online)Thedifferentialcapturerate dΓ/dEnn
F , F , and F are from diagrams (a), (b), and calculated using L1,A =6 fm3. The doted line is the LO re-
1a,1b 2 3a,3b,3c
sult. Thedashedandsolidlines,whichsitontopofeachother
(c), respectively, in Fig. 1 in this scale, are theNLO and N2LO results, respectively.
2M pγ
N
F =
1a π(M2∆2 p2E2)
N E − use the p/m to estimate the accuracyof the expansion.
π
F = γ 2MNEp∆E +ln MN∆E −pE For example, it is well-know tha1t in the nucleon-nucleon
1b πE3 M2∆2 p2E2 M ∆ +pE
(cid:20) N E − (cid:18) N E (cid:19)(cid:21) scattering,theeffectivetheorywithoutpionworksrather
4γ Ep wellatthe nucleonmomentumonthe orderof100MeV,
F = tanh−1
2 π∆ E M ∆ a value close to the pion mass. This can also be seen in
E (cid:18) N E(cid:19)
2γM2 the present calculation because the NLO result at large
F3a = π2EN2 Im G21A(1S0)(p) neutronmomentumdoesnotcompletelymodifythelead-
2γM2 n o ingorderresult,whereasthenaivepowercountingwould
F3b = π2EN2 Im G22A(1S0)(p) indicate otherwise. Fortunately, for muon capture, the
mostoftheeventsoccurwhentheneutronmomentumis
2γM2 n o
F3c = π2EN2 Im G1G2A(1S0)(p) , (8) about half of the muon mass, that is, about 50 MeV.
In Fig. 2 we show the differential rate dΓ/dE
n o nn
with the re-scattering amplitude in the singlet channel in terms of the relative motion energy Enn =
as A(1S0)(p) = −M4πN(a(n1n1S0) − ε12rn(1nS0)p2 + ip)−1, and r2e(gioMnN2wh+erpe2t−heMENF)To(/πf)twcaolcfiunlaatli-osntaitsemnoeusttrroenlisabinle.thIet
functions G1,2 are defined as G1 = tan−1(2(γE−ip)) + ispclear from the figure that the differential rate in the
ε E Ldb and G = tan−1( E )+ε E Ldb. The energy region E 10 MeV is very small, and is neg-
4MNgA 1,A 2 2(γ−ip) 4µ(1) 1 nn ≥
eMnergyEinjec(tMion intMo th).e tTwhoe-nerueltartoinvesymstoemmenitsu∆mEbe≡- lLigOib,lNeLfoOr,Eannnd≥N21L5OM, weVe.fiBnyd cgoomodpacroinngvetrhgeenrceesuolftsthoef
µ n p
− − −
tween the two final-state neutrons is 2p, with p = expansion.
MN∆E γ2 E2/4. In the case that a neutron energy cut can be imposed
− −
Power counting for the present calculation is rather on experimental data [14], it is possible to define and
p
tricky, and it would be misleading, for example, to just measure the integrated capture rate up to a threshold
4
400
Γ((sE−1nt)hn3)50 rthanegseizoefoEfntahn(,Etthhe).sizTehoisfsbh(Eownthsn)hioswabanouetrr1o.3r i∼n 1ca.5p%turoef
nn
300 rate is translated into an uncertainty of L1,A.
250
Table 1:
200 Coefficientsfunctionsa(Eth)andb(Eth)forspecificval-
nn nn
ues of two-neutronrelativeenergy Eth from the EFT(/π)
150 nn
calculation.
100
Eth(MeV) 5.0 10.0 15.0 20.0
50 nn
a(Eth)(s−1) 239.2 308.0 332.0 342.3
nn
5 10Eth(M1e5V) 20 25 30 b(Enthn)(s−1fm−3) 3.3 4.2 4.7 4.9
nn
FIG. 3: (Color online) The integrated capture rate Γ(Eth)
calculated using L1,A =6 fm3 for the relative energy Enthnnnof ingInEsFuTm(m/π)atroy,Nw2eLOca.lcTuhlaetemdatjohregµo−adl iscatpotufirxetrhaetetwuos--
the two neutrons up to 30 MeV. The description of the lines
is the same as that in the caption of Fig. 2. nucleon isovector axial coupling constant L1,A from fu-
tureprecisionexperimentaldata. Anexperimentalresult
on the integrated rate up to some neutron energy Eth
nn
energy Eth with a 2% error should be able to, through comparison
nn
with our calculation with theoretical error 2-3%, fix the
Γ(Enthn)≡− Enthn dEdΓ dEnn. (9) Lte1r,mAiwneitthheernreourt±ri2n.o0dfmeu3t.erTonhibsrienaktuurpncarollsoswssecutsiotnoadned-
Z0 nn
the pp fusion rate in the sun to 2-3%.
The result up to 30 MeV is shown in Fig. 3. In the This work was supported by the U. S. Department of
whole energy region, the NLO contribution is less than Energy via grantDE-FG02-93ER-40762and the NSC of
10%oftheLOcontribution,whiletheN2LOcontribution Taiwan. We thank S. Ando, D.W. Hertzog, P. Kammel,
is less than 1%. This small size of N2LO contribution K. Kubodera, and T.-S. Park for helpful discussions.
is accidental and does not happen for unpolarized and
F = 3/2 rates. For example, the unpolarized rate has
the expansion (for L =6 fm3)
1,A
Γ =ΓLO (1+5.3%+4.9%), (10) [1] Q.R. Ahmad et al., Phys. Rev. Lett. 87, 071301 (2001);
unpol. unpol.
89, 011301 (2002); 89, 011302 (2002).
where the NLO correction is abnormally small, and [2] J.N. Bahcall, A.M. Serenelli, and S. Basu, Astrophys.J.
NNLO is of the normal size. A similar expansion is ob- 621, L85 (2005).
tained for L = 6 fm3: [3] M.N. Butler and J.W. Chen, Nucl. Phys. A675, 575
1,A − (2000); M.N. Butler, J.W. Chen, and X. Kong, Phys.
Rev. C63, 035501 (2001).
Γ =ΓLO (1+10.9%+5.2%) (11)
unpol. unpol. [4] S.Ying,W.C.HaxtonandE.M.Henley,Phys.Rev.C45,
1982 (1992); Phys.Rev. D40, 3211 (1989)
which shows a nice convergence pattern. Based on this
[5] S. Nakamura, T. Sato, V. Gudkov and K. Kubodera,
trend, we assign a 2-3% correction at NNNLO, corre- Phys. Rev.C63, 034617 (2001)
sponding to an error in L1,A 2 fm3. This is consistent [6] X.KongandF.Ravndal,Nucl.Phys.A656,421(1999);
with the naive estimation of∼3% if the small expansion A665, 137 (2000); Phys. Lett. B470, 1 (1999); Phys.
parameter is 1/3. A calculation shows the N3LO final Rev. C64, 044002 (2001).
[7] M. Butler and J.W. Chen,Phys. Lett. B520, 87 (2001).
state P-wave re-scattering contributes only 1%. Fur-
∼ [8] M. Butler, J.W. Chen, and P. Vogel, Phys. Lett. B549
thermore, the result is insensitive to the uncertainty in
26 (2002).
a(n1nS0). Choosing L1,A =5.6 fm3, the energy dependence [9] J.W. Chen, K.M. Heeger, and R.G.H. Robertson, Phys.
ofourresultmatchestheprevioushybridcalculationvery Rev. C67, 025801 (2003).
well [16]. [10] R. Schiavilla et al. Phys.Rev. C58, 1263 (1998).
To extractL fromexperimentaldata,it is useful to [11] T.-S. Park et al., nucl-th/0106025; nucl-th/0107012.
1,A [12] G. Bardin et al., Nucl. Phys. A453591 (1986).
provide the dependence of the rate on L
1,A [13] M.Cargellietal.,in: M.Morita,H.Ejiri,H.Ohtsubo,T.
Sato (Eds.), Proceedings of the XXIII Yamada Confer-
Γ(Eth)=a(Eth)+b(Eth)L , (12)
nn nn nn 1,A ence on Nuclear Weak Processes and Nuclear Structure,
Osaka, 1989, World Scientific,Singapore, P. 115 (1989).
where L is in unit of fm3. The energy-cut dependent
1,A [14] P. Kammel et al.,nucl-ex/0202011; nucl-ex/0304019.
functions a(Enthn) and b(Enthn) for a set of Enthns are listed [15] N.Tatara, Y.Kohyama,K.Kubodera, Phys.Rev. C42,
intheTable1,fromwhichweobservethat,forthewhole 1694 (1990)
5
[16] S. Ando, T.S. Park, K.Kubodera and F. Myhrer, Phys. Phys. Rev.C71, 044321 (2005).
Lett.B533, 25 (2002). [23] J.W. Chen and M.J. Savage, Phys. Rev. C60, 065205
[17] J.W. Chen, G. Rupak, and M.J. Savage, Nucl. Phys. (1999).
A653 386 (1999) [24] G. Rupak,Nucl. Phys. A678, 405 (2000).
[18] D.B. Kaplan, Nucl.Phys. B494, 471 (1997). [25] M.J. Savage, Nucl.Phys. A695, 365 (2001).
[19] S. R. Beane and M. J. Savage, Nucl. Phys. A694, 511, [26] P. F. Bedaque and U. van Kolck, Ann. Rev. Nucl. Part.
(2001). Sci. 52, 339 (2002); E. Braaten and H.-W. Hammer,
[20] S.AndoandC.H.Hyun,Phys.Rev.C72,014008(2005) cond-mat/0410417.
[21] H.W.GriesshammerandG.Rupak,Phys.Lett.B529,57 [27] A. Sirlin, Rev.Mod. Phys. 50, 573 (1978).
(2002).
[22] J.W.Chen,X.Ji,andY.Li,Phys.Lett.B620,33(2005);
