Proton-proton fusion in pionless effective theory
aa r X i v : . [ nu c l - t h ] J a n January 28, 2008
Proton-proton fusion in pionless effective theory
S. Ando a,b , J. W. Shin a , C. H. Hyun c , S. W. Hong a , and K. Kubodera da Department of Physics, Sungkyunkwan University, Suwon 440-746,Korea b Theoretical Physics Group, School of Physics and Astronomy,The University of Manchester, Manchester, M13 9PL, UK c Department of Physics Education, Daegu University, Gyeongsan712-714, Korea d Department of Physics and Astronomy, University of South Carolina,Columbia, SC 29208, USA
The proton-proton fusion reaction, pp → de + ν , is studied in pionless effective fieldtheory (EFT) with di-baryon fields up to next-to leading order. With the aid of thedi-baryon fields, the effective range corrections are naturally resummed up to the infiniteorder and thus the calculation is greatly simplified. Furthermore, the low-energy constantwhich appears in the axial-current-di-baryon-di-baryon contact vertex is fixed through theratio of two- and one-body matrix elements which reproduces the tritium lifetime veryprecisely. As a result we can perform a parameter free calculation for the process. Wecompare our numerical result with those from the accurate potential model and previouspionless EFT calculations, and find a good agreement within the accuracy better than1%.PACS(s): 21.45.Bc, 26.20.Cd, 26.65.+t mailto:[email protected] . Introduction The proton-proton fusion process, pp → de + ν e , is a fundamental reaction for the nuclearastrophysics, especially important for the understanding of the star evolutions [1] and solarneutrinos [2, 3, 4]. However, the process has never been studied experimentally becausethe event is extremely unlikely to take place in the laboratory at the proton energies inthe sun. The calculation of the transition rate and its uncertainty has naturally become achallenge to nuclear theory. The first calculation of the process was carried out by Betheand Critchfield [5] in 1938. This estimation was improved by Salpeter [6] in 1952. Later,small corrections, such as the electromagnetic radiative corrections, were considered byBahcall and his collaborators [8, 9] in the framework of effective range theory. Recently,accurate phenomenological potential models were employed to study the process [10, 11].Furthermore, in Ref. [12] the two-nucleon current operators were calculated from heavy-baryon chiral perturbation theory (HB χ PT) up to next-to-next-to-next-to leading order(N LO), and Park et al. obtained quite an accurate estimation ( ∼ pp fusion process at the core of the sun is quite low, kT c ≃ .
18 keV, where T c is the core temperature of the sun, T c ≃ . × K, and k is the Boltzmann constant. The proton momentum at the core, p c ≃ q m p kT c ≃ . m p is the proton mass, is still significantly small compared to the pion mass, m π ≃
140 MeV. Therefore, we may regard the pion as a heavy degree of freedom for the pp fusion process. It may be convenient and suitable to employ a pionless effective fieldtheory (EFT) [15], in which the pions are integrated out of the effective Lagrangian for theprocess in question. The pp fusion process in the pionless theory has been studied by Kongand Ravndal [16] up to next-to leading order (NLO) and by Butler and Chen [17] up tofifth order (N LO). Thanks to the perturbative scheme in EFT, the accuracy of the N LOcalculation would, in principle, be ( Q/ Λ) ∼ (1 / ≃ Q/ Λ ∼ / L A which appears in the two-nucleon-axial-current contactinteraction in the pionless effective Lagrangian, an uncertainty estimated in the pionlessEFT for the pp fusion process is still significantly larger than what is expected from thecounting rules of the theory.In this work, we employ a pionless EFT with di-baryon fields [18, 19, 20]. Theamplitude for the pp fusion process at the zero proton momentum is calculated up toNLO. We introduce two di-baryon fields [24], which have the same quantum numbersas those of S -wave two-nucleon states ( S and S states), as auxiliary fields: afterintegrating out the di-baryon fields we do have the ordinary pionless theory without thedi-baryon fields. However, as have intensively been discussed in Refs. [18, 19, 25, 26], For a recent historical review, see Ref. [7]. We have employed the same formalism in the studies of the two-body processes, such as neutron-neutron fusion [21], radiative neutron capture on a proton at BBN energies [22], and neutral pion pro-duction in proton-proton collision near threshold [23]. Q much improved, and it is not necessary to employ the powerdivergence subtraction scheme [27] any longer. Furthermore, by assuming that the leadingorder (LO) contribution in the di-baryon-di-baryon-current contact interaction can bedetermined mainly from the one-body current interaction as discussed in Ref. [19], we canreproduce the results from the effective range theory [28] in the LO calculations of thepionless EFT with the di-baryon fields. The NLO correction, the di-baryon-di-baryon-current contact interaction denoted by the unknown LEC l A , is approximately presumedto be the two-body (2B) current correction in the pionful calculations. We fix the LEC l A by using the relative strength of the two-body matrix element to that of the one-bodycontribution, δ B [14], which has been determined from the accurate tritium lifetimedatum. (We discuss it in detail later.) Consequently we can make our estimation of the pp fusion amplitude free from unknown parameters. Moreover, though our calculation israther simple and is only up to NLO, we can obtain a result comparable to that from theaccurate potential model calculation within the accuracy better than ∼ pp fusion process is calculated up to NLO. We show ournumerical results in Sec. 5. In Sec. 6, discussion and conclusions are given.
2. Pionless effective Lagrangian with di-baryon fields
For the low-energy process, the weak-interaction Hamiltonian can be taken to be H = G F V ud √ l µ J µ , (1)where G F is the Fermi constant and V ud is the CKM matrix element. l µ is the leptoncurrent l µ = ¯ u e γ µ (1 − γ ) v ν , and J µ is the hadronic current. We will calculate the two-body hadronic current J µ from the pionless effective Lagrangian with di-baryon fields upto NLO.We adopt the standard counting rules of pionless EFT with di-baryon fields [18]. In-troducing an expansion scale Q < Λ( ≃ m π ), we count the magnitude of spatial part ofthe external and loop momenta, | ~p | and | ~l | , as Q , and their time components, p and l ,as Q . The nucleon and di-baryon propagators are of Q − , and a loop integral carries Q . The scattering lengths and effective ranges are counted as Q ∼ { γ, /a , /ρ d , /r } where γ , a , ρ d and r are the effective range parameters for the S -wave N N scattering; γ ≡ √ m N B , where B is the deuteron binding energy, a is the scattering length in the S channel, ρ d and r are the effective ranges in the S and S channel, respectively. Theorders of vertices and transition amplitudes are easily obtained by counting the numbersof these factors in the Lagrangian and diagrams, respectively. As discussed below, somevertices acquire factors like r and ρ d after renormalization and thus their orders can differfrom what the above naive dimensional analysis suggests. Note that we do not include3he higher order radiative corrections, such as the vacuum polarization effect [29] and theradiative corrections from one-body part [30].A pionless effective Lagrangian with di-baryon fields may be written as [18, 19] L = L N + L s + L t + L st , (2)where L N is a one-nucleon Lagrangian, L s is the spin-singlet ( S state) di-baryon La-grangian including coupling to the two-nucleon, L t is the spin-triplet ( S state) di-baryonLagrangian including coupling to the two-nucleon and L st describes the weak-interactiontransition (due to the axial current) from the S di-baryon to the S di-baryon.A pionless one-nucleon Lagrangian in the heavy-baryon formalism reads L N = N † (cid:26) iv · D − ig A S · ∆ + 12 m N h ( v · D ) − D i + · · · (cid:27) N , (3)where the ellipsis represents terms that do not appear in this calculation. v µ is the velocityvector satisfying v = 1; we choose v µ = (1 , ~ S µ is the spin operator 2 S µ = (0 , ~σ ).Covariant derivative D µ reads as D µ = ∂ µ − i ~τ · ~ V µ where ~ V µ is the external isovectorvector current, and ∆ µ = − i ~τ · ~ A µ , where ~ A µ is the external isovector axial current. g A is the axial-vector coupling constant and m N is the nucleon mass.The Lagrangians that involve the di-baryon fields are given by L s = σ s s † a (cid:20) iv · D + 14 m N [( v · D ) − D ] + ∆ s (cid:21) s a − y s h s † a ( N T P ( S ) a N ) + h.c. i , (4) L t = σ t t † i (cid:20) iv · D + 14 m N [( v · D ) − D ] + ∆ t (cid:21) t i − y t h t † i ( N T P ( S ) i N ) + h.c. i , (5) L st = − " r + ρ d √ r ρ d ! g A + l A m N √ r ρ d s † a t i A ai + h.c. i , (6)where s a and t i are the di-baryon fields for the S and S channel, respectively. Thecovariant derivative for the di-baryon field is given by D µ = ∂ µ − iC V extµ where V extµ isthe external vector field. C is the charge operator for the di-baryon field; C = 0 , , nn , np , pp channel, respectively. σ s,t is the sign factor σ s,t = ± s,t is themass difference between the di-baryon and two nucleons, m s,t = 2 m N + ∆ s,t . y s,t is thedi-baryon-two-nucleon coupling constant. P ( S ) i is the projection operator for the S = S or S channel; P ( S ) a = 1 √ σ τ τ a , P ( S ) i = 1 √ σ σ i τ , Tr (cid:16) P ( S ) † i P ( S ) j (cid:17) = 12 δ ij , (7)where σ i ( τ a ) is the spin (isospin) operator. Note that, as mentioned in the Introduction,we separate the di-baryon-di-baryon-current contact interaction in Eq. (6) into the LOand NLO terms. The LO interaction proportional to g A is determined by the one-bodyaxial-current interaction and the factor ( r + ρ d ) / √ r ρ d is included so as to reproducethe result from the effective range theory at LO. The NLO correction is parameterized4 + + + ... Figure 1: Diagrams for the dressed di-baryon propagator including the Coulomb inter-action. A double-line with a filled circle denotes the renormalized dressed di-baryonpropagator. Double-lines without the filled circle and single-curves denote the bare di-baryon propagators and nucleon propagators, respectively. Two-nucleon propagator witha shaded blob denotes the Green’s function including the Coulomb potential. A (spin-singlet) di-baryon-nucleon-nucleon ( sN N ) vertex is proportional to the LEC y s .by the LEC l A . More detailed discussion about the separation of LO and NLO contactinteraction with external probe in the di-baryon formalism can be found in Ref. [19].
3. Initial and final
N N channels
The typical energy of the pp fusion reaction is very low, as discussed in the Introduction,so we can assume that the dominant channel of the reaction is from the initial S pp state to the final S deuteron state. In this section, we fix the LECs which appear inthe initial and final two-nucleon states for the pp fusion process from the effective rangeparameters.In Fig. 1, LO diagrams for the initial pp state in S channel, i.e., the dressed S chan-nel di-baryon propagator, are shown where the two-nucleon bubble diagrams including theCoulomb interaction are summed up to the infinite order. The inverse of the propagatorin the center of mass (CM) frame is thus obtained by iD − s ( p ) = iσ s ( E + δ s ) − iy s J ( p ) , (8)with J ( p ) = Z d ~k (2 π ) d ~q (2 π ) h ~q | ˆ G (+) C ( E ) | ~k i , (9)where ˆ G (+) C is the outgoing two-nucleon Green’s function including the Coulomb potential,ˆ G (+) C ( E ) = 1 E − ˆ H − ˆ V C + iǫ . (10) E is the total CM energy, E ≃ p /m N , ˆ H is the free Hamiltonian for two-proton,ˆ H = ˆ p /m N , and ˆ V C is the repulsive Coulomb force ˆ V C = α/r : α is the fine structureconstant. Employing the dimensional regularization in d = 4 − ǫ space-time dimension,we obtain [31, 32] J ( p ) = αm N π " ǫ − C E + 2 + ln πµ α m N ! − αm N π h ( η ) − C η m N π ( ip ) , (11)5igure 2: Diagram for the S -wave pp scattering amplitude with the Coulomb and stronginteractions. See the caption of Fig. 1 for details.where µ is the scale of the dimensional regularization, C E = 0 . · · · , and h ( η ) = Re ψ ( iη ) − ln η , Re ψ ( η ) = η ∞ X ν =1 ν ( ν + η ) − C E ,C η = 2 πηe πη − , η = αm N p . (12)Thus the inverse of renormalized dressed di-baryon propagator is obtained as iD − s ( p ) = iy s m N π " πσ s ∆ Rs m N y s + 4 πσ s m N y s p + αm N h ( η ) + ip C η , (13)where ∆ Rs is the renormalized mass difference σ s ∆ Rs = σ s ∆ s − y s αm N π " ǫ − C E + 2 + ln πµ α m N ! . (14)In Fig. 2, a diagram of the S -wave pp scattering amplitude with the Coulomb andstrong interactions is shown. Thus we have the S -wave scattering amplitude as iA s = ( − iy s ψ ) iD s ( p )( − iy s ψ )= i πm N C η e iσ − πσ s ∆ Rs m N y s − πσ s p m N y s − αm N h ( η ) − ip C η , (15)with ψ = Z d ~k (2 π ) h ~k | ψ (+) ~p i = Z d ~k (2 π ) h ψ ( − ) ~p | ~k i = C η e iσ , (16)where h ~k | ψ ( ± ) ~p i are the Coulomb wave functions obtained by solving the Schr¨odinger equa-tions ( ˆ H − E ) | ψ ( ± ) ~p i = 0 with ˆ H = ˆ H + ˆ V C and represented in the | ~k i space for the twoprotons. σ is the S -wave Coulomb phase shift σ = arg Γ(1 + iη ). The S -wave amplitude A s is given in terms of the effective range parameters as iA s = i πm N C η e iσ − a C + r p + · · · − αm N h ( η ) − ip C η , (17)6 + + + ... Figure 3: Dressed di-baryon propagator without Coulomb interaction (double line witha filled circle) at leading order. A single line stands for the nucleon, while a double linerepresents the bare di-baryon.Figure 4: Diagram for the S -wave N N amplitude without Coulomb interaction at leadingorder. The double line with a filled circle represents the dressed di-baryon propagatorobtained in Fig. 3.where a C is the scattering length, r is the effective range, and the ellipsis represents thehigher order effective range corrections. Now it is easy to match the parameters σ s and y s with the effective range parameters. Thus we have σ s = − y s = ± m N s πr , D s ( p ) = m N r a C − r p + αm N h ( η ) + ip C η . (18)In Fig. 3, LO diagrams for the final deuteron channel, i.e., the dressed S channeldi-baryon propagators are depicted. Since insertion of a two-nucleon one-loop diagramdoes not alter the order of the diagram, the two-nucleon bubbles should be summed up tothe infinite order. Thus the inverse of the dressed di-baryon propagator for the deuteronchannel in the CM frame reads iD − t ( p ) = iσ t ( E + ∆ t ) + iy t m N π ( ip )= i m N y t π " πσ t ∆ t m N y t + 4 πσ t Em N y t + ip , (19)where we have used dimensional regularization for the loop integral and E is the totalenergy of the two nucleons, E ≃ p /m N . The dressed di-baryon propagators are renor-malized via the S -wave N N amplitudes. The amplitudes obtained from the diagram inFig. 4 should satisfy iA t = ( − iy t ) [ iD t ( p )] ( − iy t ) = 4 πm N i − πσ t ∆ t m N y t − πσ t m N y t p − ip , (20)7 p de + ν (a) (b) (c) Figure 5: Diagrams for the pp fusion process, pp → de + ν e , up to NLO.where A t is related to the S -wave N N scattering S -matrix via S − e iδ t − ipp cot δ t − ip = i (cid:18) pm N π (cid:19) A t . (21)Here δ t is the phase shift for the S channel. Meanwhile, effective range expansion reads p cot δ t = − γ + 12 ρ d ( γ + p ) + · · · . (22)Now, the above renormalization condition allows us to relate the LECs to the effective-range expansion parameters. For the deuteron channel, one has σ t = − y t = ± m N s πρ d , D t ( p ) = m N ρ d γ + ip − ρ d ( γ + p ) = Z d E + B + · · · , (23)where Z d is the wave function normalization factor of the deuteron at the pole E = − B ,and the ellipsis in Eq. (23) denotes corrections that are finite or vanish at E = − B . Thusone has [18] Z d = γρ d − γρ d . (24)This Z d is equal to the asymptotic S -state normalization constant. It is to be noted thatthe order of the LECs y t is now of Q / , and the deuteron state is also described by therenormalized dressed di-baryon propagator.
4. Amplitude up to NLO
Diagrams for the pp fusion process up to NLO are shown in Fig. 5. In the limit p → A = − ~ǫ ∗ ( d ) · ~ǫ ( l ) G F V ud g A T fi . (25)Here ~ǫ ∗ ( d ) is the spin polarization vector of the out-going deuteron, ~ǫ ( l ) is the spatial partof the lepton current l µ in Eq. (1), and T fi ≃ s πγ − γρ d C η e iσ γ " e χ − a C γχI ( χ ) + 14 a C ( r + ρ d ) γ + a C γ g A m p l A , (26)8here I ( χ ) = 1 χ − e χ E ( χ ) , E ( χ ) = Z ∞ χ dt e − t t , (27)with χ = αm p /γ . We note that the amplitude T fi vanishes at the p → C η . The approximation is taken by keeping p dependence in C η whileignoring higher order p/m N corrections in the remaining part. Since p/m N ∼ . p/m N terms will be sub 1 % order, which can beneglected conservatively at the uncertainty level we are considering in the present work.Introducing a “standard reduced matrix element” [16],Λ( p ) = vuut γ πC η | T fi ( p ) | , (28)we have a finite and analytic expression of the reduced matrix element Λ( p ) in the p → √ − γρ d ( e χ − a C γ [1 − χe χ E ( χ )] + 14 a C ( r + ρ d ) γ + a C γ g A m p l A ) . (29)As mentioned above, we exactly reproduce the result of the effective range theory at LO,and have a higher order correction proportional to the LEC l A at NLO in Eq. (29).
5. Numerical results
We obtain the matrix element Λ(0) in Eq. (29) in terms of the four effective range pa-rameters, a C , r , γ and ρ d , and the LEC l A . The values of the effective range parametersare well known, but three of them are slightly different in the references. In this work, wetake two sets of the values: one is a C = − . ± . r = 2 . ± .
014 fm, and ρ d = 1 . ± .
005 fm from Table VIII in Ref. [33]. The other is a C = − . ± . r = 2 . ± .
014 fm, and ρ d = 1 . ± .
008 fm from Table XIV in Ref. [34].We take an average of numerical values of Λ(0) from the two sets of the parameters forour numerical result. The value of the LEC l A should be fixed by experimental data,but there are no precise ones for the two-body system. We fix the value of the LEC l A indirectly from the relative strength of the two-body matrix element to one-body one, δ B ≡ M B / M B = (0 . ± .
05) % in Eq. (29) in Ref. [14]. This value has been obtainedfrom the accurate potential model calculation for the two-body matrix element with thecurrent operators derived from HB χ PT up to N LO where the two-body current oper-ator has been fixed from an accurate experimental datum, the tritium lifetime, for thethree-body system. Thus we have l A = − . ± . , (30)where we have used our LO amplitude as the one-body input. This is a good approxima-tion because the difference between the amplitude from the effective range theory, whichis almost the same as our LO result, and that from accurate potential model calculations9ur result KR(NLO)[16] BC(N LO)[17] Pot. model[11]Λ (0) 7.09 ± ∼ ∼ ∼ (0). The value in second column is our result. The valuesin third, fourth, and fifth column are estimated from the pionless EFT calculation up toNLO by Kong and Ravndal (KR) [16], that up to N LO by Butler and Chen (BC) [17],and an accurate phenomenological potential model calculation [11], respectively.is tiny [12]. For other well known parameters, we use B = 2 . g A = 1 . m p = 938 .
272 MeV, and m n = 939 .
565 MeV, and thus have γ = 45 .
70 MeV, χ = 0 . E ( χ ) = 1 . LO (0) = 2 . NLO1 (0) = 2 . ± .
002 from the first set of the parameter values andΛ
NLO2 (0) = 2 . ± .
003 from the second one up to NLO. Thus we have an average valueΛ
NLO (0) = 2 . ± . , (31)and Λ (0) = 7 . ± .
02 where the estimated error bars mainly come from those of theeffective ranges, r and ρ d , and the LEC l A .In Table 1, we compare our numerical result for Λ (0) with those from other theoret-ical estimations, the pionless EFT without di-baryons up to NLO by Kong and Ravndal(KR) [16], that up to N LO by Butler and Chen (BC) [17], and the accurate phenomeno-logical potential model calculation [11]. We find that our numerical result is in goodagreement with the values from the former theoretical estimations within the accuracyless than 1 %. As discussed before, the uncertainties of the estimations from the pion-less EFT without di-baryon fields are still large, ∼ ∼ LO, mainly because of the unfixed LEC L A .Though the results in the previous pionless EFT calculations have the unfixed LEC L A ,we can directly compare our result of the amplitude Λ(0) in Eq. (29) to the expressionsin Eq. (7) in Ref. [17], and fix the value of the LEC L A . Assuming the higher order LEC K A = 0, we have L A = 1 . ± .
12 fm , which is consistent with our previous estimation, L A = 1 . ± .
11 fm in Ref. [21]. When comparing our result with that from the accuratephenomenological potential model calculation, we find that our result is overestimated by ∼ LO, respectively. In our calculation with di-baryon field, higher ordercorrections to the wave functions are incorporated naturally by the summation of effectiverange contribution to infinite order, which gives Λ(0) equal to 2.64. A great advantageof the pionless EFT with di-baryon field lies in that we don’t need to care the higher10rder contribution to the wave function, and it is sufficient to take into account only thecorrections to the vertices with external probe. This advantage reduces the number ofFeynman diagrams dramatically, and makes the calculation of higher order terms verysimple.
6. Discussion and conclusions
In this work, we employed the pionless EFT with di-baryon fields including the Coulombinteraction, and calculated the analytic expression of the amplitude for the pp fusionprocess, pp → de + ν e , up to NLO. Employing the assumption to distinguish LO andNLO terms in the contact di-baryon-di-baryon-axial-current interaction, we reproducedthe expression for the amplitude of the effective range theory at LO. The LEC l A , whichappears in the contact di-baryon-di-baryon-axial-current interaction at NLO, is fixed byusing the relative strength of the two-body amplitude to the one-body one, δ B , whichhas been determined from the tritium lifetime in the HB χ PT calculation, and thus wecould perform the parameter-free-calculation for the pp fusion process. We find that ournumerical result of squared reduced amplitude Λ (0) is in good agreement with those ofthe recent theoretical calculations within the accuracy better than 1%.As mentioned in the Introduction, the current theoretical uncertainties for the pp fusionprocess is ∼ χ PT calculaiton up to N LO [14]. To improve our result toa few tenth % accuracy, it would be essential to include the higher order corrections inthe modified counting rules discussed in the neutron beta decay calculation [30]: the nexthigher order corrections would be the α order and 1 /m N corrections. It is known that thehigher α order corrections, such as the vacuum polarization effect [9] and the radiativecorrections from the one-body part [30] , are significant, whereas the corrections from the1 /m N terms would be p c /m N ∼ . S factor for the pp fusion process in a few tenth % accuracy with the pionless EFT with di-baryon fieldsincluding those higher order corrections.Another issue that we would need to clarify is the value of the LEC l A , which hasbeen fixed in this work by using the result from the HB χ PT calculation. As discussed,e.g., in Refs. [14, 35], the LECs which appear in the two-di-baryon-axial-current or four-nucleon-axial-current contact interactions, denoted by l A in the pionless EFT with di-baryon fields, L A in the pionless EFT without di-baryon fields, and ˆ d R in HB χ PT, areuniversal. In other words, those LECs are shared by the processes, such as, the pp fusionprocess ( pp → de + ν e ) [12, 13, 14, 16, 17], nn fusion process ( nn → de − ¯ ν e ) [21], neutrinodeuteron reactions ( ν e d → ppe − , ν e d → npν e ) [36, 37], muon capture on the deuteron( µ − d → nnν µ ) [38, 39], radiative pion capture on the deuteron ( π − d → nnγ [40] andits crossed partner γd → nnπ + [41]), tritium beta decay [14], and hep process ( p He → He e + ν e ) [14]. If these LECs are determined by using the experimental data from oneof the processes, the lattice simulation [42], or the renormalization group method [43],then we can predict the other processes in each of the formalisms without any unknown The radiative corrections from the one-body part are quite significant, 2 ∼
3% level, and are con-ventionally included into the renormalized Fermi constant G ′ V ≃ G F V ud and the phase factor f pp in theestimation of the S factor for the pp fusion process. l A in the same formalism,the pionless EFT with di-baryon fields, from, e.g., the tritium lifetime extending ourformalism to the three-body systems with electroweak external probes. Acknowledgments
We would like to thank T.-S. Park for communications and M. C. Birse for reading themanuscript and commenting on it. SA is supported by the Korean Research Foundationand the Korean Federation of Science and Technology Societies Grant funded by KoreanGovernment (MOEHRD, Basic Research Promotion Fund): the Brain Pool program (052-1-6) and KRF-2006-311-C00271, and by STFC grant number PP/F000448/1. Work ofJWS and SWH is supported by the Korea Science and Engineering Foundation grantfunded by the Korean Government (MOST) (No. M20608520001-07B0852-00110). Thework of KK is supported by the US National Science Foundation under Grant PHY-0457014.
References [1] H. A. Bethe, Phys. Rev. , 434 (1938).[2] J. N. Bahcall, “Neutrino Astrophysics” , Cambridge University Press, Cambridge(1989).[3] E. G. Adelberger et al. , Rev. Mod. Phys. , 1265 (1998).[4] K. Kubodera and T.-S. Park, Ann. Rev. Nucl. Part. Sci. , 19 (2004).[5] H. A. Bethe and C. L. Critchfield, Phys. Rev. , 248 (1938).[6] E. E. Salpeter, Phys. Rev. , 547 (1952).[7] E. E. Salpeter, arXiv:0711.3139.[8] J. N. Bahcall and R. M. May, Astrophys. J. , 501 (1969).[9] M. Kamionkowski and J. N. Bahcall, Astrophys. J. , 884 (1994).[10] J. Carlson et al. , Phys. Rev. C , 619 (1991).[11] R. Schiavilla et al. , Phys. Rev. C , 1263 (1998).[12] T.-S. Park, K. Kubodera, D.-P. Min and M. Rho, Astrophys. J. , 443 (1998).[13] T.-S. Park et al. , nucl-th/0106025.[14] T.-S. Park et al. , Phys. Rev. C , 055206 (2003).[15] J.-W. Chen, G. Rupak, and M. J. Savage, Nucl. Phys. A 653 , 386 (1999).1216] X. Kong and F. Ravndal, Nucl. Phys.
A 656 , 421 (1999); Phys. Lett.
B 470 , 1(1999); Phys. Rev. C , 044002 (2001).[17] M. Butler and J.-W. Chen, Phys. Lett. B 520 , 87 (2001).[18] S. R. Beane and M. J. Savage, Nucl. Phys.
A 694 , 511 (2001).[19] S. Ando and C. H. Hyun, Phys. Rev. C , 014008 (2005).[20] J. Soto and J. Tarrus, arXiv:0712.3404.[21] S. Ando and K. Kubodera, Phys. Lett. B 633 , 253 (2006).[22] S. Ando, R. H. Cyburt, S.-W. Hong, C. H. Hyun, Phys. Rev. C , 025809 (2006).[23] S. Ando, Eur. Phys. J. A 33 , 185 (2007); arXiv:0708.4088.[24] D. B. Kaplan, Nucl. Phys.
B 494 , 471 (1997).[25] F. Gabbiani, nucl-th/0104088; D. B. Kaplan and J. V. Steele, Phys. Rev. C ,0604002 (1999).[26] M. Rho, in AIP Conference Proceedings (American Institute of Physics, New York,1999), Vol. 494, p. 391; nucl-th/9908015; D. R. Phillips and T. D. Cohen, Nucl. Phys.
A 668 , 45 (2000); D. R. Phillips, G. Rupak, M. J. Savage, Phys. Lett.
B 473 , 209(2000).[27] D. B. Kaplan, M. J. Savage, M. B. Wise, Phys. Lett.
B 424 , 390 (1998); Nucl. Phys.
B 534 , 329 (1998).[28] H. A. Bethe, Phys. Rev. , 38 (1949).[29] L. Durand, III, Phys. Rev. . 1597 (1957); L. Heller, Phys. Rev. , 627 (1960).[30] S. Ando et al. , Phys. Lett. B 595 , 250 (2004).[31] X. Kong and F. Ravndal, Phys. Lett.
B 450 , 320 (1999).[32] S. Ando, J. W. Shin, C. H. Hyun, S. W. Hong, Phys. Rev. C , 064001 (2007).[33] R. B. Wiringa, V. G. J. Stokes, R. Schiavilla, Phys. Rev. C , 38 (1995).[34] R. Machleidt, Phys. Rev. C , 024001 (2001).[35] A. Gardestig and D. R. Phillips, Phys. Rev. Lett. , 232301 (2006).[36] S. Nakamura et al. , Nucl. Phys. A 707 , 561 (2002); S. Ando et al. , Phys. Lett.
B555 , 49 (2003). 1337] M. Butler and J.-W. Chen, Nucl. Phys.
A 675 , 575 (2000); M. Butler, J.-W. Chen,X. Kong, Phys. Rev. C , 035501 (2001).[38] S. Ando et al. , Phys. Lett. B 533 , 25 (2002).[39] J.-W. Chen et al. , Phys. Rev. C , 061001 (2005).[40] A. Gardestig and D. R. Phillips, Phys. Rev. C , 014002 (2006).[41] V. Lensky et al. , Eur. Phys. J. A 26 , 107 (2005).[42] W. Detmold and M. J. Savage, Nucl. Phys.
A 743 , 170 (2004).[43] M. C. Birse, J. A. McGovern, K. G. Richardson, Phys. Lett.
B 464 , 169 (1999); T.Barford and M. C. Birse, Phys. Rev. C , 064006 (2003); S. X. Nakamura and S.Ando, Phys. Rev. C74