Halo EFT calculation of charge form factor for two-neutron {}^{6}\textrm{He} halo nucleus: two-body resonant P-wave interaction
aa r X i v : . [ nu c l - t h ] F e b EPJ manuscript No. (will be inserted by the editor)
Halo EFT calculation of charge form factor for two-neutron He halo nucleus: two-body resonant P-wave interaction S. Jesri
M. Moeini Arani
S. Bayegan Department of Physics, University of Tehran, P.O.Box 14395-547, Tehran, IranReceived: date / Revised version: date
Abstract.
We take a new look at He halo nucleus and set up a halo effective field theory at low energiesto calculate the charge form factor of He system with resonant P-wave interaction. P-wave Lagrangianhas been introduced and the charge form factor of He halo nucleus has been obtained at Leading-Order.In this study, the mean-square charge radius of He nucleus relative to He core and the root-mean-square(r.m.s) charge radius of He nucleus have been estimated as h r E i = 1 .
408 fm and h r E i He = 2 .
058 fm,respectively. We have compared our results with the other available theoretical and experimental data.
Key words.
Halo Effective Field Theory, He Halo nucleus, Charge form factor,Charge radius
PACS.
Weinberg was the first one who applied the effective fieldtheory (EFT) to nuclear forces [1]. Also the concept of ap-plying EFT to nuclear forces was brought by Rho [2] andby Ord´o˜nez and van Kolck [3]. An effective field theoryincludes the appropriate degrees of freedom to describephysical phenomena occurring at a chosen length scale orenergy scale. Up to now, cold atoms and few-nucleon sys-tems at low energies have been studied by this formalism[4,5,6]. Pion-exchange effects are not resolved at low en-ergies and momenta (
E < m π M N , p ≤ m π , where m π and M N are the mass of pion and nucleon respectively), so thistheory is constructed by only short-range contact interac-tions known as pionless effective field theory (EFT( π/ )) [7,8,9].One of the major challenges for nuclear theory is thecalculation of properties of halo nuclei. These nuclei arecharacterized by a tightly bound core and one or twoweakly bound valence nucleons [10,11,12,13]. In halo EFT,either core or nucleons are treated as the fundamentalfields and one can find relations between different nuclearlow-energy observables in this EFT. On the other hand,most systems can be explained by a short-range EFT ex-panding in Ra , which R is the range of the nucleon-nucleusinteraction such that M high ∼ R and a is the two-bodyscattering length such that M low ∼ a , so it is found that a e-mail: [email protected] b e-mail: [email protected] c e-mail: [email protected](corresponding author) R ≪ a . Based on an EFT in terms of the expansion pa-rameter R | a | , 2n halo nuclei are described as an effectivethree-body system including of a core and two weakly-attached valence neutrons. Some universal properties ofthese nuclei are investigated such as the matter densityform factors and mean square radii [14,15].Investigations have been carried out into Borromeannuclei such as Li, Be ,and He [10,11,14,15]. Whilethese nuclei have only one bound state, there are no boundstates in the binary subsystems. Properties of one neutronhalo nucleus Li has been pursued by the two-body sectorhalo EFT [16,17,18]. A detailed analysis of electromag-netic properties of halo nuclei Be system investigatedwhere the low-energy E1 strength function in breakup tothe Be-neutron channel has been performed [19].The lightest nuclei with 2n-halo structure are He, Li, Be, C, and B. Hagen et al. [20] and Vanasse[21] have studied the S-wave EFT framework for Li, Be, and C nuclei and calculated corresponding chargeradius. Other two-neutron halo nucleus, B, will be dealtwith in the future works. The halo EFT we construct in-cludes the two-body P-wave interaction in the nα sub-system of He nucleus. Binding energies, radii and otherproperties of various halo nuclei of s-wave and p-wave typehave been reviewed in halo EFT [22]. Ji etal . have consid-ered a halo EFT for the three-body nnα system to explainthe He ground state [23]. An EFT with P-wave resonanceinteractions has been developed for elastic nα scatteringby Bertulani et al. [24]. Bedaque etal . [25] have suggesteda different power counting compared to [24] to describenarrow resonances in EFT and illustrated their results in S. Jesri, et al.: Halo EFT calculation of charge form factor for two-neutron He halo nucleus the case of nucleon-alpha scattering. The electric dipolestrength function distribution of the He halo nucleus hasbeen recently evaluated based on the halo EFT approachusing the particle-dimer scattering amplitudes in the nnα system and the normalized He wave function [26]. Fi-nally, the momentum-space probability density of He atleading order in halo EFT has been presented. The mo-mentum distribution of He requires the n-n and n-coret-matrices as well as a c-n-n force as input in the Faddeevequations [27].In this paper, we focus on the two-neutron halo nucleus He, calculate the electric charge form factor and find theroot-mean-square (r.m.s) charge radius of He nucleus.Therefore we introduce the strong Lagrangian including nα P-wave interaction for the halo EFT at leading order inSection 2. In Section 3, the formalism for two- and three-body propagators are completely presented. He chargeform factor is evaluated in Section 4. In Section 5, ournumerical results for the form factor and the charge ra-dius are presented and compared with experimental data.Finally, we conclude in Section 6. In Appendix A, theparticle-dimer scattering is explained and in Appendix B,some expressions for the contributions of diagrams partic-ipate into the form factor of He are presented in details.
We apply a halo effective field theory in non-relativisticformalism for the alpha core ( α ) with spin zero interact-ing with two spin half neutrons. In this method, we define Q as a low momentum scale attributed to core and neu-tron momentums. Furthermore, the high momentum pa-rameter, ¯ Λ can be scaled as ¯ Λ ∼ m π ∼ √ m α E α where m α and E α = 20 .
21 MeV refer to the mass and theexcitation energy of α particle. There are the two-bodyneutron-neutron ( nn ) and the neutron-alpha ( nα ) inter-actions in He calculation. The remarkable state in the nn is S -wave virtual bound state. A low-momentum scale Q is defined by the inverse of the di-neutron scatteringlength, a = − . fm − and the inverse of the effectiverange of this S-wave state, r =2 . fm − is considered asthe high-momentum scale ¯ Λ . So, the leading order (LO)scattering amplitude of two neutrons is constructed bythe scattering length contribution only. With respect tothese scales, the S -wave effective range expansion (ERE)for di-neutron system at the lowest-order can be given by k cot δ = − a + · · · . (1)In low-energy region, only S - and P -wave interactionsare significant in the nα system. There are three possiblepartial waves for the nα system, S / , P / and P / .We use the power counting introduced by Bertulani et al.in Ref. [24] which also applied in the Gamow shell modelcalculation of He in halo EFT [28]. This power counting specifies that nα interaction gets the LO contributionsonly from both scattering length and effective range of P / channel as1 a ∼ Q , r ∼ Q , P ∼ Λ , (2)where a = − .
95 fm , r = − .
88 fm − and P = − . P / state, respectively [29]. There-fore the lowest-order terms of effective range expansion forthe resonant P-wave nα system are given by k cot δ = − a + r k + · · · . (3) Generally, the effective field theory expansion parameteris defined by the momentum ratio Q/ ¯ Λ and it createsthe order-by-order pattern of convergence. At LO, theLagrangian for He system can be written as the sum-mation of one-, two- and three-body contributions, L = L (1) + L (2) + L (3) , where L (1) = n † (cid:16) i∂ + ~ ∇ m n (cid:17) n + φ † (cid:16) i∂ + ~ ∇ m α (cid:17) φ L (2) = ∆ d † d − g √ (cid:16) d † ( n † iσ n ) + h.c. (cid:17) + η d † h i∂ + ~ ∇ m n + m α ) − ∆ i d + g h d † ~S † · [ n~ ∇ φ − ( ~ ∇ n ) φ ] + h.c. − r [ d † ~S † · ~ ∇ ( nφ ) + h.c. ] i L (3) = Ωt † t − h √ m n g Λ h t † ( d ( iσ ) ~S · ~P n ) + h.c. i , (4)where m n is the neutron mass and n, φ, d ( d ) denote thetwo component spinor field of the neutron, the bosonic al-pha core field, the auxiliary dimer field of nn ( nα ) system.Also, t implies a spin-0 trimer auxiliary field. Moreover wehave ~P = ˜ µ −→ m i ←−∇ − ˜ µ ←− m i −→∇ , (5)that −→ m ( ←− m )implies the mass of n ( d ) field and ˜ µ denotesthe reduced mass of n − d system. η is equal to ± g = πm n and r = ( m α − m n )( m α + m n ) . Also, σ indicates the Paulimatrix so that the spin projection matrix ( i √ σ ) projectsthe two neutrons on the spin-singlet case. In Eq. (4), the S i, s are the 2 × j = 1 / j = 3 /
2. These matricessatisfy the following relations S i S † j = 23 δ ij − i ǫ ijk σ k ,S † j S i = 34 δ ij − { J / i , J / j } + i ǫ ijk J / k , (6) . Jesri, et al.: Halo EFT calculation of charge form factor for two-neutron He halo nucleus 3 where J / i are the generators of the J = 3 / ∆ shouldbe fixed from matching the pionless EFT nn scatteringamplitude to the ERE scattering amplitude of two non-relativistic nucleons. Also we have the following relations[24] g = − η πµ r , ∆ = 1 µa r , (7)where µ is the reduced mass of nα system. According tothe sign of r the sign η should be fixed to +1. Dueto gauge invariance of the non-interacting parts of La-grangian for charged alpha and d -dimer, we include elec-tromagnetic coupling with vector potential A µ . This min-imal coupling gives the covariant derivative as ∂ µ → D µ = ∂ µ + i ˆ QA µ , (8)that A µ satisfies the Coulomb gauge fixing relation as ~ ∇· ~A = 0. In Eq. (8), ˆ Q introduces the charge operatorsuch that ˆ Qφ = Z eφ , ˆ Qd = Z ed , ˆ Qn = 0, and ˆ Qd = 0,where Z is the number of protons in the alpha core. The full dimer propagators are obtained by the infinitesum of diagrams shown in Fig. 1. The solid lines indicateneutron and the dashed lines are the α particle. The bare nn -dimer propagator has been depicted by double solidlines with empty arrow, and the bare nα -dimer propagatoris observed by dashed-solid lines with empty arrow.Based on introduced power counting, the LO full dimerpropagators shown by filled arrow in Fig. (1) for auxiliaryfield d and d are obtained by the following expressions iD ( p , p ) = i a − q p − m n p − iǫ ,iD ( p , p ) ¯ β ¯ α = iη δ ¯ β ¯ α (cid:26) p − p m n + m α ) − µa r − µr (cid:0) µm n + m α p − µp − iǫ (cid:1) + iǫ (cid:27) − , (9)that the incoming and outgoing spin components of d -dimer is indicated by ¯ α and ¯ β respectively. Because of J = of d -dimer, δ ¯ β ¯ α is a 4 × The amplitude of the particle-dimer scattering process in nnα system has been calculated using the Faddeev equa-tion introduced in Appendix A. The transition amplitude
Fig. 1.
Dressing the bare nn ( nα ) dimer propagator with bub-ble diagrams leads to the full nn ( nα ) dimer propagator to allorders. The solid lines show neutron and the dashed lines arethe α particle. Double solid lines with empty (filled) arroware the bare (full) d -dimer propagators and dashed-solid lineswith empty (filled) arrow are the bare (full) d -dimer propa-gators. Fig. 2.
Feynman diagrams for the full trimer propagator t .Single, double, triple lines denote neutron, d -dimer and trimerrespectively. Triple line with empty (filled) arrow is the bare(full) trimer propagator. (T-matrix) has a pole at three-body bound state, so theT-matrix can be factorized at energy E = − B n as [20] T ( E, k, p ) = − ~ B † ( k ) ~ B ( p ) E + B n + regular terms , (10)where B n = 0 .
97 MeV [30] denotes the 2n separation en-ergy of system. ~ B ( p ) = (cid:0) B ( p ) B ( p ) (cid:1) is the bound state vector,such that B ( B ) corresponds to the φd → φd ( φd → nd ) transition of the bound state equation according toEqs. (A.11) and (A.12). Fig. 2 indicates the Feynman di-agrams contributing to the full trimer propagator t ( E ).Based on the three-body interaction introduced in Eq.(A.10), the three-body force appears only between the in-coming and outgoing n + d channels. So, only T com-ponent derived from 2 × t ( E ). Using Feynman rules andtaking into account the projection operator in Eq. (A.2),for a trimer propagator we can write it ( E ) = iΩ h m n g Λ h Ω Z Λ dq q π q D ( E, q ) − m n g Λ h Ω Z Λ dq q π Z Λ dq ′ q ′ π (cid:16) D ( E, q ′ ) q ′ q T ( E, q ′ , q ) D ( E, q ) (cid:17)i , (11)where the energy integrals have been carried out and D ( E, q ) = D ( E − q m n , q ). S. Jesri, et al.: Halo EFT calculation of charge form factor for two-neutron He halo nucleus
Fig. 3.
Diagrammatic representation of coupled integral equa-tion for the trimer-dimer-particle three point function ~ G irr . The trimer wave function renormalization constant can beextracted from the following relation [20,31] Z t = lim E →− E (3) ( E + E (3) ) t ( E ) , (12)where E (3) = B n . By neglecting the regular functions interms of E corresponding to the first and second terms inEq. (11), we substitute the last term of t ( E ) into Eq. (12)and finally obtain Z t = m n g Λ h Ω lim E →− E (3) h − ( E + E (3) ) Z Λ dq q π Z Λ dq ′ q ′ π × (cid:16) D ( E, q ′ ) T ( E, q ′ , q ) D ( E, q ) (cid:17)i . (13)For the incoming and outgoing n + d channels, insertingEq. (10) into Eq. (13) yields Z t = m n g Λ h Ω Z Λ dq q π Z Λ dq ′ q ′ π × (cid:16) D ( − B n , q ′ ) B † ( q ′ ) B ( q ) D ( − B n , q ) (cid:17) = m n g Λ h Ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z Λ dq q π D ( − B n , q ) B ( q ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (14) ~ G irr The calculation of the charge form factor of He halo nu-cleus requires the trimer-dimer-particle three point func-tion ~ G irr . The LO ~ G irr function is illustrated by the cou-pled integral equation in Fig. 3. ~ G irr is a vector with twocomponents as ~ G irr ( E, p ) = (cid:0) G irr G irr (cid:1) , where G irr ( G irr ) isthe three point function for trimer constructed by α − d ( n − d ) system. Using introduced Lagrangian in Eq. (4)and Feynman rules, we can obtain the relation for twocomponents of ~ G irr as i G irri ( E, k ) = i √ m n Λ g h √ h δ i ~S † · ~k σ − Z Λ dq q π (cid:16) ~S † · ~q σ (cid:17) D ( E, q ) T i ( E, q, k ) (cid:12)(cid:12) H =0 i , (15) where the three-body force H has been introduced in Eq.(A.10). All trimer-reducible contributions are neglected bysetting H = 0 in Eq. (15) [20]. Taking into consideration T i ( E, q, k ) | H =0 component of Eq. (A.3), we have G irri ( E, k ) = √ m n Λ g h √ δ i ~S † · ~k σ − X j =0 Z Λ dq q π G irrj ( E, q ) D j ( E, q ) R ji ( E, q, k ) , (16)where i, j = 0 , R , R , R , and R that have beenderived in Eqs. (A.4)-(A.7). Taking into account the pro-jection operator based on Eq. (A.2), we haveTr (cid:16)r σ ( ˆ e k · ~S ) ~S † · ~k σ (cid:17) = r k, (17)therefore, the matrix integral equation for the P-wave irre-ducible trimer-dimer-particle three point function is givenby ~ G irr ( E, k ) = Z t √ m n Λ g h √ k ˆ υ − Z Λ dq q π (cid:16) R ( E, q, k ) D ( E, q ) ~ G irr ( E, q ) (cid:17) , (18)where 2 × R has been defined in Eq. (A.4)and we have D ( E, q ) ≡ (cid:18) D ( E, q ) 00 D ( E, q ) (cid:19) , (19)with D ( E, q ) = D ( E − q m α , q ) and D ( E, q ) = D ( E − q m n , q ).Two components of Eq. (18) that enter into the calcu-lation of form factor for the He halo nucleus are derivedas G irri ( E, k ) = 1 √ m n g Λ | β H ( Λ ) | h k δ i − Z Λ dq q π (cid:16) q D ( E, q ) T i ( E, q, k ) (cid:12)(cid:12) H =0 (cid:17)i , (20)where β = Z Λ dq q π q D ( − B n , q ) B ( q ) , (21)and the calculation of H ( Λ ) = h Ω requires the normalized He wave function which is obtained by solving the boundstate equation corresponding to the homogeneous part ofEq. (A.3) with E = − B n . We should mention that (ˆ υ ) i = δ i and after multiplying √ Z t in Eq. (18) ~ G irr has thecutoff dependence which is small enough to render ourpredictions renormalized. . Jesri, et al.: Halo EFT calculation of charge form factor for two-neutron He halo nucleus 5
Fig. 4.
Diagrammatic representation of the matrix element Γ ( ~Q ) in Eq. (23) falling into four different classes (a), (b), (c)and (d). The wavy lines show minimally photon coupled andthe vertex depicted by filled circle in diagram (d) indicatesminimally coupled photon to the P-wave d -dimer. He halo nucleus We present a formalism for form factor calculation of Hehalo nucleus with shallow P-wave interaction. We initiallyemphasize that all calculations have been performed inthe Breit frame in which no energy is carried by the pho-ton. This implies P = K and ~P = ~K where ~P ( ~K ) de-notes the incoming (outgoing) three momentum of trimer.The He charge form factor only depends on the three-momentum of the photon ~Q = ( ~K − ~P ) according to[32,33] h t ( K , ~K ) | j | t ( P , ~P ) i = ( − ie Z ) F E ( ~Q ) = Z t iΓ ( ~Q ) , (22)where Z implies the atomic number of He nucleus and j is the zeroth component of the electromagnetic current.At LO, Γ ( ~Q ) in Eq. (22) introduces the sum of alldiagrams with external trimer lines and minimally coupledphoton to the alpha and d -dimer as shown in Fig. 5, sowe have iΓ ( ~Q ) = Z d p (2 π ) Z d k (2 π ) × i ~ G irr ( E, ~P , p , ~p ) T iΓ ( E, ~P , p , ~p, ~K, k , ~k ) × i ~ G irr ( E, ~K, k , ~k ) . (23)The matrix element Γ in Eq. (23) is defined by the sumof diagrams that are depicted in Fig. 4. The energy quan-tity is defined by E = P − ~P M tot , where the kinetic energyof the He system is subtracted and M tot = m α + 2 m n .Therefore, the He charge form factor at LO is found bythe sum of diagrams in Fig. 4 as F E = F ( a ) E + F ( b ) E + F ( c ) E + F ( d ) E . The wavy lines show minimally coupled photon andthe vertex depicted by filled circle in diagram (d) indicatesminimally coupled photon to the P-wave d -dimer.For calculating of the charge form factor, it is necessaryto define the relation between G irri ( E, ~P , p , ~p ) and center-of-mass ( c.m. ) quantity G irri ( E, p ) from Eq. (20) via the following integral equation G irri ( E, ~P , p , ~p ) = 1 √ m n g Λ | β H ( Λ ) | p δ i − X j =0 Z Λ dq q π R ij (cid:16) M i M tot E + p − ~P · ~pM tot + p m i , p, q (cid:17) × D j ( E, q ) G irrj ( E, q ) , (24)where m i = m n , m α for i = 0 ,
1, respectively, and M = 2 m n , M = m α + m n . (25)After performing calculations according to Eq. (4) andusing Feynman rules, we obtain the following final rela-tions for charge form factor contributions of diagrams (a),(b), (c) and (d) in Fig. 5 F ( a ) ( Q ) = Z Λ p π dp Z Λ k π dk ~ G irr ( p ) T × D ( p ) Υ ( a ) ( Q, p, k ) D ( k ) ~ G irr ( k ) , (26) F ( b ) ( Q ) = Z Λ (cid:16) − p π (cid:17) dp Z Λ (cid:16) − k π (cid:17) dk × ~ G irr ( p ) T D ( p ) Υ ( b ) ( Q, p, k ) D ( k ) ~ G irr ( k ) , (27) F ( c ) ( Q ) = Z Λ (cid:16) − p π (cid:17) dp Z Λ (cid:16) − k π (cid:17) dk ~ G irr ( p ) T × D ( p ) Υ ( c ) ( Q, p, k ) D ( k ) ~ G irr ( k )+2 Z Λ (cid:16) − p π (cid:17) dp ~ G irr ( p ) T D ( p ) ~Υ ( c ) ( Q, p ) + Υ ( c )0 ( Q ) , (28)and F ( d ) ( Q ) = Z Λ (cid:16) − p π (cid:17) dp Z Λ (cid:16) − k π (cid:17) dk × ~ G irr ( p ) T D ( p ) Υ ( d ) ( Q, p, k ) D ( k ) ~ G irr ( k )+2 Z Λ (cid:16) − p π (cid:17) dp ~ G irr ( p ) T D ( p ) ~Υ ( d ) ( Q, p )+ Υ ( d )0 ( Q ) . (29)The detailed derivations of Eqs. (26)-(29) including thedefinitions of the used functions have been explained inAppendix B. As we know at Q = 0, the charge form factor is nor-malized to one because of conservation of current. The S. Jesri, et al.: Halo EFT calculation of charge form factor for two-neutron He halo nucleus
Fig. 5.
Diagrams for the charge form factor of the He nucleusat LO corresponding to F E ( ~Q ) in Eq.(22). Notations are asin Fig. (4). expansion of the form factor in powers of Q leads to F E ( Q ) = 1 − h r E i Q + · · · , (30)where h r E i is the mean-square charge radius of the Hehalo system relative to He mean-square charge radius.By taking the limit Q → + , h r E i can be extracted as h r E i = − Q → + d F E dQ , (31)and we can obtain the mean-square charge radius of Hehalo nucleus by the following relation h r E i He = h r E i + h r E i He . (32)It is necessary to point out that we have neglected thesmall negative mean-square charge radius of the neutron h r E i n = − .
115 fm [34] in our calculation. In this section,we apply our P-wave halo EFT formalism to calculate theform factor and the mean-square charge radius of He nu-cleus relative to He core ( h r E i ) according to Eq. (31). Wecompare our EFT evaluation with other available theoret-ical results. Our formalism applies directly to two-neutronhalo nucleus, He with J P = 0 + .Fitting the three-body binding energy of He nucleusto B n = 0 .
97 MeV, the three-body force can be deter-mined at leading order. Using this determined three-bodyforce [26], the renormalized He wave function and sothe renormalized trimer-dimer-particle three point func-tion is obtained. The effects of the cutoff dependence forthe two components of ~ G function are shown in Fig.6. Theplots represent the cutoff variations of the P-wave irre-ducible trimer-dimer-particle three point function between Λ = 600 MeV and Λ = 1200 MeV with the three-bodyforce which is introduced by Eq. (A.10). As depicted inFig.6, by considering the three-body force, the cutoff vari-ations of the results are acceptable in comparison withthe LO systematical uncertainty. Therefore our numericalresults for the trimer-dimer-particle three point functionare properly renormalized.As mentioned in Section 2.1, we concentrate on thepower counting which is suggested by Bertulani and col-laborators in Ref. [24] for the nα interaction. Investiga-tion of the full propagator of the d field in this power Table 1.
The mean-square charge radius of He nucleus rel-ative to He from Eq. (31) and the r.m.s charge radius of Henucleus according to Eq. (32) that have been compared to theother theoretical and experimental results.This Experimental Theoreticalwork Results Results h r E i [fm ] 1.408 1 . ± .
058 [38] 1.426(38) [42]1 . ± .
034 [39]2.060(8) [40] 2.06(1) [42] h r E i He [fm] 2 .
058 2 . ± .
014 [38] 2.12(1) [42]2.147 [36]2.586 [41] counting discloses in addition to the physical resonance(shallow na resonance), there also exists one spurious pole(the unphysical He bound state) around p ∼
99 MeVwith negative residue and a deeper binding energy. Usingthis power counting, the Gamow shell model calculationof He in Ref. [28] removed the spurious pole in the nα T-matrix by constructing bi-orthogonal complete basis.In the Halo EFT analysis of the He system, in orderto get rid of this spurious pole, one can also treat theunitarity term ik in the denominator of nα propagatoras a perturbation. This method was recently applied to He in Ref. [23,35]. One of fundamental drawbacks of thismethod is that unitarity is lost at LO, which is actuallya requirement for the form p cot δ l − ip that was chosen forthe scattering amplitude. In fact the loss of unitarity atLO is not problematic in Ref. [23], since only bound stateobservable is considered in the related bound-state three-body calculation. In the Faddeev equation, the resonancepole of nα scattering, which requires the unitary term,was never crossed. However, the unitary term matters ifone wants to calculate a resonance state in He.Generally, in the three-body sector, for solving Fad-deev integral equation, analogous to the Skornyakov-Ter-Martyrosian (STM) equation for S-wave contact interac-tions, one can solve Faddeev integral equation for resonantP-wave interactions. In order to discard spurious pole onecan use the contour deformation suggested by Hethering-ton and Schick, namely a rotation p → pe − iΦ ( Φ >
0) asapplied in Ref. [26] for the positive energies.In this paper we are concerned with the homogeneouspart of the integral equations projected onto the bound0+ ground state of He. The position of spurious boundstate of a P-wave propagator is on the real axes but for thenegative energies E = E B = − B n , Eq. (A.11). One canhandle this unphysical deep bound state with similar p → pe − iΦ ( Φ >
0) analytical continuation by contour rotationof the real axes. In this simpler contour path integral, thereis no logarithmic singularities in the loop momentum incomparison with logarithmic singularities in the Legendrefunctions of second kind in the positive energies on thereal axes.In Fig. 7, our calculation for the charge form factorof He with Λ = 700 MeV is depicted as a function of . Jesri, et al.: Halo EFT calculation of charge form factor for two-neutron He halo nucleus 7 L= L= L= L= L= L= G H E , p L L= L= L= L= L= L= @ MeV D G H E , p L Fig. 6. (color online) Two components of the trimer-dimer-particle three point function for the different cutoff values from 600MeV to 1200 MeV.
Table 2.
The mean-square charge radius of He nucleus relative to He from Eq. (31) and the r.m.s charge radius of Henucleus according to Eq. (32) for different cutoff values. Λ (MeV) 600 700 800 900 1000 1200 h r E i [fm ] 1.40822 1.40807 1.40739 1.40786 1.40801 1.40777 h r E i He [fm] 2.05766 2.05763 2.05746 2.05758 2.05761 2.05755 the photon momentum ( ~Q = ~K − ~P ) in halo EFT. Ourresults have been compared with distorted wave Born ap-proximation (DWBA) [36] large scale shell model (LSSM)calculations. This model-dependent approach studies Henucleus as a six-body system (not three-body halo one)by using a Woods-Saxon single-particle wave function ba-sis. The difference in the calculated form factors appearsas the Q is increasing. As we expect EFT is a model-independent and precision-controlled approach, so includ-ing the higher-order corrections can give us better judg-ment in comparing our three-particle halo formalism withthe full six-body DWBA results.Using Eq. (31), we fit our data for form factor via thestandard interpolation method and take the second orderderivatives of the fitted function with respect to Q in thelimit Q → + to evaluate the mean-square charge radiusof He nucleus relative to He core. Ingo Sick has stud-ied the world data on elastic electron-Helium scatteringto determine a precise value for He r.m.s charge radiusand has obtained the value (1 . ± . He nucleus relativeto He from Eq. (31) and the r.m.s charge radius of Henucleus in comparison with other theoretical and experi-mental results.As discussed in the introduction, the expansion pa-rameter of our theory R core /R halo is roughly R/a . In or-
Halo EFTDWBA Q ( MeV ) F E Q Fig. 7. (color online) The charge form factor of the He nucleuswith Λ = 700 MeV at LO corresponding to F E ( ~Q ) in Eq. (22).The shaded region implies a criterion of estimated theoreticalartifacts. der to obtain better estimates, we compare the typicalenergy scales E halo and E core of the two-neutron haloand the core, respectively. To estimate E halo , we choosethe two-neutron separation energy B n . The energy scaleof the core is estimated by the excitation energy of thealpha particle E α . The square root of the energy ratio R core /R halo ∼ p E halo /E core then yields an estimate forthe expansion parameter of the effective theory. The two- S. Jesri, et al.: Halo EFT calculation of charge form factor for two-neutron He halo nucleus neutron separation energy of He is 0.97 MeV and thefirst excitation energy of the alpha particle is 20.21 MeV.The expansion parameter and the error can be estimatedas R core /R halo ∼ p B n /E α ∼ .
22. So, the calculatedmean-square charge radius and the r.m.s charge radius of He nucleus in Table 1 have LO systematic errors of orderof δ h r E i = 0 .
308 fm and δ h r E i He = 0 .
555 fm, respec-tively. The expansion parameter R core /R halo is typicallynot much smaller than 1. As a consequence, the main un-certainty in our calculation is from the next-to-leadingorder corrections in the effective theory.The small and negligible cutoff variation in the cal-culated values of the r.m.s charge radius of He nucleusas presented in Table 2 shows that our EFT results havebeen properly renormalized. The r.m.s charge radius of He nucleus has been determined to a precision of 0 . . ± . h r c i He − h r c i He has been eval-uated (1 . ± . in a laser spectroscopic measure-ment at Argonne National Laboratory [38]. Isotope shiftsof the matter radii have been deduced via scattering ofGeV/nucleon nuclei on Hydrogen in inverse kinematics.This approach leads to the value (1 . ± . forthe mean-square charge radius of He isotope relative to He [39]. The first direct mass measurement of He hasbeen performed with the TITAN Penning trap mass spec-trometer at the ISAC facility [40]. The obtained mass is m ( He) = 6 . u . With this new mass valueand the previously measured atomic isotope shifts, theyhave obtained the r.m.s charge radii of 2 . He[40]. Antonov et al. have also calculated the value 2 .
147 fmfor the r.m.s charge radius of He nucleus using LSSMdensities [36]. R.m.s radius in fm for He has calculated2 .
586 fm using the shell model wave functions and thespecified single particle wave functions [41]. Our resultsare consistent to the Monte Carlo calculation based onAV18+IL2 three-body potential that reports 2 . He [42]. Using AV18+UIXthree-body potential, the r.m.s charge radius of He hasbeen obtained 2 . In the present halo EFT formalism, we have describedthe electromagnetic structure of He halo nucleus. Thetrimer propagator and the trimer wave function renor-malization ( Z t ) are obtained in details. The trimer-dimer-particle three point function ~ G irr that is required for cal-culations of form factor is discussed completely. The mainpurposes of the present work are the calculation of thecharge form factor and the r.m.s charge radius of He.The charge form factor of He has been obtained by thesummation of four different diagrams depicted in Fig. 5.We have presented our EFT results for form factor in Fig.7 and we have shown the shaded region that implies acriterion of estimated theoretical artifacts in our calcu-lations. The mean-square charge radius of He nucleusrelative to He core and the r.m.s charge radius of He nucleus have been evaluated as h r E i = 1 .
408 fm and h r E i He = 2 .
058 fm, respectively with remarkable agree-ment with other experimental and theoretical results. Inthe future works, this formalism can be expanded to next-to-leading order (NLO) in order to reduce EFT theoreticalerror. The B nucleus can be also described using this P-wave halo EFT approach in the future.
Acknowledgement
We would like to thank A. N. Antonov and M. K. Gaidarovfor providing the data of their DWBA calculation.
Appendix A: The Faddeev equation of theparticle-dimer scattering process in nnα system
Since He nucleus has spin-parity J P = 0 + in the groundstate, we apply the Faddeev equation (T-matrix) of theparticle-dimer scattering process in nnα system with J P =0 + . This integral equation is shown in Fig. 8. According toLagrangian in Eq. (4) we use two different dimers, d and d , so there are four possible transitions between particle-dimer states n + d −→ n + d , n + d −→ φ + d ,φ + d −→ n + d , φ + d −→ φ + d . (A.1)In the c.m. frame, on-shell T-matrix depends on the totalenergy E and the incoming (outgoing) three-momentumsof the φ + d and n + d systems which indicated by k ( p )and k ( p ) respectively.In the cluster-configuration space, the projection op-erator of He channel is obtained by( P + ) β ,β ¯ α,α = β α q (cid:16) σ ˆ e · ~S (cid:17) β ¯ α ! , (A.2)where is the 2 × e denotes the unitvector of c.m. momentum of the n + d system [26]. Ap-plying the projection operator according to Eq. (A.2), theresulting 2 × T ( E, k , p , k , p ) = h R ( E, k , p , k , p ) + H ( k , p , Λ ) i − π Z Λ q dq h(cid:16) R ( E, k , q, k , q ) + H ( k , q, Λ ) (cid:17) · D ( E, q ) · T ( E, q, p , q, p ) i , (A.3)where Λ is an ultraviolet cutoff. The kernel R ( E, k , q , k , q )is a 2 × R ( E, k , q , k , q ) ≡ (cid:18) R ( E, k , q ) R ( E, k , q ) R ( E, k , q ) (cid:19) , (A.4) . Jesri, et al.: Halo EFT calculation of charge form factor for two-neutron He halo nucleus 9
Fig. 8.
Representation of the integral equation for the T-matrix. Single, double, triple lines denote particles, dimers andtrimers respectively. Triple lines with empty arrow are the baretrimer propagators. that R ( E, k, q ) = − g m α h − r ) k + q kq Q ( ε ( E, k, q ))+ 83 Q ( ε ( E, k, q )) + ( 43 + (1 − r ) ) Q ( ε ( E, k, q )) i , (A.5) R ( E, k, q ) = − g g m n √ h k Q ( ε ( E, k, q ))+ 1 + rq Q ( ε ( E, k, q )) i , (A.6) R ( E, k, q ) = − g g m n √ h q Q ( ε ( E, k, q ))+ 1 + rk Q ( ε ( E, k, q )) i . (A.7)The relation between the L − th Legendre function of thefirst kind P L ( z ) and the second kind Q L ( z ) is written by Q L ( z ) = R − dt P L ( t ) z − t , therefore Q ( z ) = 12 ln (cid:16) z + 1 z − (cid:17) ,Q ( z ) = 12 z ln (cid:16) z + 1 z − (cid:17) − ,Q ( z ) = − z + 14 (3 z − (cid:16) z + 1 z − (cid:17) . (A.8)The functions of ε , ε , and ε in the above equationare defined by [26] ε ( E, k, q ) = m α E − m α µ ( k + q ) kq ,ε ( E, k, q ) = m n E − k − m n µ q kq ,ε ( E, k, q ) = m n E − m n µ k − q kq . (A.9)In Eq. (A.3), the three-body force H shown by a baretrimer with external particle-dimer lines in Fig. 8 is givenby the following relation H ( k, q, Λ ) ≡ (cid:18) − m n g kqH ( Λ ) Λ (cid:19) , (A.10) which connects only the incoming and outgoing n + d channels [23,26,28]. The bound state equation is writtenas T He ( p ) = − π Z Λ q dq h R ( − B n , k, q, k, q )+ H ( k, q, Λ ) i · D ( − B n , q ) · T He ( q ) , (A.11)The transition xX → yY ( x, y = φ, n and X, Y = d , d )contributes to construction of He such that T He ( q ) ≡ (cid:18) T He, φd → φd ( q ) T He, nd → φd ( q ) T He, φd → nd ( q ) T He, nd → nd ( q ) (cid:19) . (A.12)For the incoming φ + d channel, the proper normalizationcondition for the solution of Eq. (A.11) is [43] (cid:16) D ~ B (cid:17) T ⊗ ddE (cid:16) I − K (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) E = − B n ⊗ (cid:16) D ~ B (cid:17) = , (A.13)where D matrix is given by Eq. (19), ~ B = T He (cid:0) (cid:1) = (cid:0) B B (cid:1) is the bound state vector, and K is given by K ( E, q, q ′ ) = R ( E, q, q ′ , q, q ′ ) + H ( q, q ′ , Λ ). We must define the inversepropagators matrix I = diag( I , I ) with I , = 2 π q δ ( q − q ′ ) D , ( E, q ) − . (A.14)We have defined here the short notation [43] A ⊗ B ≡ π Z Λ q dq A ( · · · , q ) B ( q, · · · ) . (A.15)Therefore for each cutoff Λ , we have fixed the H ( Λ ) = h Ω parameter such that Eq. (A.11) is satisfied at experimentalvalue of E = − B n = − .
97 MeV.
Appendix B: The contribution of diagrams ( a ) , ( b ) , ( c ) and ( d ) to charge form factor In this appendix, we introduce explicitly the relations offour different diagrams (a), (b), (c) and (d) that contributeto the charge form factor as shown in Fig. 5.
B.1 Contribution F ( a ) E After performing the energy integral analytically, usingEqs. (4), (22)-(24) and Feynman rules, the contributionof diagram (a) in Fig. 5 is given by F ( a ) ( Q ) = Z Λ p π dp Z Λ k π dk ~ G irr ( p ) T × D ( p ) Υ ( a ) ( Q, p, k ) D ( k ) ~ G irr ( k ) , (B.1) He halo nucleus where the components of 2 × Υ ( a ) ( Q, p, k ) aregiven by Υ ( a ) ij ( Q, p, k ) = g µ ) π Z − dx Z − dy Z π dφ × (cid:26) p + k + r ′ − ky + m n M px ) r ′ + 2 m n M kp ( p − x p − y cos φ + xy ) + 2 µB n (cid:27) − × (cid:26) p + k + r ′ + 2( px + m n M ky ) r ′ + 2 m n M kp ( p − x p − y cos φ + xy ) + 2 µB n (cid:27) − × U ( p, k, Q, x, y, φ ) δ i δ j , (B.2)where ~r ′ = m n M tot ~Q , µ = m n m α m n + m α , the polar angles x = cos( ∠ ( ~Q, ~p )), y = cos( ∠ ( ~Q, ~k )),cos( ∠ ( ~p, ~k )) = p (1 − x ) p (1 − y ) cos φ + xy and U ( p, k, Q, x, y, φ ) = − r ′ xy + 43 pyr ′ (cid:16) m n M (cid:17) − kxr ′ (cid:16) m n M (cid:17) − pxr ′ ( p − x p − y cos φ + xy )+ 203 kyr ′ ( p − x p − y cos φ + xy )+ 163 m n M p ( p − x p − y cos φ + xy )+ 163 m n M k ( p − x p − y cos φ + xy ) − r ′ ( p − x p − y cos φ + xy )+ 203 kp ( p − x p − y cos φ + xy ) − pk (cid:16) − m n M (cid:17) . (B.3) B.2 Contribution F ( b ) E For the contribution of the diagram (b) in Fig. 5, calcu-lating the energy integral analytically, applying Eqs. (4),(22)-(24) and Feynman rules lead to the following rela-tions F ( b ) ( Q ) = Z Λ (cid:16) − p π (cid:17) dp Z Λ (cid:16) − k π (cid:17) dk × ~ G irr ( p ) T D ( p ) Υ ( b ) ( Q, p, k ) D ( k ) ~ G irr ( k ) , (B.4)where Υ ( b ) ij ( Q, p, k ) = m α Z Λ (cid:16) q π (cid:17) dqq Z − dx ′ x ′ χ ( b ) ij ( r ′ , q, x ′ , p, k ) , (B.5) Fig. 9.
Diagrammatic representation of diagram (c). so that χ ( b ) ij ( r ′ , q, x ′ , p, k ) = 1 Q n R i (cid:16) − B n , p, d ( q, r ′ , x ′ ) (cid:17) × D (cid:16) − B n − m n m α M tot Q − qQx ′ m α , q (cid:17) × R j (cid:16) − B n − qQx ′ m α , d ( q, r ′ , − x ′ ) , k (cid:17) − R i (cid:16) − B n + qQx ′ m α , p, d ( q, r ′ , x ′ ) (cid:17) × D (cid:16) − B n − m n m α M tot Q + qQx ′ m α , q (cid:17) × R j (cid:16) − B n , d ( q, r ′ , − x ′ ) , k (cid:17)o δ i δ j , (B.6)with d ( q, r ′ , ± x ′ ) = p q + r ′ ± qr ′ x ′ . B.3 Contribution F ( c ) E The leading contribution to charge form factor in the Hehalo nucleus comes from diagram (c) in Fig. 5 by couplingthe photon to α core inside a nα bubble. For calculatingthe contribution of the form factor presented in Fig. 9, westart with the four-momentum integration as F ( c ) ( Q ) = ( − ie Z ) − Z k<Λ d k (2 π ) Z p<Λ d p (2 π ) (2 π ) × δ ( k − p ) δ (3) ( ~k − ~p − m n M tot ~Q ) × ( − ie Z ) n i G irr ( P , ~P , p , ~P ) T × i m n M tot P − p − ( mnMtot ~P − ~p ) m n + iε × h − iΣ (cid:16) M M tot ¯ P + ¯ p, M M tot ¯ K + ¯ k (cid:17)i × iD (cid:16) M M tot K + k , M M tot ~K + ~k (cid:17) × iD (cid:16) M M tot P + p , M M tot ~P + ~p (cid:17) × i G irr ( K , ~K, k , ~k ) o , (B.7) . Jesri, et al.: Halo EFT calculation of charge form factor for two-neutron He halo nucleus 11 where Σ (cid:16) M M tot ¯ P + ¯ p, M M tot ¯ K +¯ k (cid:17) is the nα bubble contribu-tion in Fig. 9. One of the two four-momentum integrationsin Eq. (B.7) is absorbed by a delta-function, so we obtain F ( c ) ( Q ) = Z p<Λ d p (2 π ) n G irr ( P , ~P , p , ~P ) T × i m n M tot P − p − ( mnMtot ~P − ~p ) m n + iε × h − iΣ (cid:16) M M tot ¯ P + ¯ p, M M tot ¯ K + ¯ k (cid:17)i × D (cid:16) M M tot K + p , M M tot ~K + ~p + m n M tot ~Q (cid:17) × D (cid:16) M M tot P + p , M M tot ~P + ~p (cid:17) × G irr ( K , ~K, p , ~p + m n M tot ~Q ) o . (B.8)Using the rescaled four-momentum ¯ s = m n M tot ¯ Q and theshifted loop momentum according to ¯ p ¯ q − ¯ s , we have F ( c ) ( Q ) = Z q<Λ d ~q (2 π ) Z + ∞−∞ dq π n G irr ( P , ~P , ¯ q − ¯ s ) T × i m n M tot P − q − ( mnMtot ~P − ~q + ~s ) m n + iε × h − iΣ (cid:16) M M tot ¯ P + ¯ q − ¯ s, M M tot ¯ K + ¯ q + ¯ s (cid:17)i × D (cid:16) M M tot K + q , M M tot ~K + ~q + ~s (cid:17) × D (cid:16) M M tot P + q , M M tot ~P + ~q − ~s (cid:17) × G irr ( K , ~K, ¯ q + ¯ s ) o . (B.9)After performing the q integration according to the pole q = m n M tot P − ( mnMtot ~P − ~q + ~s ) m n + iε , we obtain F ( c ) ( Q ) = Z q<Λ d ~q (2 π ) n G irr ( P , ~P , ¯ q − ¯ s ) T × h − iΣ (cid:16) M M tot ¯ P + ¯ q − ¯ s, M M tot ¯ K + ¯ q + ¯ s (cid:17)i × D (cid:16) M M tot K + q , M M tot ~K + ~q + ~s (cid:17) × D (cid:16) M M tot P + q , M M tot ~P + ~q − ~s (cid:17) × G irr ( K , ~K, ¯ q + ¯ s ) o . (B.10) Fig. 10.
Diagrammatic representation of the bubble contribu-tion − iΣ (¯ p, ¯ k ). Considering Eq. (24) and by substituting the pole q , wehave G irr ( P , ~P , ¯ q − ¯ s ) T = 1 √ m n g Λ | β H ( Λ ) | d ( q, s, − x ′ ) − X i =0 Z Λ dp p π G irri ( E, p ) D i ( E, p ) × R i (cid:16) E, p, d ( q, s, − x ′ ) (cid:17) , (B.11) G irr ( K , ~K, ¯ q + ¯ s ) = 1 √ m n g Λ | β H ( Λ ) | d ( q, s, x ′ ) − X j =0 Z Λ dk k π R j (cid:16) E, d ( q, s, x ′ ) , k (cid:17) × D j ( E, k ) G irrj ( E, k ) , (B.12)with d ( q, s, ± x ′ ) = p q + s ± qsx ′ and the polar angle x ′ = cos( ∠ ( ~Q, ~q )). Using Eq. (9), Eq. (19) and insertingthe pole q , we can redefine the two propagators in Eq.(B.9) as D (cid:16) M M tot K + q , M M tot ~K + ~q + ~s (cid:17) = D (cid:16) E − m n M M tot Q − qQx ′ M , q (cid:17) , (B.13) D (cid:16) M M tot P + q , M M tot ~P + ~q − ~s (cid:17) = D (cid:16) E − m n M M tot Q + qQx ′ M , q (cid:17) . (B.14) B.3.1 Bubble diagram
We now calculate the term − iΣ (cid:16) M M tot ¯ P + ¯ q − ¯ s, M M tot ¯ K + ¯ q +¯ s (cid:17) for the bubble diagram depicted in Fig. 10. For generalincoming (outgoing) four-momenta ¯ p (¯ k ) and according to He halo nucleus
Eq. (4), we obtain − iΣ (¯ p, ¯ k ) = − Z d ~q ′ (2 π ) (cid:16) i g (cid:17) × h q ′ m n + ( ~k + ~q ′ ) m α − k × q ′ m n + ( ~p + ~q ′ ) m α − p × ( ~k (1 − r ) + 2 ~q ′ ) j ( ~p (1 − r ) + 2 ~q ′ ) i Tr( S † j S i ) i . (B.15)According to Eq. (6), we derive Tr( S † j S i ) = δ ij . Fur-thermore, using the relation a a = R dx [ a x + a (1 − x )] anddefining the rescaled loop momentum ~b := ~q ′ + µmα [ x~p +(1 − x ) ~k ] µmα Q and replacing our kinematics ¯ p M M tot ¯ P + ¯ q − ¯ s and¯ k M M tot ¯ K + ¯ q + ¯ s in Eq. (B.15), we can obtain thefollowing relation for the bubble contribution, Σ , as − iΣ (cid:16) M M tot ¯ P + ¯ q − ¯ s, M M tot ¯ K + ¯ q + ¯ s (cid:17) = 43 µ m α Q g Z dx Z d ~b (2 π ) ( b − A | q − s | , | q + s | ( x )) × "(cid:16) M M tot ~K + ~q + ~s (cid:17) (1 − r )+ 2 µm α (cid:16) Q~b − h x (cid:16) M M tot ~P + ~q − ~s (cid:17) +(1 − x ) (cid:16) M M tot ~K + ~q + ~s (cid:17)i(cid:17) i × "(cid:16) M M tot ~P + ~q − ~s (cid:17) (1 − r )+ 2 µm α (cid:16) Q~b − h x (cid:16) M M tot ~P + ~q − ~s (cid:17) +(1 − x ) (cid:16) M M tot ~K + ~q + ~s (cid:17)i(cid:17) i ) , (B.16)where A | q − s | , | q + s | ( x ) = x − x (cid:16) C P ,q − s − C K,q + s (cid:17) − C K,q + s , (B.17)with C P ,q − s = 2 µ h B n + ( ~q − ~s ) µ i(cid:16) M M tot (cid:17) s ,C K,q + s = 2 µ h B n + ( ~q + ~s ) µ i(cid:16) M M tot (cid:17) s , (B.18)and ˜ µ = m n M M tot . Finally, after some derivations, the re-lation of the bubble contribution in Eq. (B.16) converts to − iΣ (cid:16) M M tot ¯ P + ¯ q − ¯ s, M M tot ¯ K + ¯ q + ¯ s (cid:17) = 43 µm α Q g Z dx ( µ m α Q Z d ~b (2 π ) b [ b − A | q − s | , | q + s | ( x )] +16 Qs (cid:16) µ m α M M tot (cid:17) ( x − x ) Z d ~b (2 π ) b − A | q − s | , | q + s | ( x )] +16 µ m α s ( x − x ) Z d ~b (2 π ) b − A | q − s | , | q + s | ( x )] +4 (cid:16) µ m α M M tot (cid:17) Q ( x − x ) Z d ~b (2 π ) b − A | q − s | , | q + s | ( x )] ) = 2 i π µ m α g Q [ − I + I − I ] , (B.19)where the functions I , I and I in the last line are givenusing the relations Z d ~b (2 π ) b − A | q − s | , | q + s | ( x )] = i π p A | q − s | , | q + s | ( x ) , (B.20)and Z d ~b (2 π ) b [ b − A | q − s | , | q + s | ( x )] = 3 i π q A | q − s | , | q + s | ( x ) , (B.21)as I = Z dx x p A | q − s | , | q + s | ( x )= i (cid:16)q C P ,q − s − q C K,q + s (cid:17) + (1 + C P ,q − s − C K,q + s )2 I , (B.22) I = Z dx x p A | q − s | , | q + s | ( x )= 14 h (1 + C K,q + s − C P ,q − s ) i q C P ,q − s +(1 + C P ,q − s − C K,q + s ) i q C K,q + s i + 12 (cid:16) C K,q + s + (1 + C P ,q − s − C K,q + s ) (cid:17) I +(1 + C P ,q − s − C K,q + s ) i (cid:16)q C P ,q − s − q C K,q + s (cid:17) + (1 + C P ,q − s − C K,q + s ) I , (B.23) . Jesri, et al.: Halo EFT calculation of charge form factor for two-neutron He halo nucleus 13 I = Z dx q A | q − s | , | q + s | ( x )= 14 h (1 + C K,q + s − C P ,q − s ) i q C P ,q − s +(1 + C P ,q − s − C K,q + s ) i q C K,q + s i − (cid:16) C K,q + s + (1 + C P ,q − s − C K,q + s ) (cid:17) I . (B.24)The I function in Eqs. (B.22)-(B.24) is defined by theexpression I = Z dx p A | q − s | , | q + s | ( x )= − i " arctan M tot M s + m α M qx ′ r µ (cid:16) B n + ( ~q − ~s ) µ (cid:17) ! + arctan M tot M s − m α M qx ′ r µ (cid:16) B n + ( ~q + ~s ) µ (cid:17) ! . (B.25) B.3.2 Final representation of F ( c ) ( Q )In the final step, by inserting Eqs. (B.11)-(B.14) into Eq.(B.10), we find F ( c ) ( Q ) = Z Λ (cid:16) − p π (cid:17) dp Z Λ (cid:16) − k π (cid:17) dk ~ G irr ( p ) T × D ( p ) Υ ( c ) ( Q, p, k ) D ( k ) ~ G irr ( k )+2 Z Λ (cid:16) − p π (cid:17) dp ~ G irr ( p ) T D ( p ) ~Υ ( c ) ( Q, p ) + Υ ( c )0 ( Q ) , (B.26)where Υ ( c ) ij ( Q, p, k ) = 1(2 π ) Z Λ q dq Z − dx ′ Z π dφ × χ ( c ) ij ( m n M tot Q, p, k, q, x ′ , φ ) ,Υ ( c ) i ( Q, p ) = 1(2 π ) Z Λ q dq Z − dx ′ Z π dφ × χ ( c ) i ( m n M tot Q, p, q, x ′ , φ ) ,Υ ( c )0 ( Q ) = 1(2 π ) Z Λ q dq Z − dx ′ Z π dφ × χ ( c )0 ( m n M tot Q, q, x ′ , φ ) , (B.27) with the following relations χ ( c ) ij ( m n M tot Q, q, x, p, k ) = R i (cid:16) − B n , p, d ( q, s, − x ′ ) (cid:17) × D (cid:16) − B n − m n M M tot Q − qQx ′ M , q (cid:17) × h − i Σ (cid:16) M M tot ¯ P + ¯ q − ¯ s, M M tot ¯ K + ¯ q + ¯ s (cid:17)i × D (cid:16) − B n − m n M M tot Q + qQx ′ M , q (cid:17) × R j (cid:16) − B n , d ( q, s, x ′ ) , k (cid:17) , (B.28) χ ( c ) i ( m n M tot Q, p, q, x ′ , φ ) = 1 √ m n g Λ | βH ( Λ ) | d ( q, s, x ′ ) × R i (cid:16) − B n , p, d ( q, s, − x ′ ) (cid:17) × D (cid:16) − B n − m n M M tot Q − qQx ′ M , q (cid:17) × h − i Σ (cid:16) M M tot ¯ P + ¯ q − ¯ s, M M tot ¯ K + ¯ q + ¯ s (cid:17)i × D (cid:16) − B n − m n M M tot Q + qQx ′ M , q (cid:17) , (B.29) χ ( c )0 ( m n M tot Q, q, x ′ , φ ) = 1 √ m n g Λ | βH ( Λ ) | d ( q, s, − x ′ ) × D (cid:16) − B n − m n M M tot Q − qQx ′ M , q (cid:17) × h − i Σ (cid:16) M M tot ¯ P + ¯ q − ¯ s, M M tot ¯ K + ¯ q + ¯ s (cid:17)i × D (cid:16) − B n − m n M M tot Q + qQx ′ M , q (cid:17) × √ m n g Λ | β H ( Λ ) | d ( q, s, x ′ ) . (B.30)As it mentioned all calculations have been performed inBreit frame, so we have substituted E = − B n in Eqs.(B.28)-(B.30) and drop this energy variable in ~ G irr andmatrix D in Eq. (B.26). B.4 Contribution F ( d ) E Diagram (d) in Fig. 5 is the same as diagram (c) by con-verting the nα bubble to the vertex of photon- d coupling,therefore the contribution of the diagram (d) is given by F ( d ) ( Q ) = Z Λ (cid:16) − p π (cid:17) dp Z Λ (cid:16) − k π (cid:17) dk × ~ G irr ( p ) T D ( p ) Υ ( d ) ( Q, p, k ) D ( k ) ~ G irr ( k )+2 Z Λ (cid:16) − p π (cid:17) dp ~ G irr ( p ) T D ( p ) ~Υ ( d ) ( Q, p )+ Υ ( d )0 ( Q ) , (B.31) He halo nucleus where Υ ( d ) ij ( Q, p, k ) = 1(2 π ) Z Λ q dq Z − dx ′ Z π dφ × χ ( d ) ij ( m n M tot Q, p, k, q, x ′ , φ ) ,~Υ ( d ) i ( Q, p ) = 1(2 π ) Z Λ q dq Z − dx ′ Z π dφ × χ ( d ) i ( m n M tot Q, p, q, x ′ , φ ) ,Υ ( d )0 ( Q ) = 1(2 π ) Z Λ q dq Z − dx ′ Z π dφ × χ ( d )0 ( m n M tot Q, q, x ′ , φ ) , (B.32)with the following relations χ ( d ) ij ( m n M tot Q, q, x ′ , p, k ) = R i (cid:16) − B n , p, d ( q, s, − x ′ ) (cid:17) × D (cid:16) − B n − m n M M tot Q − qQx ′ M , q (cid:17) × D (cid:16) − B n − m n M M tot Q + qQx ′ M , q (cid:17) × R j (cid:16) − B n , d ( q, s, x ′ ) , k (cid:17) ,χ ( d ) i ( m n M tot Q, p, q, x ′ , φ ) = 1 √ m n g Λ | β H ( Λ ) |× d ( q, s, x ′ ) R i (cid:16) − B n , p, d ( q, s, − x ′ ) (cid:17) × D (cid:16) − B n − m n M M tot Q − qQx ′ M , q (cid:17) × D (cid:16) − B n − m n M M tot Q + qQx ′ M , q (cid:17) ,χ ( d )0 ( m n M tot Q, q, x ′ , φ ) = 1 √ m n g Λ | β H ( Λ ) |× d ( q, s, − x ′ ) × D (cid:16) − B n − m n M M tot Q − qQx ′ M , q (cid:17) × D (cid:16) − B n − m n M M tot Q + qQx ′ M , q (cid:17) × √ m n g Λ | β H ( Λ ) | d ( q, s, x ′ ) . 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