Electroweak Processes in Few-Nucleon Systems
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1 Electroweak Processes in Few-Nucleon Systems M. Viviani INFN, Sezione di Pisa & Department of Physics, University of Pisa Pisa (Italy) Electron Nucleon Scattering XI, June 25-29, 212, Marciana Marina, Isola d Elba M. Viviani (INFN-Pisa) Electroweak Processes Elba, June 28, / 31
2 Outline 1 EFT approach 2 EM Processes 3 Compton scattering 4 Weak interactions 5 Outlook Collaborators F. Spadoni Graduate student, Pisa R. Schiavilla Jefferson Lab. & ODU, Norfolk (VA, USA) S. Pastore ANL (USA) L. Girlanda University of Salento & INFN-Lecce, Lecce (Italy) A. Kievsky & L.E. Marcucci - INFN-Pisa & Pisa University, Pisa (Italy) M. Viviani (INFN-Pisa) Electroweak Processes Elba, June 28, / 31
3 Chiral symmetry - QCD with u and d quarks only u q = d «q R/L = (1 ± «γ5 ) ur/l q = 2 d R/L q R = Rq R = exp i θ R τ/2 q R q L = Lq L = exp i θ L τ/2 q L θ R = θ L = θ V : isospin transformation θ R = θ L = θ A : axial transformation L QCD (almost) invariant under the L, R transformations since m u, m d small also for locals transformations introducing external currents L = L QCD + q L γµ`l µ(x) v(s) µ(x) q L + q R γ µ`r µ(x) v(s) µ(x) q R q R (x)`s(x) + ip(x) q L (x) q L (x)`s(x) ip(x) q R (x) r µ(x) r µ (x) = R(x)rµ(x)R (x) + ir(x) µr (x), etc The external current are related to A µ(x) and W µ ± (x) to reproduce the EM and weak interactions of the quarks Example r µ(x) = l µ(x) = e τz Aµ(x) v(s) µ (x) = e 2 2 Aµ(x) 2 L em = ea µ 3 uγµ u 1 «3 dγµ d M. Viviani (INFN-Pisa) Electroweak Processes Elba, June 28, / 31
4 Chiral Symmetry - Hadrons Non-linear realization of the chiral symmetry for hadrons [Weinberg, 1968, 199],[CCWZ, 1969],[Gasser & Leutwyler, 1984],... Compensator field h u = exp(i π τ/2f π) u = Luh = hur h h(l, R, π) N = hn Nucleons However ( µn) does not transform covariantly u µ = i[u ( µ ir µ)u u( µ il µ)u ] D µ = µ [u ( µ ir µ)u + u( µ il µ)u ] iv (s) µ Transformations: u µ = hu µh (D µn) = hd µn Lagrangian L πn = N `iγ µ D µ m N + g A 2 γ µ γ 5 u µ N + + CS NNNN + it contains an infinite number of LECs Contributions organized as an expansion over (Q/Λ χ) ν [Λ χ 1 GeV] M. Viviani (INFN-Pisa) Electroweak Processes Elba, June 28, / 31
![] D µ = µ + 1 2 [u ( µ ir µ)u + u( µ il µ)u ] iv (s) µ Transformations: u µ = hu µh (D µn) = hd µn Lagrangian L πn = N `iγ µ D µ m N + g A 2 γ µ γ 5 u µ N + + CS](/docs-images/48/19237894/images/page_4.jpg "NNNN + it contains an infinite number of LECs Contributions organized as an expansion over (Q/Λ χ) ν [Λ χ 1 GeV] M.")
5 NN, 3N,..., potentials from the EFT NN Potential V Two methods: S-matrix: for a given process NN NN define V so that (on-shell) NN T EFT NN NN T V NN Unitary transformation: find U in order to decouple NN Hilbert space from NNπ, etc. Realization thanks to the chiral counting: all terms can be organized as powers of Q/Λ χ, Q small momenta or the pion mass Alternatively: Lattice χeft [Lee et al., 21] Example T EFT = T V V + VG V + G = (E H + iǫ) 1 T EFT T () EFT + T (1) EFT + T (2) EFT... V V() + V (1) + V (2)... T (n) EFT, V(n) Q n p 1 p 2 V(n ) G V (n) p 1 p 2 = X p 1 p 2 p 1 p 2 V(n ) p 1 p 2 Then V () = T () EFT V (1) = T (1) EFT V() G V (), etc p 1 p 2 V(n) p 1 p 2 Q n+n +1 E p1 + E p2 E p 1 E p 2 + iǫ M. Viviani (INFN-Pisa) Electroweak Processes Elba, June 28, / 31
6 NN, 3N,..., potentials from the EFT NN Potential V Two methods: S-matrix: for a given process NN NN define V so that (on-shell) NN T EFT NN NN T V NN Unitary transformation: find U in order to decouple NN Hilbert space from NNπ, etc. Realization thanks to the chiral counting: all terms can be organized as powers of Q/Λ χ, Q small momenta or the pion mass Alternatively: Lattice χeft [Lee et al., 21] Example T EFT = T V V + VG V + G = (E H + iǫ) 1 T EFT T () EFT + T (1) EFT + T (2) EFT... V V() + V (1) + V (2)... T (n) EFT, V(n) Q n p 1 p 2 V(n ) G V (n) p 1 p 2 = X p 1 p 2 p 1 p 2 V(n ) p 1 p 2 Then V () = T () EFT V (1) = T (1) EFT V() G V (), etc p 1 p 2 V(n) p 1 p 2 Q n+n +1 E p1 + E p2 E p 1 E p 2 + iǫ M. Viviani (INFN-Pisa) Electroweak Processes Elba, June 28, / 31
7 NN & 3N interaction For more information see for example [Epelbaum et al., NPA 714, 535 (23)] NN interaction J-N3LO [Epelbaum and Coll, ] I-N3LO [Entem & Machleidt, 23] Part of the LEC s fitted to the NN database or πn database 3N interaction J-N2LO [Epelbaum et al, 22] N-N2LO [Navratil, 27] 3N force at N3LO [see Kreb s talk] At N2LO there are two LECS c D and c E : fitted to some 3N data (see later) At N3LO no new parameters At N4LO 1 new LECs [Girlanda et al., 211] M. Viviani (INFN-Pisa) Electroweak Processes Elba, June 28, / 31
![database or πn database 3N interaction J-N2LO [Epelbaum et al, 22] N-N2LO [Navratil, 27] 3N force at N3LO [see Kreb s talk] At N2LO there](/docs-images/48/19237894/images/page_7.jpg "are two LECS c D and c E : fitted to some 3N data (see later) At N3LO no new parameters At N4LO 1 new LECs [Girlanda et al., 211] M.")
8 EM current Basic problem: transition α + γ β β H e.m. α; qλ = Ψ β K 1 Ψ α K 1 = e Z 2ωΩ dx e iq x bǫ qλ Ĵ(x) K 1 acts only on the nucleons d.o.f. α, β initial & final nuclear states, Ψ α, Ψ β corresponding w.f. q, ω, ˆǫ qλ = momentum, energy, polarization of the emitted photon for virtual photons, one needs also the m.e. of ˆq bj and ρ Z J µ (q) = dx e iq x b J µ (x) µ =, 1, 2, 3 M. Viviani (INFN-Pisa) Electroweak Processes Elba, June 28, / 31
9 Meson exchange currents bj(x) = P i b j i (x) + 2B + 3B +... Current conservation Ĵ(x) = i[h, ρ(x)] Strict interplay between H, b J and bρ bρ(x) = AX i=1 1 + τ z(i) δ(r i x) 2 [Buchmann et al, 1985] [Riska, 1989], [Schiavilla et al, 199] EFT approach: H and J µ derived from the same Lagrangian. M. Viviani (INFN-Pisa) Electroweak Processes Elba, June 28, / 31
![i=1 1 + τ z(i) δ(r i x) 2 [Buchmann et al, 1985] [Riska, 1989], [Schiavilla et al, 199] EFT](/docs-images/48/19237894/images/page_9.jpg "approach: H and J µ derived from the same Lagrangian. M.")
10 Current at N3LO [Park et al, 1993], [Kolling et al, 29], [Pastore et al, 29] N3LO (Q 1 ) terms LO (Q 2 ) NLO (Q 1 ) NNLO (Q ) black square= (Q/M N ) 2 relativistic correction to the NNγ vertex Note: NNγ vertex = (e N /2M N )(p+p )+i(e N +κ N )µ N (σ q) it takes into account the Pauli term + pion loop corrections 2 new LECs black dot= three (Q/Λ χ) 2 vertices 3 new LECs Most of the LECs enter also the NN potential. There are 5 uncostrained LECs ( µ d, µ 3 H, µ3 He, etc.) M. Viviani (INFN-Pisa) Electroweak Processes Elba, June 28, / 31
11 Wave functions HH variational method: A. Kievsky, S. Rosati, MV, L.E. Marcucci, and L. Girlanda J. Phys. G, 35, 6311 (28) A benchmark for A = 4 AGS: Deltuva & Fonseca, PRL (27) FY: Lazauskas & Carbonell, PRC 7, 442 (24) n 3 H & p 3 He elastic scattering E c.m. B 3 B MeV NN interaction models: AV18 [Wiringa, Stoks & Schiavilla (1995)] I-N3LO [Entem & Machleidt (23)] V low q [Bogner, Kuo & Schwenk, (23)] (derived from the CD-Bonn potential [Machleidt (21)]) Results reported in [MV et al., 211] M. Viviani (INFN-Pisa) Electroweak Processes Elba, June 28, / 31
12 n 3 H scattering (I-N3LO pot.) dσ/dω [mb/sr] MeV AGS 2 MeV 3.5 MeV 6 MeV 18,4 A y,2,2 18 A y,1 -,1 18 M. Viviani (INFN-Pisa) Electroweak Processes Elba, June 28, / 31
13 n 3 H scattering (I-N3LO pot.) dσ/dω [mb/sr] MeV AGS HH 2 MeV 3.5 MeV 6 MeV 18,4 A y,2,2 18 A y,1 -,1 18 M. Viviani (INFN-Pisa) Electroweak Processes Elba, June 28, / 31
14 n 3 H scattering (I-N3LO pot.) dσ/dω [mb/sr] MeV AGS HH FY 2 MeV 3.5 MeV 6 MeV 18,4 A y,2,2 18 A y,1 -,1 18 M. Viviani (INFN-Pisa) Electroweak Processes Elba, June 28, / 31
15 p 3 He scattering (I-N3LO pot.) dσ/dω [mb/sr] MeV Famularo 1954 Fisher MeV Mcdonald 1964 Fisher MeV Mcdonald A y,4,2 Fisher 26 George 21 Fisher 26 Alley 1993 A y,2,1 Daniels 21 Daniels Alley 1993 Daniels M. Viviani (INFN-Pisa) Electroweak Processes Elba, June 28, / 31
16 Predictions by different potentials dσ/dω [mb/sr] MeV Famularo 1954 Fisher 26 I-N3LO AV18 low-k 4.5 MeV Mcdonald 1964 Fisher MeV Mcdonald ,4 Fisher 26 George 21 Fisher 26 Alley 1993 A y,2 A y,2,1 Daniels 21 Daniels Alley 1993 Daniels M. Viviani (INFN-Pisa) Electroweak Processes Elba, June 28, / 31
17 ,4 2.25MeV 4.5MeV 5.54 MeV,2 Daniels 21 Daniels 21 Daniels 21 Alley 1993 A yy -,2 -,4 -, ,2 Daniels 21 Daniels 21 Alley 1993 A xx,1 -, ,4,3,2,1 -,1 -,2 A xz Alley MeV A zx 5.54 MeV Alley 1993 A zz Alley θ [c.m.] θ [c.m.] θ [c.m.] M. Viviani (INFN-Pisa) Electroweak Processes Elba, June 28, / 31
18 Results for A = 3, 4 (1) 4 p-d E c.m. = 2. MeV 4 p- 3 He E c.m. =4.15 MeV 3 AV18 AV18/UIX I-N3LO I-N3LO/N-N2LO 3 I-N3LO AV18 AV18/UIX I-N3LO/N-N2LO dσ/dω [mb/sr] 2 dσ/dω [mb/sr] M. Viviani (INFN-Pisa) Electroweak Processes Elba, June 28, / 31
19 Results for A = 3, 4 (2),6 A y p-d E c.m. =2 MeV A y p- 3 He E c.m. =4.15 MeV,5,4,5,4 Alley 1993a Alley 1993b AV18 I-N3LO I-N3LO/N-N2LO A y,3,3,2,1 AV18 AV18/UIX I-N3LO I-N3LO/N-N2LO,2, Study of the 3N force in A = 4 scattering in progress M. Viviani (INFN-Pisa) Electroweak Processes Elba, June 28, / 31
20 Fit of the LECS Fit of the LECs Current at N3LO (O(Q)) NN potential at NLO (O(Q 2 )) q bj(q) = [H, ρ(q)] We have constructed a NN potential at NLO and fitted the corresponding LECs to the NN database: [Pastore et al., 29] In J there are 5 additional LECs: fitted to the A = 2, 3 magnetic moments & n p capture cross section at thermal energies using the I-N3LO NN potential The model depends on a cutoff Λ (Λ = 5 6 MeV) the dependence on Λ is used to test the convergence [Girlanda et al., 21] n.m..9 µ d exp LO NLO N 2 LO N 3 LO(S-L) σ γ np mb n.m. n.m. µ S ( 3 He/ 3 H) µ V ( 3 He/ 3 H) Λ (MeV) Λ(MeV) M. Viviani (INFN-Pisa) Electroweak Processes Elba, June 28, / 31
![, 29] In J there are 5 additional LECs: fitted to the A = 2, 3 magnetic moments & n p capture cross section at thermal energies using the I-N3LO NN potential The model depends on](/docs-images/48/19237894/images/page_20.jpg "a cutoff Λ (Λ = 5 6 MeV) the dependence on Λ is used to test the convergence [Girlanda et al., 21] n.m..9 µ d exp LO NLO N 2 LO N 3 LO(S-L) σ γ np 36 34 32 mb.85 3.45-2. n.m. n.m. µ S ( 3 He/ 3 H) µ V ( 3 He/ 3 H).")
21 Deuteron-photodisintegration Wave functions calculated using I-N3LO for Λ = 5 & 6 MeV Observable dominated by the E1 transitions.28 Deuteron photo-disintegration total cross section (mb).24.2 EXP I-N3LO5 JEFT1(FULL) I-N3LO5 JEFT1(E1) σ dis (fm 2 ) H(γ,n) 1 H E Lab (MeV) M. Viviani (INFN-Pisa) Electroweak Processes Elba, June 28, / 31
22 n d & n 3 He radiative captures at thermal energies n d capture from the 2 S 1/2 & 4 S 3/2 waves n 3 He capture from the 3 S 1 wave Scattering lenghts Case I-N3LO/N-N2LO Expt. a nd doublet (1) a nd quartet (2) a n 3 He doublet (53) n d & n 3 He capture cross sections Order σ n d [mb] σ n 3 He [µb] LO NLO N2LO N3LO (loops) N3LO (LECs) Expt.58(15) 52(4) mb σ γ nd R c Λ (MeV) σ γ n 3 He Λ (MeV) exp LO NLO N 2 LO N 3 LO(S-L) N 3 LO(LECs) SNPA SNPA* µb M. Viviani (INFN-Pisa) Electroweak Processes Elba, June 28, / 31
23 Compton scattering (1) Nucleon Polarizabilities Induced dipoles by an EM field: d = αe µ = βb» H eff = 2π αe 2 + βb 2 + γ E1E1 σ E E «+ t Experimental status [Griesshammer et al., 212] proton: from γp γp experiments (MAMI [de Lèon et al., 21],...) α p = (1.7 ±.3(stat) ±.2(Baldin) ±.8(theory)) 1 4 fm 3 β p = (3.1 ±.3(stat) ±.2(Baldin) ±.8(theory)) 1 4 fm 3 neutron: from γd γd experiments or with other methods Data sparse and not accurate [Illinois (1994), SAL (2), Lund (23)] α n = (11.1 ± 1.8(stat) ±.4(Baldin) ±.8(theory)) 1 4 fm 3 β n = ( (stat) ±.4(Baldin) ±.8(theory)) 1 4 fm 3 Theory input needed to separate: 1) structure effects 2) MEC effects New experiments on d, 3 He, 6 Li planned/in progress at TUNL/HγGS, MaxLab (Lund), S-DALINAC (Darmstaad) M. Viviani (INFN-Pisa) Electroweak Processes Elba, June 28, / 31
24 Compton scattering (2) Status of the calculations First applications to γn: [Bernard et al, 1992] Recent applications to γd: [Beane et al, 24]: NNLO, no rescattering [Griesshammer & Shukla, 29]: NLO, rescattering calculated with AV18 Review: [Griesshammer et al., 212] Only a few applications to γ 3 He Aims of the new calculation NNγ NN transition operators derived from the EFT at N3LO [Pastore et al., 29] NN interaction derived from the same EFT (at present we have used the I-N3LO potential by Entem & Machleidt) Future: applications to 3 He and 6 Li M. Viviani (INFN-Pisa) Electroweak Processes Elba, June 28, / 31
25 General framework K 1 K1 K 1 K1 + V V K K1 K 1 K1 K 1 } {{ } termini dispersivi d f γ f T d i γ i = Ψ d f K 2 + K 1 GK 1 + K 1 GK 1 Ψd i {z } Dispersive part The Green function G = (E H + iǫ) 1 describes the rescattering of the NN pair between the two EM vertices NN interaction from [Entem & Machleidt, 23] The irriducible kernel K 2 derived from the EFT at NLO ( Q 2 ) (PRELIMNARY) In literature K 2 is derived up to NNLO [Griesshammer et al, 212] inclusion of d.o.f. [Hildebrandt Ph.D. Thesis, München, 25] M. Viviani (INFN-Pisa) Electroweak Processes Elba, June 28, / 31
26 Diagrams (1) Diagrams (a): contribution to K 2 seagull (SG) & spin-orbit (SO) P =q q K (SG) 2 = X j e j 2M ǫ i ǫ f ei(q q ) r j, e j = 1 + τ j z 2 P= (b2) (b1) (a) (b3) (b4) (b5) Note: the SG can be derived from H NR = (1/2M)(p ea) 2 SG: order Q 3, SO=corrections to the SG Q 2 Diagrams (b): contribution to K 2 from the polarization of the nucleon They can be used to estimate α and β [Bernard et al, 1992] We ll consider α and β as free parameters (α j = α p(1 + τz)/2 j + α n(1 τz)/2) j K (αβ) = X h 2πα 2 j ǫ i ǫ f qq j 2πβ j (q ǫ i ) (q ǫ f ) ie i(q q ) r j M. Viviani (INFN-Pisa) Electroweak Processes Elba, June 28, / 31
27 Diagrams (2) Diagrams (c-f): Contributions taken into account by the dispersive part Φ f d K 1 GK 1 + K 1 GK 1 Φi d (d1) (c2) (c1) V (e1) (e2) (f1) V V V (f2) (g1) (g2) (c1), (f1) Q 3, (e1), (f2) Q 4 Exact resummation [Ishikawa et al., 1998] Ψ 1 = GK 1 Φ i d Ψ 2 = GK 1 Φi d (E H + iǫ) Ψ 1 = K 1 Φ i d (E = q B d > ) (E H + iǫ) Ψ 2 = K 1 Φi d (E = q B d < ) Diagrams (g): contribution taken into account by the fact that Φ d are solution of the Schroedinger equation Diagrams (h): Contribution to K 2 (h1) (h2) (h3) (h4) M. Viviani (INFN-Pisa) Electroweak Processes Elba, June 28, / 31
28 Test Photodisintegration I[ Φ f d K 1 GK 1 Φ i d σ γ+d n+p In our calculation [ Φ f d K 1 GK 1 Φ i d = Φf d K 1 Ψ 1 At E γ = 2 MeV σ γ+d n+p = 54.7 µb: we find µb (I-N3LO + JEFT Λ = 5 MeV) Thomson limit For E γ, the calculation should reproduce the Thomson limit Compton amplitude M (TL) = e 2 /M d (note: M (SG) = e 2 /M 2M (TL) ) True if V, K 1, and K 2 are consistent (current conservation) I-N3LO6 JEFT2 SG SG+Dh SG+Dh+DS(NLO) SG+Dh+DS(N3LO) limite di Thomson dσ/dω[nb/sr] θ M. Viviani (INFN-Pisa) Electroweak Processes Elba, June 28, / 31
29 Results (1) 4 49 MeV -- I-N3LO + JEFT (Λ=5 MeV) 35 dσ/dω [nb/sr] Illinois Data SG +SO +DS +Dh +αβ 1 5 (α p +α n )/2=11.5 (β p +β n )/2= θ M. Viviani (INFN-Pisa) Electroweak Processes Elba, June 28, / 31
30 Results (2) 4 66 MeV -- I-N3LO + JEFT (Λ=5 MeV) 35 dσ/dω [nb/sr] Lundin (2) SG +SO +DS (N3LO) +Dh +Pol. (standard) θ M. Viviani (INFN-Pisa) Electroweak Processes Elba, June 28, / 31
31 Sensitivity to α n & β n 4 49 MeV -- I-N3LO + JEFT (Λ=5 MeV) 35 3 Illinois Data α n =2, β n =2 dσ/dω [nb/sr] (α p +α n )/2=11.5 (β p +β n )/2= θ M. Viviani (INFN-Pisa) Electroweak Processes Elba, June 28, / 31
32 β-decay and the 3N force e [Gardestig & Phillips, 26], [Gazit et al., 29] c D c E d R d R = Mn Λ χg A c D M N(c 3 + 2c 4 ) New fit of c D and c E using 3 H binding energy and tritium β-decay lifetime New versions of the 3N at N2LO: first application for µ-capture on d and 3 He [Marcucci et al., 212] M. Viviani (INFN-Pisa) Electroweak Processes Elba, June 28, / 31
33 pp capture at astrophysical energies Aim: new calculation of the astrophysical factor S 11 of the p + p 2 H + e + ν e reaction S 11 (E) = S 11 + S 11 E S 11 E2 + Preliminary Results units 1 25 MeV b I-N3LO NN interaction + weak transition operator derived from EFT S 11 S 11 /S 11 [MeV 1 ] S [MeV 2 ] LO Λ = LO Λ = Full Λ = Full Λ = M. Viviani (INFN-Pisa) Electroweak Processes Elba, June 28, / 31
34 Outlook Motivations of this work consistent calculations for a variety of processes using potential/current/wave functions derived from the same EFT Applications Main interest: test of 3N interaction in A = 3, 4 systems, study of reactions of astrophysical interest p d & d d captures, form factors of light nuclei, ldots) Compton scattering on 3 He & 6 Li (in the near future new data at HIγS & Lund) weak transitions (pp capture, µ-capture, parity-violation in nuclei,...) M. Viviani (INFN-Pisa) Electroweak Processes Elba, June 28, / 31
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