aa r X i v : . [ nu c l - t h ] A p r Effective field theory and electro-weak processes
F. Myhrer
Dept. Physics and Astronomy, Univ. South Carolina, Columbia, SC 29208, USA
Abstract.
Heavy baryon chiral perturbation theory is applied to one- and two nucleon processes.
Keywords:
Chiral Perturbation Theory, One- and two-nucleon weak processes.
PACS:
Introduction
A careful and systematic study of low-energy weak- and strong interaction reactionsis desirable in order to enhance our understanding of some fundamental astro-physicalprocesses. Since low energy processes are insensitive to details of the short distancestructures of the hadrons, we can make use of an effective field theory like Chiral Per-turbation Theory (ChPT), which allows a unified approach to weak- and strong interac-tion processes. The ChPT Lagrangian, which reflects the symmetries and the symme-try breaking pattern of the underlying theory of QCD, also gives a model-independent,gauge-invariant evaluation of radiative QED corrections to these reactions.We know that the QCD lagrangian is chirally symmetric provided the u and d quarks are massless. Furthermore, it is established that chiral symmetry is sponta-neously broken, which implies the existence of massless Goldstone Bosons (pions).The quarks have non-zero masses which however are small compared to the QCDscale, m u ≃ m d ≪ L QCD . Therefore, a perturbative treatment of the explicit chiral sym-metry breaking appears reasonable. In ChPT these considerations are reflected in thehadronic scale, L ch ≃ ≃ m N , being much larger than the corresponding pionmass m p ( (cid:181) √ m quark ) ≪ L ch . ChPT assumes that we consider only low-energy reac-tions which only allow low momentum probes. As a result we will consider the fol-lowing (perturbative) expansion parameter in Heavy Baryon Chiral Perturbation Theory(HBChPT): Q / L ch ≪
1, where Q denotes either the typical 4-momentum involved inthe process under consideration, or m p . The HBChPT Lagrangian L ch is written as an expansion in powers of Q / L ch , see e.g. the reviews [1, 2] L ch = L ( ) p N + L ( ) p N + L ( ) pp + L ( ) p N + · · · where L ( n ) contains terms of order ( Q / L ch ) n . We assume that the terms in the lowestorder Lagrangian give the dominant contributions to a process. The higher order termsresumably give smaller perturbative corrections. In HBChPT the pions are treatedrelativistically, whereas the nucleons are treated non-relativistically. In reality we havetwo simultaneous expansion parameters, ( Q / L ch ) n and ( Q / m N ) n , which for pragmaticpurposes are considered simultaneously. The lowest order pion Lagrangian is: L ( ) pp = f p Tr n (cid:209) m U † (cid:209) m U + c † U + c U † o , (1)where (cid:209) m U = ¶ m U − i ( v m + a m ) U + iU ( v m − a m ) . Here v m and a m are external currents,and c (cid:181) (cid:18) m u m d (cid:19) . In the evaluations of specific processes the U -field is expanded inpowers of the pion field: U = uu = exp ( i ~ t · ~ f ) / f p ≃ + ( i ~ t · ~ f ) / f p + · · · . This expansiongives the familiar first two terms in L ( ) pp : L ( ) pp = (cid:16) ¶ m ~ f (cid:17) − m p ~ f + · · · The lowest order heavy nucleon Lagrangian is: L ( ) p N = ¯ N n i ( u · D ) + g A ( S · u ) o N , (2)where D m = ¶ m + [ u † , ¶ m u ] − i u † ( v m + a m ) u − i u ( v m − a m ) u † . If we choose the nu-cleon velocity u m = ( ,~ ) , then the nucleon spin is: S m = ( , ~ s ) . By again expandingthe U -field we find the following first three terms: L ( ) p N = ¯ N i ¶¶ t − ~ t · (cid:16) ~ f × ~ ˙ f (cid:17) f p + g A f p ~ t · (cid:16) s · (cid:209) ~ f (cid:17) N + · · · In effective field theory the lagrangian contains low energy constants (LECs), whichparametrize the short-distance physics not probed at long wave-lengths. In principle aLEC should be evaluated from QCD but in practice LECs are determined by reproducingthe experimental values of appropriate observables. The nucleon axial coupling constant, g A ≃ .
27, in Eq.(2) is an example of a LEC. Once the LECs are determined the theoryhas predictive power.
In the next order heavy nucleon Lagrangian with the expanded U -field, L ( ) p N = ¯ N ( ( v · ¶ ) − ¶ m N + · · · ) N , we display only the heavy nucleon kinetic operator (“the Schrödinger kinetic operator") ~ (cid:209) m N . This nucleon kinetic operator is a “recoil" correction to the leading terms; in otherwords the heavy nucleon expansion is different from the Foldy-Wouthuysen expansionas discussed in [3]. In the following we will give some examples of one- and two-nucleonelectroweak processes which have been evaluated in HBChPT. pecific processes The following one-nucleon processes , ordinary muon capture: m − + p → n m + n (OMC), radiative muon capture: m − + p → n m + n + g (RMC), and the radiative correc-tions to n → p + e − + n e and ¯ n e + p → e + + n (the CHOOZ process), have all been inves-tigated in HBChPT. Since in all these weak-interaction processes the momentum trans-fers are small Q ≪ m W , the effective interaction lagrangian is the “Fermi" Lagrangian: L Fermi = G F √ J b ( lepton ) · J b ( hadron ) where J b ( lepton ) = ¯ u n g b ( − g ) u l and J b ( hadron ) = v had b − a had b . Traditionally thehadronic currents v had b and a had b are written as: v had b = ¯ Y ( G V ( q ) g b + G M ( q ) i s bd q d m N + ) Y a had b = ¯ Y (cid:26) G A ( q ) g b g + G P ( q ) q b g m N + (cid:27) Y . When we expand the nucleon form-factors including the q terms, the LECs are de-termined by the nucleon’s r.m.s. radius, axial radius, anomalous nucleon magnetic mo-ments, i.e. G M ( q ) = k p − k n , and the Goldberger-Treiman discrepancy. The pseudo-scalar form factor is derived in ChPT (including one-loop corrections) and found to be G P ( q ) m N = − f p g p NN q − m p − g A m N < r A > , (3)where the values of all parameters in Eq.(3) have been determined from other reactions.This expression for G P ( q ) was derived some time ago by Adler and Dothan [4] andWolfenstein [5]. N. Kaiser used HBChPT to show that the next order corrections to G P are very small [6]. One challenge exists: Can G P ( q ) be measured in some process inorder to confirm this theoretical prediction?Two processes can determine G P , OMC and RMC. The m − p capture rate has recentlybeen measured at PSI by Andreev et al. [7]. Instead of the standard liquid Hydrogentarget they [7] used a gas target in order to minimize the molecular complications inthe capture process, see e.g. Refs. [8, 9]. The initial results are consistent with theChPT prediction. Forthcoming final experimental results are expected at 1% accuracy.The radiative muon capture has the advantage that q changes with the photon energy, E g , meaning RMC could determine G P ( q ) via the pion-pole dominance of Eq.(3). ATRIUMF team was able to measure the extremely low RMC rate, d G / d E g , for photonenergies E g >
60 MeV [10, 11]. It was a big surprise that the RMC experimental resultsdisagreed with the HBChPT prediction.he advantage of the systematic ChPT expansion can be illustrated by the followingorder by order expression for the m − p spin-singlet capture rate taken from Ref. [12] G = (cid:18) − m N + (cid:20) . m N − . (cid:21)(cid:19) s − The near cancellation of the two terms in the square bracket, originating from “recoil"(1 / m N ) and q form-factor contributions, testifies to the value of the systematic pertur-bative expansion of HBChPT. The radiative corrections to neutron b -decay and the CHOOZ process are ofcritical importance since in the coming decade the processes n → p + e − + ¯ n e and¯ n e + p → e + + n will be measured very precisely. The precise measurements of neutron b -decay aim at an accurate value for V ud . To extract V ud requires an updated understand-ing of the radiative corrections (RC). The second reaction, the CHOOZ process [13],will be used to determine neutrino oscillation parameters. Why a new investigation ofthese RC? A systematic reevaluation of RC [14] to the CHOOZ process is possiblewithin HBChPT, which allows a model-independent, gauge-invariant evaluation of RC.The short distance physics is again well defined in the HBChPT lagrangian by theradiative LECs, which are determined in, e.g., neutron beta-decay RC evaluation [15]. The two-nucleon processes to be discussed are connected to fundamental astro-physical reactions; muon capture on the deuteron: m − + d → n m + n + n , the charged-and neutral currents (CC and NC) of the Sudbury Neutrino Observatory (SNO) reac-tions: n e + d → e − + p + p and n x + d → n x + p + n , and the radiative pion capture onthe deuteron: p − + d → g + n + n or the crossing symmetric process g + d → p + + n + n .The ChPT evaluation of these reactions include one unknown axial two-nucleon LEC,ˆ d R , which also enters in the evaluations of the following few-nucleon reactions [16]; tri-ton b -decay: H → He + e + + n e , solar pp fusion: p + p → d + e + + n e , the solar Hepprocess: He + p → He + e + + n e , and the modern three-nucleon potential ( ˆ d R is re-lated to c D , one of the two unknown LEC parameters in the ChPT-derived three-nucleonpotential [17]). The Hep process produces the highest energy solar neutrinos and has tobe carefully evaluated [18] in order to extract accurately the Be solar neutrino spectrumdetected at, e.g., SuperKamiokande and SNO. A precise evaluation of Hep is howeverdifficult since leading contributions almost cancel as discussed in e.g. [19].Ideally the two-nucleon reactions should be evaluated using transition operators andnucleon wave functions obtained from ChPT. For pragmatic reasons however a hybrid
ChPT called
EF T ∗ has been used in the two- and more nucleon processes. In EFT ∗ we use the one- and two-nucleon transition operators from ChPT, whereas the nuclearwave functions are evaluated using modern “high precision" NN potentials V NN , e.g.,Argonne V , CD-Bonn, V low − k , etc. In EF T ∗ calculations a Gaussian cut-off L G wasintroduced in the nuclear wave functions in order to limit the contributions from the highmomentum components of the wave functions generated by V NN . These high momentumcomponents in the nuclear wave functions generated by, e.g., the Argonne V potential,go beyond the relevant limited momentum range of ChPT, Q ≪ L ch . This Gaussiancut-off procedure is therefore in accordance with one of the principal assumptions ofhPT allowing only a limited low Q range. As a consequence however the axial two-nucleon LEC will ˆ d R depend on L G . The observables should be independent of thismomentum cut-off, and we find that the measurable rates and cross-sections have lessthan 1% variations for 500 MeV < L G <
800 MeV.Presently ˆ d R is is determined from tritium b -decay rate. It is however desirable toavoid the complexity of a three-nucleon system in determining the two-nucleon axialcoupling ˆ d R , so that two-nucleon processes can be calculated self-consistently withinthe framework of ChPT. Avoiding the inherent uncertainties of the three-nucleon systemwill also allow a more reliable evaluation of the uncertainties involved in two-nucleonreactions. The rate of muon capture on a deuteron ( m − d ) is being measured (2009-2011)at PSI by the MuSun collaboration with a projected error of 1.5% [20]. We are presentlyre-evaluating our m − d ChPT calculation to match this expected experimental precision.Once the m − d capture rate is accurately measured, the following three reaction can beevaluated model independently with the same accuracy: (i) the solar pp fusion reaction ,the primary energy source in the sun, (ii) the SNO neutrino-deuteron reactions , whichprovided convincing evidence for neutrino oscillation, and (iii) the reaction p − + d → g + n + n [21] or g + d → p + + n + n [22] which can be used to determine the neutron-neutron scattering length a nn , see the review [23] for a discussion. Furthermore, oneof the LEC in the three-nucleon potential, c D , which is an axial two-nucleon LEC, isdetermined once the value of ˆ d R is fixed by the m − d capture reaction. In other words,only one unknown three-nucleon LEC, c E , remains in the ChPT three-nucleon potential.The expected accurate measurements of the m − p and m − d capture rates will requirea re-examination of the radiative corrections to these two processes. The estimatedradiative corrections are larger than the expected experimental errors from the MuCapand MuSun collaborations and a renewed evaluation of the RC is in progress. Supernova Explosion
Computer simulations of the supernova have not been very successful in generatingthe explosion. This is possibly due to the neutrino luminosity being too small. We haveidentified new processes which generate neutrinos in the proto-neutron star at the centerof the supernova explosion. These reactions might affect the explosion-simulation dueto an (estimated) increased in the neutrino flux [24].
Conclusions
The low-energy effective theory, ChPT, allows a systematic evaluation of electro-weak and strong interaction processes. HBChPT predicts accurately the analytic expres-sion for G P ( q ) . The predicted value for G P in the m − p process is confirmed by recentMuCap data [7]. The published MuCap experimental m − p capture rate is also compat-ible with the HBChPT prediction. The advantage utilizing ChPT is that ChPT providesanalytic expressions for both the m − p and m − d capture operators at each perturbativeorder, and ChPT permits us to make a reasonable estimate the theoretical uncertaintyof the calculated observable. Once we have a measurable quantity evaluated at “or-der" ( Q / L ch ) v , an estimated uncertainty is given by the magnitude of the next ordercontribution ( Q / L ch ) v + . Two-nucleons reactions (including the energy dependence of + p → d + g which is important in cosmology) are well described by EFT ∗ (one ex-ception is the measured RMC rate versus the photon energy). The m − d capture processbeing measured by the MuSun collaboration at PSI will allow a more accurate value forˆ d R . This MuSun measurement will permit us to make more accurate model-independentpredictions for the solar pp fusion and the n d SNO reactions. However, improved ra-diative corrections are needed to “compete" with the expected MuCap and MuSun m − -capture data. ChPT is ideally suited for an evaluation of these radiative corrections. ACKNOWLEDGMENTS
This work was supported in part by the NSF grant PHY-0758114.
REFERENCES
1. V. Bernard, N. Kaiser, U.-G. Meissner,
Int. J. Mod. Phys. E , 193 (1995)2. V. Bernard, prog. Part. Nucl. Phys. , 82 (2008)3. A. Gårdestig, K. Kubodera, F. Myhrer, Phys. Rev. C , 014005 (2007).4. S.L. Adler and Y. Dothan, Phys. Rev. , 1267 (1966); , 2062(E).5. L. Wolfenstein, in
High-Energy Physics and Nuclear Structure , edited by S. Devons (Plenum, NewYork, 1970), p 661.6. N. Kaiser,
Phys. Rev. C , 027002 (2003)7. V.A. Andreev et al. , MuCap Collaboration, Phys. Rev. Letters , (2007)8. S. Ando, F. Myhrer, K. Kubodera, Phys. Rev. C , 015203 (2000); Phys. Rev. C , 048501 (2002).9. V. Bernard, L. Elouadrhiri, U.-G. Meissner, J. Phys. G: Nucl. Part. Phys. , R1 (2002)10. G. Jonkmans et al. , Phys. Rev. Letters , 4512 (1996)11. D.H. Wright et al. , Phys. Rev. C , 373 (1998)12. V. Bernard, T. Hemmert and U.-G. Meissner, Nucl. Phys. A , 290 (2001)13. M. Apollonio et al. , Eur. Phys. J. C , 331 (2003), and Y. Déclais, priv. comm.14. P. Vogel, Phys. Rev. D , 1918 (1984)15. S. Ando et al. , Phys. Lett. B (2004) and ref. therein.16. T.-S. Park et al. , Phys. Rev. C , 055206 (2003).17. A. Nogga, S.K. Bogner, A. Schwenk, Phys. Rev. C , 061002(R) (2004);D. Gazit, S. Quagliono, P. Navrátil, Phys. Rev. Letters , 102502 (2009).18. K. Kubodera and T.-S. Park,
Ann. Rev. Nucl. Part. Sci. , 19 (2004).19. J. Carlson et al. , Phys. Rev. C , 619 (1991).20. P. Kammel and K. Kubodera, Ann. Rev. Nucl. Part. Sci. to appear (2010); V.A. Andreev et al. ,arXiv:1004.1754 (nucl-exp).21. A. Gårdestig and D.R. Phillips,
Phys. Rev. C , 014002 (2006)22. V. Lensky et al. , Eur. Phys. J. A , 107 (2005)23. A. Gårdestig, J. Phys. G: Nucl. Part. Phys. , 053001 (2009).24. T. Sato et al.et al.