aa r X i v : . [ nu c l - t h ] O c t Pion reactions with few-nucleon systems
Vadim Baru ∗ , Forschungszentrum Jülich, Institut für Kernphysik (Theorie) and Jülich Center for HadronPhysics, D-52425 Jülich, Germany andInstitute for Theoretical and Experimental Physics, B. Cheremushkinskaya 25,117218 Moscow, RussiaE-mail: [email protected]
We report about the recent results for s- and p-wave pion production in NN → NN p within ef-fective field theory and discuss how the charge symmetry breaking in pn → d p can be used toextract the strong contribution to the neutron-proton mass difference. ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ ion reactions with few-nucleon systems
Vadim Baru
1. Introduction
With the advent of chiral perturbation theory (ChPT), the low-energy effective field theory(EFT) of QCD, high accuracy calculations for hadronic reactions with a controlled error estimationhave become possible [1, 2]. The framework has been successfully applied to study, in particular, pp [3] and p N [4] scattering observables as well as nuclear forces [5]. In this contribution wediscuss an application of ChPT to the reactions involving pion production on two-nucleon systems.This type of reactions allows one to test predictions of ChPT in the process with the large momen-tum transfer typical for the production process. As was first advocated in Refs. [6, 7], the initialnucleon momentum in the threshold kinematics sets the new "small" scale in the problem, namely, p ≃ √ m N M p ≃
360 MeV ( c ≃ p / m N ≃ p M p / m N ), where M p ( m N ) is the pion (nucleon) mass.The proper way to include this scale in the power counting was presented in Ref. [8] and imple-mented in Ref. [9], see Ref. [10] for a review article. As a consequence, the hierarchy of diagramschanges in the modified power counting scheme of Ref. [8], and some loops start to contribute al-ready at NLO for s -wave pion production. On the other hand, p -wave pion production is governedby the tree-level diagrams up to NNLO. In what follows we discuss the status of the theory for pionproduction in the isospin conserving and isospin violating case. It was first argued in Ref. [11] thatcharge symmetry breaking (CSB) effects recently observed experimentally in pn → d p [12] pro-vide an access to the neutron-proton mass difference which is the fundamental quantity of QCD.To take this challenge, however, one needs to have the isospin conserving case fully under control.In sec.2 we briefly discuss the theoretical status for s-wave pion production. Sec. 3 is devoted toa more detailed discussion of the recent results for p-wave pion production. In sec. 4 we brieflyhighlight the recent developments in the study of charge symmetry breaking effects in pn → d p .We finalize with the summary of the latest results.
2. s-wave pion production and the concept of reducibility
NLO2m(cid:25) ~p 0~p 0 ~p 0~pb d1 d2 ~p 0~p ~p 0~pa1 a2~p 0~p ~p 0~p ~p 0~pLO ~p~p
Figure 1:
Complete set of nucleonic diagrams up to NLO. Note that sum of all loops at NLO vanishes.
A method how to calculate processes on few nucleon systems with external probes was pro-posed by Weinberg [13]:1. the perturbative transition (production) operators have to be calculated systematically usingChPT. They should consist of irreducible graphs only.2. the transition operators have to be convoluted with the non-perturbative NN wave functions.2 ion reactions with few-nucleon systems Vadim Baru h =k p /m p s / h [ m b ] Figure 2:
Comparison of our results to experimen-tal data for NN → d p . The data sets are fromRefs. [17–21]. Note that the green diamond and theblue triangle correspond to the most resent measure-ments from pionic deuterium atom [20, 21] at h = Therefore it is necessary to disentangle those diagrams that are part of the wave function from thosethat are part of the transition operator. In complete analogy to NN scattering, the former are calledreducible and the latter irreducible. The distinction stems from whether the diagram shows a two-nucleon cut or not. Thus, in accordance to this rule, the one-loop diagrams shown in Fig. 1(b)–(d)are irreducible, whereas diagrams (a) seem to be reducible. This logic was used in Ref. [9] tosingle out the irreducible loops contributing at NLO. The findings of Ref. [9] were: • For the channel pp → pp p the sum of diagrams (b)–(d) of Fig. 1 vanished due to a cancel-lation between individual diagrams • For the channel pp → d p + the same sum gave a finite answer : A b + c + dpp → d p + = g A f p ( − + + ) | ~ q | = g A | ~ q | f p , (2.1)where f p denotes the pion decay constant and g A is the axial-vector coupling of the nucleon.The latter amplitude grows linearly with increasing final NN –relative momentum | ~ q | , which leadsto a large sensitivity to the final NN wave function, once the convolution of those with the transitionoperators is evaluated. However, the problem is that such a sensitivity is not allowed in a consistentfield theory as was stated in Ref. [15]. The solution of this problem was presented in Ref. [14](see also Ref. [16]). It was pointed out in Ref. [14] that the diagrams that look formally reduciblecan acquire irreducible contributions in the presence of the energy-dependent vertices (or time-dependent Lagrangian densities). Specifically, the energy dependent part of the leading Weinberg-Tomozawa (WT) p N → p N vertex cancels one of the intermediate nucleon propagators (see theone with the red square in Fig. 1), resulting in an additional irreducible contribution at NLO. Itturned out that this additional irreducible contribution compensates the linear growth of diagrams(b)–(d) thus solving the problem. Thus, up to NLO, only the diagrams appearing at LO (see Fig.1), contribute to pp → d p + , with the rule that the p N → p N vertex is put on–shell. The latterstems from the observation that in addition to the leading WT-term ( ∼ M p /
2) the nucleon recoilcorrection to the WT p N vertex ( ∼ M p /
2) also contributes at LO. As a result the dominant p N -rescattering amplitude is enhanced by a factor of 4 / ∼ M p / pp → d p + (see Fig. 2). The connection of the amplitude A to the observables is given, e.g., in Ref [14] ion reactions with few-nucleon systems Vadim Baru N LOLO NLO
Figure 3:
Diagrams that contribute to the p -wave amplitudes of NN → NN p up to NNLO. q p -0.3-0.2-0.10 A y d=3d=-3d=0 h=0.14 h -0.5-0.4-0.3-0.2-0.10 A y ( ) KorkmazMathieHeimberg
Figure 4:
Results for the analyzing power at h =0.14 (left panel) and the analyzing power at 90 degrees(right panel) for the reaction pp → d p + for different values of the LEC d (in units 1 / ( f p M N ) ) of the ( ¯ NN ) p contact operator. Shown are d = d = d = − Note, however, that the relatively large theoretical uncertainty of about 2 M p / m N ≈
30% for thecross section requires a carefull study of higher order effects.
3. p-wave pion production
Diagrams that contribute to p-wave pion production up to NNLO in the modified power count-ing are shown in Fig.3. In particular, at NNLO there are subleading rescattering and direct pionproduction operators as well as the ( ¯ NN ) p contact term. Notice that it is the same contact termthat also contributes to the three-nucleon force [8, 23], to the processes g d → p NN [24, 25] and p d → g NN [26, 27] as well as to weak reactions such as, e.g., tritium beta decay and proton-proton ( pp ) fusion [28, 29]. Therefore it provides an important connection between different low-energyreactions. It is getting even more intriguing once one realizes that this operator appears in the abovereactions in very different kinematics, ranging from very low energies for both incoming and out-going NN pairs in pd scattering and the weak reactions up to relatively high initial energies for the NN induced pion production. As a part of this connection in Ref. [30] it was shown that both the H and He binding energies and the triton b -decay can be described with the same contact term.However, an apparent discrepancy between the strength of the contact term needed in pp → pn p + and in pp → de + n e was reported in Ref. [31]. If the latter observation were true, it would questionthe applicability of chiral EFT to the reactions NN → NN p . To better understand the discrepancyreported in Ref. [31], in the recent paper [32] we simultaneously analyzed different pion produc-4 ion reactions with few-nucleon systems Vadim Baru q p (deg) d s / d Wp d M ( pb / M e V s r) q p (deg) -1-0.500.51 A y Figure 5:
Results for d s / d W p dM pp (left panel) and A y (right panel) for pn → pp ( S ) p − . Shown are theresults for d = d = d = − tion channels. In particular, we calculated the p -wave amplitudes for the reactions pn → pp p − , pp → pn p + , and pp → d p + . Note that even in these channels the contact term occurs in entirelydifferent dynamical regimes. For the first channel p -wave pion is produced along with the slowlymoving protons in the S final state whereas for the other two channels the S pp state is to beevaluated at the relatively large initial momentum. In practice, we varied the value of the low-energy constant (LEC) d , which represents the strength of the contact operator, in such a way to getthe best simultaneous qualitative description of all channels of NN → NN p . It should be stressed,however, that the value of d depends on the NN interaction employed and on the method used toregularize the overlap integrals. It therefore does not make much sense to compare values for d asfound in different calculations. Instead one should compare results on the level of observables andthis is what we do below (see also Ref. [32]). In Fig.4 we compare our results for various valuesof d with the experimental data for the analyzing power for the reaction pp → d p + . We find thatthe data prefer a positive value of d of about 3. A similar pattern can be observed in the reaction pn → pp p − as illustrated in Fig. 5. Again the data show a clear preference of the positive valuefor LEC d – our fit resulted in d = h = .
66) where the conclusion may be spoiled dueto the onset of pion d-waves. Fortunately, a new measurement for the same observables at lowerenergies is currently ongoing at COSY [41] which will soon allow a quantitative extraction of thevalue of the LEC d . We now turn to the reaction pp → pn p + – this channel was used in the anal-ysis of Ref. [31]. The reaction pp → pn p + should have, in principle, the same information onthe LEC d as contained in the deuteron channel. However, it is much more difficult to extract thepertinent information unambiguously from this reaction. In particular, the final NN-system mightbe not only in S - but also in P -wave both for isospin-zero and for isospin-one NN states. At theenergies considered in the experimental investigation, h = Pp amplitudesmay contribute significantly [43–45]. In the current study these states are disregarded. The resultsof our calculation for the magnitude A are given in the left panel of Fig. 6 . One finds again The coefficients A i are related to the unpolarized differential cross section via d s d W = A + A P ( x ) , with P ( x ) being the second Legendre polynomial ion reactions with few-nucleon systems Vadim Baru h A ( m b ) h -0.6-0.4-0.200.2 a Figure 6:
Results for the mag-nitude A (left panel) and thepartial wave amplitude a ( p m b ) representing the relevant transition S → S p (right panel) for pp → pn p + for different values of theLEC d . The notation is the sameas in Fig. 5, gray band correspondsto the results of Ref. [31]. The dataare from Ref. [42]. that the positive LEC d ≃ NN → NN p can be described simultaneously with the same value of the LEC d . Coming back tothe problem raised in Ref. [31] it should be pointed out that the results of this work were not com-pared directly to the observables in pp → pn p + . Instead, they were compared to the results of thepartial wave analysis (PWA) performed in Ref. [42], as demonstrated in the right panel of Fig. 6.It is based on this discrepancy between data and theory it was concluded in Ref. [31] about thefailure of simultaneous description of the weak processes and NN → NN p . However, the partialwave analysis of Ref. [42] seems to be not correct. Here we refer the interested reader to Ref. [32]where the drawbacks of this PWA are discussed in detail. To illustrate the problems of the PWAin the right panel of Fig. 6 we also show the results of our calculation for the relevant partial wave a which corresponds to the transition S → S p where the contact term acts. Clearly, althoughall data presented in Ref. [42] are in a good agreement with our calculation (see left panel in Fig. 6and also Ref. [32] for more details), the partial wave amplitude a is not at all described. Thus, wethink that the problem with the simultaneous description of pp → de + n e and pp → pn p + , raisedin Ref. [31], is due to the drawbacks of the partial wave analysis of Ref. [42].
4. CSB effects in pn → d p Recently, experimental evidence for CSB was found in reactions involving the production ofneutral pions. At IUCF non-zero values for the dd → ap cross section were established [46].At TRIUMF a forward-backward asymmetry of the differential cross section for pn → d p wasreported which amounts to A f b = [ . ± ( stat . ) ± . ( sys . )] × − [12]. In a charge symmetricworld the initial pn pair would consist of identical nucleons in a pure isospin one state. Thus theapparent forward–backward asymmetry is due to charge symmetry breaking.The neutron–proton mass difference is due to strong and electromagnetic interactions [47],i.e. d m N = m n − m p = d m str N + d m em N . It was stressed and exploited in Ref. [11] that the strengthof the rescattering CSB operator at LO in pn → d p (see Fig. 4(a)) is proportional to a differentcombination of d m str N and d m em N ( see also [48, 49] for related works). Thus the analysis of CSBeffects in pn → d p should allow to determine these important quantities individually. It was,however, quite surprising to find that, using the values for d m str N and d m em N from Ref. [47], theleading order calculation of the forward-backward asymmetry [11] over-predicted the experimentalvalue by about a factor of 3 — a consistent description would call for an agreement with data withinthe theoretical uncertainty of 15% for this kind of calculation. The evaluation of certain higher6 ion reactions with few-nucleon systems Vadim Baru (a) (b)
Figure 7:
Leading order diagrams for the isospin violating s -wave amplitudes of pn → d p . Diagram (a) corresponds toisospin violation in the p N scattering vertex explicitly whereasdiagram (b) indicates an isospin-violating contribution due to theneutron–proton mass difference in conjunction with the time-dependent Weinberg-Tomozawa operator. order corrections performed in Ref. [11] and in a recent study [50] did not change the situationsufficiently — the significant overestimation of the data persisted. In the recent work [51] we haveshown that there is one more rescattering operator that contributes at LO (see diagram (b) in Fig. 4).In full analogy to isospin conserving s-wave pion production, the idea was based on the fact that theenergy-dependent WT p N vertex acquires an additional contribution proportional to d m N as soonas we distinguish between the proton and the neutron. We evaluated this new LO operator andrecalculated the forward-backward assymetry at LO. It should be pointed out at this stage that A f b at LO is proportional to the interference of the s-wave pion CSB amplitude at LO and the p-wavepion isospin conserving amplitude. The latter is calculated up to NNLO, as discussed in Sec.3, andexhibits very good description of data, which is a necessary pre-requisite for studying CSB effects.The complete LO calculation gives [51] A LOfb = ( . ± . ) × − d m str N MeV (4.1)which agrees nicely with the experimental data if one uses the value of d m str N from Ref. [47]. Wemay also use (Eq. 4.1) to extract d m str N from the above expression using the data, which yields d m str N = ( . ± . ( exp . ) ± . ( th . )) MeV . (4.2)This result reveals a very good agreement with the one based on the Cottingham sum rule [47], d m str N = . ± . d m str N = . ± . ± . ± .
10 MeV.
5. Summary
We surveyed the recent developments for NN → NN p . We showed, in particular, (see Ref.[14,16]) that the s-wave pion production amplitudes calculated up to NLO for pp → d p + provide agood qualitative understanding of the pion dynamics. However, the relatively large theoretical un-certainty of about 2 M p / m N ≈
30% for the cross section requires a computation of loops at NNLO.The latter are also absolutely necessary for pp → pp p , see Ref. [53] for the first results in thisdirection. Recently, we have studied p -wave pion production up to NNLO [32]. In particular, weshowed that it is possible to describe simultaneously the p -wave amplitudes in the pn → pp p − , pp → pn p + , pp → d p + channels by adjusting a single low-energy constant accompanying theshort-range ( ¯ NN ) p operator available at NNLO. We also demonstrated that the problem with thesimultaneous description of the weak proton-proton fusion process and pp → pn p + , reported inRef. [31], is most probably due to the drawbacks of the partial wave analysis of Ref. [42] used inRef. [31]. Based on good understanding of the pion production mechanisms in the isospin con-serving case we studied charge symmetry breaking effects in pn → d p . We performed a complete7 ion reactions with few-nucleon systems Vadim Baru calculation of CSB effects at LO and extracted the strong contribution to the neutron-proton massdifference from this analysis. The value obtained, d m str N = ( . ± . ( exp . ) ± . ( th . )) MeV, isconsistent with the result based on the Cottingham sum rule and with the recent lattice calculations.At present the uncertainty in this results is dominated by the experimental error bars – an improve-ment on this side would be very important. On the other hand, a calculation of higher order effectsis also called for to confirm the theoretical uncertainty estimate.
Acknowledgments
I would like to thank E. Epelbaum, A. Filin, J. Haidenbauer, C. Hanhart, A. Kudryavtsev,V. Lensky and U.-G. Meißner for fruitful and enjoyable collaboration. I thank the organizers forthe well-organized conference and for the invitation to give this talk. Work supported in parts byfunds provided from the Helmholtz Association (grants VH-NG-222, VH-VI-231) and by the DFG(SFB/TR 16 and DFG-RFBR grant 436 RUS 113/991/0-1) and the EU HadronPhysics2 project. Iacknowledge the support of the Federal Agency of Atomic Research of the Russian Federation.
References [1] S. Weinberg, Physica A (1979) 327.[2] J. Gasser and H. Leutwyler. Ann. Phys. (1984) 142.[3] G. Colangelo, J. Gasser and H. Leutwyler, Nucl. Phys. B (2001) 125 [arXiv:hep-ph/0103088].[4] V. Bernard and U.-G. Meißner, Ann. Rev. Nucl. Part. Sci. (2007) 33 [arXiv:hep-ph/0611231].[5] P. F. Bedaque and U. van Kolck, Ann. Rev. Nucl. Part. Sci. (2002) 339; [arXiv:nucl-th/0203055];E. Epelbaum, Prog. Part. Nucl. Phys. (2006) 654; [arXiv:nucl-th/0509032]; E. Epelbaum,H.-W. Hammer and U.-G. Meißner, arXiv:0811.1338 [nucl-th], Rev. Mod. Phys., in print.[6] T.D. Cohen et al., Phys. Rev. C (1996) 2661 [arXiv:nucl-th/9512036].[7] C. da Rocha, G. Miller and U. van Kolck, Phys. Rev. C (2000) 034613 [arXiv:nucl-th/9904031].[8] C. Hanhart, U. van Kolck, and G.A. Miller, Phys. Rev. Lett. (2000) 2905 [arXiv:nucl-th/0004033].[9] C. Hanhart and N. Kaiser, Phys. Rev. C (2002) 054005 [arXiv:nucl-th/0208050].[10] C. Hanhart, Phys. Rept. (2004) 155 [arXiv:hep-ph/0311341].[11] U. van Kolck, J. A. Niskanen and G. A. Miller, Phys. Lett. B (2000) 65 [arXiv:nucl-th/0006042].[12] A. K. Opper et al. , Phys. Rev. Lett. (2003) 212302 [arXiv:nucl-ex/0306027].[13] S. Weinberg, Phys. Lett. B , 114 (1992).[14] V. Lensky, et al., Eur. Phys. J. A (2006) 37, [arXiv:nucl-th/0511054].[15] A. Gårdestig, D. R. Phillips and C. Elster, Phys. Rev. C , 024002 (2006) [arXiv:nucl-th/0511042].[16] V. Baru et al., Proceedings of MENU 2007, Julich, Germany, 10-14 Sep 2007, pp 128[arXiv:0711.2748 [nucl-th]].[17] D. A. Hutcheon et al. , Nucl. Phys. A , 618 (1991).[18] P. Heimberg et al. , Phys. Rev. Lett. , 1012 (1996).[19] M. Drochner et al. , Nucl. Phys. A , 55 (1998). ion reactions with few-nucleon systems Vadim Baru[20] P. Hauser et al. , Phys. Rev. C , 1869 (1998).[21] Th. Strauch et al. , In Proceedings of EXA08, September 2008, Vienna, Austria, published inHyperfine Interactions; Th. Strauch, PhD thesis, Cologne, 2009.[22] D. Koltun and A. Reitan, Phys. Rev. , 1413 (1966).[23] E. Epelbaum et al., Phys. Rev. C (2002) 064001 [arXiv:nucl-th/0208023].[24] V. Lensky et al., Eur. Phys. J. A (2005) 107 [arXiv:nucl-th/0505039].[25] V. Lensky et al., Eur. Phys. J. A (2007) 339 [arXiv:0704.0443 [nucl-th]].[26] A. Gårdestig and D. R. Phillips, Phys. Rev. C (2006) 014002 [arXiv:nucl-th/0501049].[27] A. Gårdestig, Phys. Rev. C (2006) 017001 [arXiv:nucl-th/0604035].[28] T. S. Park et al. , Phys. Rev. C (2003) 055206.[29] A. Gårdestig and D. R. Phillips, Phys. Rev. Lett. (2006) 232301.[30] D. Gazit, S. Quaglioni and P. Navratil, P.R.L. , 102502 (2009) [arXiv:0812.4444 [nucl-th]].[31] S. X. Nakamura, Phys. Rev. C (2008) 054001.[32] V. Baru et al, arXiv:0907.3911 [nucl-th], accepted for publication in Phys. Rev. C.[33] B. G. Ritchie et al. , Phys. Rev. C (1993) 21.[34] P. Heimberg et al. , Phys. Rev. Lett. (1996) 1012.[35] M. Drochner et al. [GEM Collaboration], Nucl. Phys. A (1998) 55.[36] E. Korkmaz et al. , Nucl. Phys. A (1991) 637.[37] E. L. Mathie et al. , Nucl. Phys. A (1983) 469.[38] H. Hahn et al. , Phys. Rev. Lett. (1999) 2258.[39] F. Duncan et al. , Phys. Rev. Lett. (1998) 4390.[40] M. Daum et al. , Eur. Phys. J. C (2002) 55.[41] A. Kacharava et al. , “Spin physics from COSY to FAIR,” arXiv:nucl-ex/0511028.[42] R.W. Flammang et al., Phys. Rev. C (1998) 916.[43] R. Bilger et al. , Nucl. Phys. A (2001) 633.[44] H.O. Meyer et al., Phys. Rev. Lett. (1999) 5439; Phys. Rev. C (2001) 064002.[45] P. N. Deepak, J. Haidenbauer and C. Hanhart, Phys. Rev. C (2005) 024004.[46] E. J. Stephenson et al. , Phys. Rev. Lett. (2003) 142302 .[47] J. Gasser and H. Leutwyler, Phys. Rept. (1982) 77.[48] U.-G. Meißner and S. Steininger, Phys. Lett. B (1998) 403.[49] N. Fettes, U.-G. Meißner and S. Steininger, Phys. Lett. B (1999) 233; N. Fettes andU.-G. Meißner, Phys. Rev. C (2001) 045201; N. Fettes and U.-G. Meißner, Nucl. Phys. A (2001) 693; M. Hoferichter, B. Kubis and U.-G. Meißner, Phys. Lett. B (2009) 65.[50] D. R. Bolton and G. A. Miller, arXiv:0907.0254 [nucl-th].[51] A. Filin et al., arXiv:0907.4671 [nucl-th].[52] S. R. Beane, K. Orginos and M. J. Savage, Nucl. Phys. B (2007) 38.[53] Y. Kim et al., Phys. Rev. C , 015206 (2009) [arXiv:0810.2774 [nucl-th]]., 015206 (2009) [arXiv:0810.2774 [nucl-th]].