aa r X i v : . [ nu c l - t h ] A ug Extracting σ πN from pionic atoms ∗ Eliahu Friedman and Avraham Gal
Racah Institute of Physics, The Hebrew UniversityJerusalem 91904, IsraelWe discuss a recent extraction of the πN σ term σ πN from a large-scalefit of pionic-atom strong-interaction data across the periodic table. Thevalue thus derived, σ FG πN = 57 ± πN isovectorscattering amplitude near threshold. It compares well with the value de-rived recently by the Bern-Bonn-J¨ulich group, σ RS πN = 58 ± πN isoscalar scattering amplitude to zero pion mass.
1. Introduction
The πN σ term σ πN = ¯ m q m N X u,d h N | ¯ qq | N i , ¯ m q = 12 ( m u + m d ) , (1)sometimes called the nucleon σ term σ N , records the contribution of ex-plicit chiral symmetry breaking to the nucleon mass m N arising from thenon-zero value of the u and d quark masses in QCD. Early calculationsyielded a wide range of values, σ πN ∼ (20 −
80) MeV [3]. Recent calcula-tions use two distinct approaches: (i) pion-nucleon low-energy phenomenol-ogy guided by chiral EFT, with or without solving Roy-Steiner equations,result in values of σ πN ∼ (50 −
60) MeV [4, 5, 6, 7, 8], the most recent ofwhich is 58 ± σ πN ∼ (30 −
50) MeV [9, 10, 11, 12, 13, 14, 15], the most recent ofwhich is 41.6 ± ∼
50 MeV, see e.g. Refs. [16, 17, 18, 19].Ambiguities in chiral extrapolations of LQCD calculations to the physicalpion mass are demonstrated on the right panel of Fig. 1. ∗ Presented in June 2019 at the 15th MENU Conf., Pittsburgh [1], and at the 3rd Jagiel-lonian Symposium on Fundamental and Applied Subatomic Physics, Krak´ow [2]. (1) krk19fg printed on August 11, 2020
JLQCD(2018)ETM(2016) χ QCD(2016)RQCD(2016)Ling et al.(2017)BMW(2016)Lutz et al.(2014)QCDSF-UKQCD(2012)Ruiz de Elvira et al.(2017)Yao et al.(2016)Hoferichter et al.(2015)Alarcon et al.(2012) 0 10 20 30 40 50 60 70 σ π N (MeV) σ π N ( M e V ) m π (GeV ) Linear fit of σ π N Quadratic fit of σ π N Eq.(33)Eq.(34)Linear fit of g Ss Ruiz de Elvira et al.(2017)
Fig. 1. Left: values of σ πN from recent calculations, based on πN phenomenology(in green) and in LQCD (other colors). Right: chiral extrapolations of LQCDderived σ πN values to the physical pion mass. Figure adapted from Ref. [14]. A third approach for evaluating σ πN was recently proposed by us [20]focusing on the σ πN -dependent in-medium renormalization of the πN isovec-tor scattering length b , determined from a wealth of strong-interaction levelshifts and widths data in pionic atoms across the periodic table [21]. Thiscontrasts with extrapolating the vanishigly small πN isoscalar scatteringlength b from m π ≈
138 MeV to the Cheng-Dashen point or nearby at m π ∼
0, as done in the first approach. To demonstrate the issues involvedin comparing these two methodologies, we cite from a recent work by theBern-Bonn-J¨ulich group [22] an expression relating the expected departureof the evaluated σ πN from their value of 59 ± b and b : σ πN ≈ (59 ±
3) MeV + 1 .
116 ∆ b free0 + 0 .
390 ∆ b free1 , (2)where ∆ b free j , j = 0 ,
1, is the difference between the values of b free j (in units of10 − m − π ) used in a given specific model and those used in the calculationof Ref. [6]. Eq. (2) suggests that the uncertainty in the determination of σ πN incurred by the model dependence of b free0 is roughly three times largerthan that incurred by the model dependence of b free1 . Regarding the modeldependence of these free-space scattering lengths we note the two sets ofinput scattering lengths ( b free0 , b free1 ) discussed in Ref. [22],( − . , − . × − m − π , (+7 . , − . × − m − π , (3)differing from each other by whether or not charge dependent effects areincorporated into the values of scattering lengths derived from π − H and π − d atoms by Baru et al. [23]. It is evident that the charge dependence ofthe near threshold πN interation affects dominantly the isoscalar b free0 while rk19fg printed on August 11, 2020 leaving the isovector b free1 basically intact. This makes an approach basedon b free1 quite attractive.To set the stage for how the third approach works we note that the πN scattering lengths [23] are well approximated by the Tomozawa-Weinbergleading-order (LO) chiral limit [24] b LO0 = 0 , b
LO1 = − µ πN πf π = − × − m − π , (4)where µ πN is the πN reduced mass and f π = 92 . b suggeststhat its in-medium renormalization is directly connected to that of f π , givento first order in the nuclear density ρ by the Gell-Mann - Oakes - Renner(GMOR) expression [25] f π ( ρ ) f π = < ¯ qq > ρ < ¯ qq > ≃ − σ πN m π f π ρ, (5)where < ¯ qq > ρ stands for the in-medium quark condensate. The decrease of < ¯ qq > ρ with density in Eq. (5) marks the leading low-density behavior ofthe order parameter of the spontaneously broken chiral symmetry, see e.g.Ref. [26]. Recalling the f π dependence of b LO1 in Eq. (4), Eq. (5) suggeststhe following density dependence for the in-medium b : b = b free1 (cid:18) − σ πN m π f π ρ (cid:19) − . (6)In this model, introduced by Weise [27, 28], the explicitly density-dependent b ( ρ ) of Eq. (6) figures directly in the pion-nucleus s -wave near-thresholdpotential. Studies of pionic atoms [29] and low-energy pion-nucleus scat-tering [30, 31] confirmed that the πN isovector s -wave interaction term isindeed renormalized in agreement with Eq. (6). It is this in-medium renor-malization that brings in σ πN to the interpretation of pionic-atom data.However, the value of σ πN was held fixed around 50 MeV in these studies,with no attempt to determine its optimal value.In our recent work [20] we kept to the πN isovector s -wave amplitude b renormalization given by Eq. (6), but varied also σ πN in fits to a com-prehensive set of pionic atoms data across the periodic table. Other real πN interaction parameters varied together with σ πN converged at expectedfree-space values. Holding these parameters fixed at the converged values,except for the tiny isoscalar s -wave single-nucleon amplitude b which isrenormalized primarily by a double-scattering term (see below), we obtaineda best-fit value of σ FG πN = 57 ± σ πN is reviewed in the next section, followed by resultsand discussion in subsequent sections. krk19fg printed on August 11, 2020
2. Pionic atoms optical potentials
The starting point in our most recent optical-potential analysis of pionicatoms [29] is the in-medium pion self-energy Π(
E, ~p, ρ ) that enters the in-medium pion dispersion relation E − ~p − m π − Π( E, ~p, ρ ) = 0 , (7)where ~p and E are the pion momentum and energy, respectively, in nuclearmatter of density ρ . The resulting pion-nuclear optical potential V opt , de-fined by Π( E, ~p, ρ ) = 2 EV opt , enters the near-threshold pion wave equation (cid:2) ∇ − µ ( B + V opt + V c ) + ( V c + B ) (cid:3) ψ = 0 , (8)where ~ = c = 1. Here µ is the pion-nucleus reduced mass, B is the complexbinding energy, V c is the finite-size Coulomb interaction of the pion with thenucleus, including vacuum-polarization terms, all added according to theminimal substitution principle E → E − V c . Interaction terms negligiblewith respect to 2 µV opt , i.e. 2 V c V opt and 2 BV opt , are omitted. We use theEricson-Ericson form [32]2 µV opt ( r ) = q ( r ) + ~ ∇ · α ( r )1 + ξα ( r ) + α ( r ) ! ~ ∇ , (9)with s -wave part q ( r ) and p -wave part, α ( r ) and α ( r ), given by [21] q ( r ) = − π (1 + µm N ) { b [ ρ n ( r ) + ρ p ( r )] + b [ ρ n ( r ) − ρ p ( r )] }− π (1 + µ m N )4 B ρ n ( r ) ρ p ( r ) , (10) α ( r ) = 4 π (1 + µm N ) − { c [˜ ρ n ( r ) + ˜ ρ p ( r )] + c [˜ ρ n ( r ) − ˜ ρ p ( r )] } , (11) α ( r ) = 4 π (1 + µ m N ) − C ˜ ρ n ( r )˜ ρ p ( r ) , (12)augmented by p -wave angle-transformation terms of order O ( m π /m N ). Here ρ n and ρ p are neutron and proton density distributions normalized to thenumber of neutrons N and number of protons Z , respectively, and ˜ ρ n and˜ ρ p are obtained from ρ n and ρ p by folding a πN ∆ form factor [33]. Thecoefficients b , b in Eq. (10) are effective density-dependent pion-nucleonisoscalar and isovector s -wave scattering amplitudes, respectively, evolvingfrom the free-space scattering lengths, and are essentially real near thresh-old. Similarly, the coefficients c , c in Eq. (11) are effective p -wave scat-tering amplitudes which, since the p -wave part of V opt acts mostly near the rk19fg printed on August 11, 2020 nuclear surface, are close to the free-space scattering volumes provided ξ = 1is applied in the Lorentz-Lorenz renormalization of α in Eq. (9). The pa-rameters B and C represent multi-nucleon absorption and therefore havean imaginary part. Their real parts stand for dispersive contributions whichoften are absorbed into the respective single-nucleon amplitudes [34]. Belowwe focus on the s -wave part q ( r ) of V opt .Regarding the isoscalar amplitude b , since the free-space value b free0 isexceptionally small, it is customary in the analysis of pionic atoms to sup-plement it by double-scattering contributions induced by Pauli correlations.For completeness we also include similar contributions to b which decreaseits value, although by only less than 10%. Thus, the single-nucleon b and b terms in Eq. (10) are extended to account also for double-scattering [32, 35],˜ b → ˜ b − π (˜ b + 2˜ b ) p F , ˜ b → ˜ b + 32 π (˜ b − b ˜ b ) p F , (13)where ˜ b j ≡ (1 + m π m N ) b j , and p F is the local Fermi momentum correspondingto the local nuclear density ρ = 2 p F / (3 π ).Regarding the isovector amplitude b , it affects primarily level shifts inpionic atoms with N − Z = 0. However, it affects also N = Z pionic atomsthrough the dominant quadratic b contribution to b of Eq. (13). Thisdominance follows already at the level of b free1 from a systematic expansionof the pion self-energy up to O ( p ) in nucleon and pion momenta withinchiral perturbation theory [36]. Following Ref. [37] it can be argued thatit is the in-medium b Eq. (6) that enters the Pauli-correlation double-scattering contribution in Eq. (13). This approach has been practised innumerous global fits to pionic atoms by us [21, 29] as well as by othergroups, e.g. Geissel et al. [38], using a fixed value of σ πN . To study therole of a variable σ πN as per Eq. (6) we extended b wherever appearing inEq. (13) by substituting b → b (cid:18) − σ πN m π f π ρ (cid:19) − . (14)Regarding the nuclear densities ρ p and ρ n that enter the potential,Eqs. (10)–(12), two-parameter Fermi distributions with the same diffuse-ness parameter for protons and neutrons were used [21, 39] yielding lowervalues of χ than other shapes do for pions. With proton densities de-termined from nuclear charge densities, the neutron densities were varied,searching for best agreement with the pionic atoms data by assuming alinear dependence of r n − r p , the difference between the root-mean-square(rms) radii, on the neutron excess ratio ( N − Z ) /A : r n − r p = γ N − ZA + δ , (15) krk19fg printed on August 11, 2020 with γ close to 1.0 fm and δ close to zero. Here we used δ = − .
035 fmand varied γ . For example, γ =1 fm means r n − r p = 0 .
177 fm in
Pb, avalue compatible with several analyses of pion strong and electromagneticinteractions in
Pb [40, 41], and with other determinations of the so called‘neutron skin’.
3. Results
Following the optical potential approach described in the preceding sec-tion, and more extensively in Refs. [21, 29], global fits to strong interactionlevel shifts and widths from Ne to U were made over a wide range of valuesfor the neutron-skin parameter γ as shown in Fig. 2. γ (fm)−0.20.00.2 pa r a m e t e r s ( m π n ) Χ ( ) b c c b b c c γ (fm)−0.4−0.20.0 pa r a m e t e r s ( m π n ) Χ ( ) b b ReB /ImB ReC /ImC Fig. 2. Fits to 98 pionic atoms data points for σ πN = 0 as a function of theneutron-skin parameter γ , with χ values plotted in the upper panels and fittedvalues of some of the π -nucleus optical potential parameters plotted in the lowerpanels. No χ minimum is reached in the 8-parameter left-panel fits, but fixingthe p -wave parameters c and c at their SAID [42] threshold values 0.23 and 0.16 m − π , respectively, produces the fits shown in the right panels. The fitted 98 data points include ‘deeply bound’ states in Sn isotopesand in
Pb. Varying all eight parameters (real b , b , c , c ; complex B , C ) in Eqs. (10)–(12) produces good χ fits, χ ∼ χ ( γ ) minimum as clearly seen in the upper left panel ofFig. 2. The lower left panel shows that the single-nucleon parameters arewell determined and vary smoothly with γ . rk19fg printed on August 11, 2020 Holding the p -wave single-nucleon parameters c , c fixed at their SAIDfree-space threshold values marked by dashed horizontal lines, thereby re-ducing the number of fitted parameters to six, a χ minimum around γ = 1to 1.1 fm was reached as shown in the upper right panel of Fig. 2. In thesesix-parameter fits, Im B and Im C (not shown) come out well-determined,with values almost independent of γ , but Re B and Re C are poorly de-termined as seen in the lower right panel of the figure. In all the fits shownhere in Fig. 2, b was treated as a free parameter regardless of any possi-ble functional dependence on σ πN , thereby corresponding to σ πN = 0 inEq. (14). The fitted values of b disagree then over a broad range of γ s withthe value b free1 marked by a dashed horizontal line. γ (fm)−0.20.00.2 pa r a m e t e r s ( m π n ) Χ ( ) b c c b b c c γ (fm)020406080100 σ π N ( M e V ) Χ ( ) Fig. 3. Left: 6-parameter fits with σ πN = 50 MeV where Re B and Re C arekept zero. Right: 4-parameter fits where c and c , additionally, are kept at theirSAID [42] threshold values 0.23 and 0.16 m − π , respectively. Of the 4 varied pa-rameters ( b , b , Im B , Im C ) b is related to σ πN by Eq. (6). Resulting valuesof σ πN are plotted in the lower right panel. Introducing the in-medium density dependence of b given by Eq. (14)in terms of σ πN we first demonstrate the effect of using a fixed value of σ πN = 50 MeV, as practised in all of our past works [29], on the fittedparameters. This is shown within six-parameter fits in the left panels ofFig. 3. Rather than keeping the p -wave single-nucleon parameters c and c to their SAID free-space threshold values, as done in the σ πN = 0 fitsshown in the right panels of Fig. 2, here we kept Re B and Re C to zerovalues thereby producing as good fits to the data as by letting them vary. krk19fg printed on August 11, 2020 In particular, suppressing Re B in pionic atoms fits amounts to absorbingit into an effective b parameter [34]. The fitted c and c , particularly c ,are clearly seen in the lower panel to come out close to the respective free-space values. As for the s -wave single-nucleon parameters b and b , thedominance of b with respect to b is also clearly seen. The introduction ofa nonzero value of σ πN allows b to reach its free-space value b free1 beginningat a neutron-skin parameter γ value of 1.1 fm.Holding now the p -wave single-nucleon parameters c and c at their free-space SAID threshold values 0.23 and 0.16 m − π , respectively, and keepingas before Re B and Re C to zero values, we show in the right panels ofFig. 3 four-parameter fits where the varied parameters are b , σ πN for b using Eq. (6), Im B and Im C . A minimum value of χ = 167 . γ ≈ . σ πN assumes a value of σ FG πN = 56 . ± . γ ≈ . Pb [41]. To check the dependence of σ πN on b we repeatedfits with b kept fixed at either one of the two free-space threshold valueslisted in Eq. (3), varying then also Re B and Re C . Typical χ valuesincreased by 20 to 30, but the χ minima remained at γ = 1 . σ πN decreasing at most by 3 MeV. We alsonote that the resulting value of σ πN is identical with that derived in ourrecently published work [20] where the effect on the derived value of σ πN of form-factor folding, ρ n,p → ˜ ρ n,p in the p -wave terms (11,12) of the pion-nucleus optical potential, was shown to be negligibly small.
4. Discussion and summary
The pionic atoms fits and the value of the πN σ term σ πN extractedin the present work are based on the in-medium renormalization of thenear-threshold πN isovector scattering amplitude b as given by Eq. (6),derived at LO from Eqs. (4) and (5) for the in-medium decrease of the piondecay constant f π associated via the GMOR expression with the in-mediumdecrease of the quark condensate < ¯ qq > . Higher order corrections to thissimple form have been proposed in the literature and were discussed by usin Ref. [20]. Briefly, one may classify two such corrections arising from: (i) N N correlation contributions [43] from one- and two-pion interaction terms,increasing the fitted σ πN value by about 7 MeV (or by a smaller amountfollowing a chiral approach at NLO [44]); and (ii) an upward shift of thein-medium pion mass m π ( ρ ) in symmetric nuclear matter from its free-spacevalue [45], decreasing the fitted σ πN value by a similar amount, and also byadding corrections of order ρ / [46, 47] which at a typical nuclear density ρ eff = 0 . − [34] are negligible. Interestingly but perhaps fortuitously,these two higher-order effects largely cancel each other. rk19fg printed on August 11, 2020 In conclusion, we have derived in this work a value of σ FG πN = 57 ± σ πN values reported in recent studies based on modern hadronic πN phenomenology [8], but in disagreement with the considerably lower σ πN values reached in some of the recent modern lattice QCD calculations,e.g. [14]. Our derivation is based on the model introduced by Weise andcollaborators [27, 28, 37] for the in-medium renormalization of the πN near-threshold isovector scattering amplitude, using its leading density depen-dence Eq. (6), and was found robust in fitting the wealth of pionic atomsdata against variation of other πN interaction parameters that enter thelow-energy pion self-energy operator. The two types of model correctionsbeyond the leading density dependence considered here were found to berelatively small, a few MeV each, and partly canceling each other. Furthermodel studies are desirable in order to confirm this conclusion. Acknowledgments
We are grateful to Norbert Kaiser, Wolfram Weise and Nodoka Ya-manaka for useful correspondence on the subject of the present work.REFERENCES [1] E. Friedman, A. Gal, AIP Conf. Proc. , 030015 (2020).[2] E. Friedman, A. Gal, Acta Phys. Pol. B , 45 (2020).[3] M.E. Sainio, πN Newsletter , 138 (2002) [arXiv:hep-ph/0110413].[4] J.M. Alarc´on, J.M. Camalich, J.A. Oller, Phys. Rev. D , 051503 (2012).[5] Y.-H. Chen, D.-L. Yao, H.Q. Zheng, Phys. Rev. D , 054019 (2013).[6] M. Hoferichter, J. Ruiz de Elvira, B. Kubis, U.-G. Meißner, Phys. Rev. Lett. , 092301 (2015).[7] V. Dmitraˇsinovi´c, H.-X. Chen, A. Hosaka, Phys. Rev. C , 065208 (2016).[8] J. Ruiz de Elvira, M. Hoferichter, B. Kubis, U.-G. Meißner, J. Phys. G ,024001 (2018).[9] R. Horsley et al. (QCDSF-UKQCD Collab.), Phys. Rev. D , 034506 (2012).[10] S. Durr et al. (BMW Collab.), Phys. Rev. Lett. , 172001 (2016).[11] Y.-B. Yang, A. Alexandru, T. Draper, K.-F. Liu ( χ QCD Collab.), Phys. Rev.D , 054503 (2016).[12] A. Abdel-Rehim et al. (ETM Collab.), Phys. Rev. Lett. , 252001 (2016).[13] G.S. Bali et al. (RQCD Collab.), Phys. Rev. D , 094504 (2016).[14] N. Yamanaka, S. Hashimoto, T. Kaneko, H. Ohki (JLQCD Collab.), Phys.Rev. D , 054516 (2018).0 krk19fg printed on August 11, 2020 [15] C. Alexandrou et al. (ETM Collab.), arXiv:1909.00485v1.[16] D.B. Leinweber, A.W. Thomas, S.V. Wright, Phys. Lett. B , 109 (2000).[17] L. Alvarez-Ruso, T. Ledwig, J.M. Camalich, M.J. Vicente-Vacas, Phys. Rev.D , 054507 (2013).[18] X.-L. Ren, L.-S. Geng, J. Meng, Phys. Rev. D , 051502(R) (2015).[19] X.-L. Ren, X.-Z. Ling, L.-S. Geng, Phys. Lett. B , 7 (2018).[20] E. Friedman, A. Gal, Phys. Lett. , 340 (2019).[21] E. Friedman, A. Gal, Phys. Rep. , 89 (2007).[22] M. Hoferichter, J. Ruiz de Elvira, B. Kubis, U.-G. Meißner, Phys. Lett. B , 74 (2016).[23] V. Baru et al. , Phys. Lett. B , 473 (2011); Nucl. Phys. A , 69 (2011).[24] Y. Tomozawa, Nuovo Cim. A , 707 (1966); S. Weinberg, Phys. Rev. Lett. , 616 (1966).[25] M. Gell-Mann, R.J. Oakes, B. Renner, Phys. Rev. , 2195 (1968).[26] T.D. Cohen, R.J. Furnstahl, D.K. Griegel, Phys. Rev. C , 1881 (1992).[27] W. Weise, Acta Phys. Pol. B , 2715 (2000).[28] W. Weise, Nucl. Phys. A , 98c (2001).[29] E. Friedman, A. Gal, Nucl. Phys. A , 128 (2014), and references thereinto earlier work on pionic atoms.[30] E. Friedman et al. , Phys. Rev. Lett. , 122302 (2004).[31] E. Friedman et al. , Phys. Rev. C , 034609 (2005).[32] M. Ericson, T.E.O. Ericson, Ann. Phys. , 323 (1966).[33] A. Gal, H. Garcilazo, Nucl. Phys. A , 153 (2011); see Fig. 2 & Tab. 3.[34] R. Seki, K. Masutani, Phys. Rev. C , 2799 (1983).[35] M. Krell, T.E.O. Ericson, Nucl. Phys. B , 521 (1969).[36] N. Kaiser, W. Weise, Phys. Lett. B , 283 (2001).[37] E.E. Kolomeitsev, N. Kaiser, W. Weise, Phys. Rev. Lett. , 092501 (2003).[38] H. Geissel et al. , Phys. Lett. B , 64 (2002); Phys. Rev. Lett. , 122301(2002).[39] E. Friedman, Hyperfine Interact. , 33 (2009).[40] E. Friedman, Nucl. Phys. A , 46 (2012).[41] C.M. Tarbert et al. , Phys. Rev. Lett. , 242502 (2014).[42] R.A. Arndt, W.J. Briscoe, I.I. Strakovsky, R.L. Workman, Phys. Rev. C ,045205 (2006); evolving SAID program http://gwdac.phys.gwu.edu/[43] N. Kaiser, P. de Homont, W. Weise, Phys. Rev. C , 025204 (2008).[44] A. Lacour, J.A. Oller, U.-G. Meißner, J. Phys. G , 125002 (2010).[45] D. Jido, T. Hatsuda, T. Kunihiro, Phys. Lett. B , 109 (2008).[46] S. Goda, D. Jido, Phys. Rev. C , 065204 (2013).[47] S. Goda, D. Jido, Prog. Theor. Exp. Phys.2014