T39 Physics Group - HADRON POLARIZABILITIES · 2013. 12. 12. · the magnetic dipole excitation from the ground state 10) at energy E o into an excited state In> at exci- tation energy

Compton Scattering Workshop ECT* Trento 29 July 2013

HADRON POLARIZABILITIES

Wolfram Weise

European Centre for Theoretical Studies in Nuclear Physics and Related Areas

and Technische UniversitätMünchen

1. History and Progress

2. Nucleon Polarizabilities: Chiral EFT and the Delta

3. Axial Polarizability and NN Van der Waals Force

4. Pion Polarizabilities

W. KniJpfer and A. Richter: Scaling of the Paramagnetic Susceptibility of Atoms, Nuclei and Nucleons 257

m

1012 <

a. • 10 8

1 0 4 s O

t/) 10 ~ u

c O)

CJ CJ

12.

I I I

Nucleon Q~

l , l , l

10 ~ 10 4 10 8 E v (eV)

Fig. 2. Paramagnetic susceptibility of atoms (metals), nuclei and nucleons as function of the corresponding Fermi energy of the three systems. The error bars associated with each point indicate the typical range of values of ZPA~A for metals and nuclei and the uncertainty of the derived value for )~PARA of the nucleons, respectively

A N 2 1_ 4 - 1 0 , 3 ( 2t )~PARA/)~PARA = \ ~ N ] p n 2 A (gs -gs ) ee in agreement with the experimental result of (10). Our first conclusion is therefore that the ratio of the atomic and nuclear paramagnetic susceptibilities is determined by the square of the ratio of the Bohr and nuclear magnetons and the ratio of the Fermi energies of the systems, respectively. (This conclusion is different from the one stated in [-1] under the subheading Nuclear Paramagnetism). Secondly, we can in analogy to atoms and nuclei

identify also in the nucleon a Fermi energy and associate it with the ground state energy of a quark in a bag model, e.g. the MIT-bag model [-26]. For a nucleon bag radius of 1fro we would have then ev e ~ 300 MeV. We finally compare the atomic, nuclear and nucleonic susceptibilities derived in this paper by plot- ting in Fig. 2 )~PARA as a function of the respective Fermi energies. The error bars associated with each point reflect the typical range in the values of ZPARA for metals and nuclei and the uncertainty of the nucleonic point. The susceptibilities of the three systems follow a straight line when they are interpreted in terms of a Fermi gas model. This behaviour re- flects partially the dependence of )~PARA on the in- volved masses in the Bohr and nuclear moments as well as the Fermi energies, in contrast to the belief in [-1] that only the masses are important.

We thank J. Hiifner for his encouragement to investigate the problem dealt with in this paper. One of us (A.R.) is grateful to B. Elschner for his continuous interest in our experiments on nuclear magnetism at the DALINAC and for many pleasant discussions over the years on similarities and differences in our respective fields of magnetic properties of nuclei and solids.

References

1. See, e.g. Kittel, C.: Introduction to solid state physics. New York: John Wiley and Sons 1971; or any other textbook on solid state physics

2. Ericson, T.E.O., Hiifner, J.: Nucl. Phys. B47, 205 (1972); Eric- son, T.E.O.: In: Interaction Studies in Nuclei. Jochim, H., Ziegler, B. (eds.), p. 577-597. Amsterdam: North-Holland 1975

3. Iachello, F., Lande, A.: Phys. Lett. 35B, 205 (1971) 4. Kniipfer, W., Richter, A.: Phys. Lett. 101B, 375 (1981) 5. Kniipfer, W., Richter, A. : Phys. Lett. 107B, 325 (1981) 6. Ericson, T.E.O., Hiifner, J.: Nucl. Phys. B57, 604 (1973) 7. Tzara, C.: J. Phys. Lett. 41, L-221 (1980) 8. See, e.g. Chrien, R., Hofmann, A., Molinari, A.: Phys. Rep. 64,

249 (1980) 9. Eulenberg, G., Sober, D.I., Steffen, W., Gr~if, H.-D., Kiichler,

G., Richter, A., Spamer, E., Metsch, B.C., KniJpfer, W.: Phys. Lett. l16B, 113 (1982)

10. Bohle, D., Richter, A., Steffen, W., Dieperink, A.E.L., Lo Iudice, N., Palumbo, F., Scholten, O.: Phys. Lett. 137B, 27 (1984)

11. Kiichler, G., Richter, A., Spamer, E., Steffen, W., Kniipfer, W.: Nucl. Phys. A406, 473 (1983)

12. Gross, W., Meuer, D., Richter, A., Spamer, E., Titze, O., Kniipfer, W.: Phys. Lett. 84B, 296 (1979)

13. Fagg, L.W.: Rev. Mod. Phys. 47, 683 (1975) 14. Schneider, R., Richter, A., Schwierczinski, A., Spamer, E.,

Titze, O., KniJpfer, W.: Nuci. Phys. A323, 13 (1979) 15. Steffen, W., Griif, H.-D., Richter, A., H~irting, A., Weise, W.,

Deutschmann, U., Lahm, G., Neuhausen, R.: Nucl. Phys. A404, 413 (1983)

16. Bohr, A., Mottelson, B.R.: Nuclear structure. Vol. I, p. 210. New York: Benjamin 1969

17. Goldanskii, V.I., Karpukhin, O.A., Kutsenko, A.V., Pavlov- skaya, V.V.: Nucl. Phys. 18, 473 (1960)

18. Baranov, P., Buinov, G., Godin, V., Kusnetzova, V., Pet- runkin, V., Tatarinskaya, L., Shirthenko, V., Shtarkov, L., Yur- tchenko, V., Yanulis, Yu.: Phys. Lett. 52B, 122 (1974)

19. Ragusa, S.: Phys. Rev. D l l , 1536 (1975) 20. Bernabeu, J., Ericson, T.E.O., Ferro Fontan, C.: Phys. Lett.

49B, 381 (1974) 21. Dattoli, G., Matone, G., Prosperi, D.: Lett. Nuovo Cimento

19, 601 (1977) 22. Hecking, P.C., Bertsch, G.F.: Phys. Lett. 99B, 237 (1981) 23. Petrunkin, V.A.: Soy. J. Part. Nucl. 12, 278 (1981) 24. Drechsel, D., Russo, A.: Phys. Lett. 137B, 294 (1984) 25. Alberico, W.M., Ericson, M., Molinari, A.: Nucl. Phys. A386,

412 (1982) 26. Grand, T.A. De, Jaffe, R.J.: Ann. Phys. NY 100, 425 (1976);

101,396 (1976)

W. Knfipfer A. Richter Siemens AG Institut ftir Kernphysik Bereich Medizinische Technik TH Darmstadt, Hochschulstrasse 9 D-8520 Erlangen D-6100 Darmstadt Federal Republic of Germany Federal Republic of Germany

POLARIZABILITIES of ATOMS, NUCLEI and NUCLEONS

Scales and orders of magnitude

Early history: T.E.O. Ericson, J. Hüfner: Nucl. Phys. B 47 (1972) 201

Quantum mechanical polarizabilties / susceptibilities

χE = 2∑

n

|〈n|E1|g.s.〉|2

En − E0 = −Ze2

6m〈r2〉 + 2

∑

n

|〈n|M1|g.s.〉|2

En − E0

χM = χdia + χpara

W. Knüpfer, A. RichterZ. Physik

320 (1985) 253

Examples: paramagnetic susceptibilities of Fermion systems

254 W. Kniipfer and A. Richter: Scaling of the Paramagnetic Susceptibility of Atoms, Nuclei and Nucleons

For a physical system of non-interacting identical atoms (nuclei) the specific susceptibility, i.e. the susceptibility per atom (nucleon) taken in cm 3 (fm3), contains two terms

% = XDIA At- ZPARA

Ze2 v l<nl M1 IO512 6mc 2(r2>+2@ E _Eo (2)

The first term with a negative sign as a manifes- tation of Lenz' law is the diamagnetic contribution to X, where Z is the charge number, m the mass of the particles composing the system and ( r 2) its mean square radius. (Note, that in nuclei there is a small additional term contributing to )~DIA which is pro- portional to 1/.4 with A being the mass of the system and which represents a center of mass contribution [2]. We neglect this term here.) The second term in (2) is the paramagnetic contribution to g, with (nl M110) representing the matrix element of the magnetic dipole excitation from the ground state 10) at energy E o into an excited state In> at excitation energy E n. In solid state physics this term is sometimes called Van Vleck paramagnetism [1]. It is clear - as will be shown below in detail for nuclei - that the matrix element determining ZPARA on the r.h.s, of (2) in an atom (nucleus) is much dependent

on the atomic (nuclear)shell structure and in particular on the occupation number of the shells determining the magnetic properties. Note, that in the above (2) we neglect the so-called Langevin paramagnetism which is temperature dependent (Curie law), specific for ions and roughly a factor of 10-100 larger than the Van Vleck susceptibility. Its corresponding counterpart is not easily identifiable in nuclei and nucleons. For the compari- son of the values of XPARA in the different phases of matter, we therefore select metals which about obey (2) as representatives of atomic systems. We even go one step further and associate later the second term in (2) with the Pauli paramagnetism [1].

2. Nuclear Paramagnetic Susceptibility An expression of N ZPARA for nuclei has been derived by Ericson and Htifner [2] in the frame of quantum mechanics. The magnetic polarizability is connected to unretarded virtual magnetic dipole excitations of the nucleus. The contributions of the orbital and the spin currents of the individual i nucleons to the total nuclear current lead to

t<n[ Y. (gll + giss)10>l z N 2 i

ZPARA = 2#N 2 n E , - E o (3)

>

E e4 z ~u -~<

n~ z ~

x

m <

z ~ x

< n,- <

z n x

10

10

10 -1

~ ( r T l l T - -

( l d 512) (v lg 912) ( v l h n t 2 )

o

0 5 0 1 0 0 150 2 0 0 2 5 0

Nuclear Massnumber A

Fig. 1. Nuclear mass number A dependent paramagnetic susceptibility (upper part). Experimental data points (open circles) are compared to an independent particle shell model (IPM) prediction (solid line). Note the strong shell effects in )~PARA'N In the lower part the ratio of the nuclear paramagnetic and diamagnetic susceptibilities is plotted as a function of mass number indicating that nuclear diamagnetism is overwhelming in heavy nuclei

χatompara

χnucpara

∼

(

µB

µN

)2εnucF

εatomF

∼ 1013

POLARIZABILITIES of the NUCLEON

2 Will be inserted by the editor

2 Polarizabilities in RCS

The physical content of the nucleon polarizabilities can be visualized best by effectivemultipole interactions for the coupling of the electric (E) and magnetic (H) fieldsof a photon with the internal structure of the nucleon. When expanding the RCSamplitude in the energy of the photon, the zeroth and first order terms follow froma low-energy theorem and can be expressed solely in terms of the charge, mass, andanomalous magnetic moment of the nucleon. The second order terms in the photonenergy describe the response of the nucleon’s internal structure to an electric ormagnetic dipole field. They are given by the following effective interaction :

H(2)eff = −4π

12 αE1 E

2 + 12 βM1 H

2, (1)

where the proportionality coefficients are the electric (αE1) and magnetic (βM1) scalardipole polarizabilities, respectively. These global structure coefficients are propor-tional to the electric and magnetic dipole moments of the nucleon which are inducedby the applied electric and magnetic fields. The polarizabilities have been measuredextensively by use of unpolarized Compton scattering. A global fit to all modernlow-energy proton Compton scattering data leads to the results [6]

αpE1 = 12.1± 0.3 (stat.)∓ 0.4 (syst.)± 0.3 (mod.) ,

βpM1 = 1.6± 0.4 (stat.)± 0.4 (syst.)± 0.4 (mod.) . (2)

Here and in the following the scalar polarizabilities are given in units of 10−4 fm3, andthe indicated errors denote the statistical, systematical and model-dependent errors.Equation 2 shows the dominance of the electric polarizability αp

E1. The tiny valueof the magnetic polarizability, βp

M1, originates from a strong cancelation betweenthe large paramagnetic contribution of the N → ∆ spin-flip transition and a nearlyequally large diamagnetic contribution, mostly due to pion-loop effects.

The internal spin structure of the nucleon appears at third order in an expansionof the Compton scattering amplitude. It is described by the effective interaction

H(3)eff = −4π

12γE1E1 σ · (E× E) + 1

2γM1M1 σ · (H× H)

−γM1E2 Eij σiHj + γE1M2 Hij σiEj ] , (3)

which involves one derivative of the fields with regard to either time or space, E = ∂tEand Eij = 1

2 (∇iEj + ∇jEi), respectively. The four spin (or vector) polarizabilitiesγE1E1, γM1M1, γM1E2, and γE1M2 describing the nucleon spin response at third order,can be related to a multipole expansion [7], as is reflected in the subscript notation. Forexample, γM1E2 corresponds to the excitation of the nucleon by an electric quadrupole(E2) field and its de-excitation by a magnetic dipole (M1) field. Expanding theCompton scattering amplitude to higher orders in the energy, one obtains higherorder polarizabilities to the respective order, e.g., the quadrupole polarizabilities atfourth order [7,8].

On the experimental side, much less is known about the spin polarizabilities,except for the forward (γ0) and backward (γπ) spin polarizabilities of the proton,given by the following linear combinations of the polarizabilities in Eq. (3):

γ0 = −γE1E1 − γM1M1 − γE1M2 − γM1E2 , (4)

γπ = −γE1E1 + γM1M1 − γE1M2 + γM1E2 . (5)

The forward spin polarizability has been determined by the Gerasimov-Drell-Hearnsum rule experiments at MAMI and ELSA [9,10,11]. A recent analysis of these data




H(2)eff = −4π

12 αE1 E

2 + 12 βM1 H

2, (1)


αpE1 = 12.1± 0.3 (stat.)∓ 0.4 (syst.)± 0.3 (mod.) ,

βpM1 = 1.6± 0.4 (stat.)± 0.4 (syst.)± 0.4 (mod.) . (2)





H(3)eff = −4π

12γE1E1 σ · (E× E) + 1

2γM1M1 σ · (H× H)





γ0 = −γE1E1 − γM1M1 − γE1M2 − γM1E2 , (4)

γπ = −γE1E1 + γM1M1 − γE1M2 + γM1E2 . (5)





H(2)eff = −4π

12 αE1 E

2 + 12 βM1 H

2, (1)


αpE1 = 12.1± 0.3 (stat.)∓ 0.4 (syst.)± 0.3 (mod.) ,

βpM1 = 1.6± 0.4 (stat.)± 0.4 (syst.)± 0.4 (mod.) . (2)





H(3)eff = −4π

12γE1E1 σ · (E× E) + 1

2γM1M1 σ · (H× H)





γ0 = −γE1E1 − γM1M1 − γE1M2 − γM1E2 , (4)

γπ = −γE1E1 + γM1M1 − γE1M2 + γM1E2 . (5)


F(ω, θ = 0) = f(ω) #ε · #ε ′∗− ig(ω)#σ · (#ε × #ε ′∗)

f(ω) = −Z2 e2

4πM+ (αE1 + βM1) ω2 + O(ω4)

g(ω) = −e2

κ2

8πM2ω + γ ω

3 + O(ω5)

Forward Compton scattering amplitude

Low-energy expansion

κp = 1.793 . . . proton

κn = −1.913 . . . neutron

Effective Hamiltonian form

B. Pasquini, D. Drechsel, M. Vanderhaeghen

arXiv:1105.4454 [hep-ph]

H. W. Grießhammer,D.R. Phillips, J.A. McGovernarXiv:1306.2200 [nucl-th]

[26] D. Babusci et al., Phys. Rev. C 58, 1013 (1998).[27] D. Drechsel et al., Phys. Rev. C 61, 015204 (2000).[28] V. Bernard et al., Int. J. Mod. Phys. E4, 193 (1995).[29] T.R. Hemmert et al., Phys. Rev. D 57, 5746 (1998).[30] X. Ji et al., Phys. Lett. B 472, 1 (2000).[31] K.B. Vijaya Kumar et al., Phys. Lett. B 479, 167 (2000).[32] G.C. Gellas et al., Phys. Rev. Lett. 85, 14 (2000).

0

100

200

300

400

500

600

100 200 300 400 500 600 700 800

E! (MeV)

"to

t (µ

ba

rn)

Armstrong et. al.

MacCormick et. al.

This work

SAID (N#)

HDT (N#)

UIM (N#+N##+$)

FIG. 1. The unpolarized total photoabsorption cross section on 1H obtained in this work is compared to previous results [9](open circles), [17] (stars) and to the HDT [18], SAID [19] and UIM [20] analyses. The statistical error bars are smaller thanthe size of the symbols.

-200

-100

0

100

200

300

400

500

600

100 200 300 400 500 600 700 800

E! (MeV)

"3/2

- "

1/2

(µ

ba

rn)

This workRef.[1] (p#

0+n#

+)

HDT (N#)

SAID (N#)

UIM (N#+N##+$)

FIG. 2. The total cross section difference (σ3/2 − σ1/2) on 1H is compared to previous results [1] (open circles) and to thepredictions of the HDT [18], SAID [19] and UIM [20] analyses. Only statistical errors are shown.

5

∆

Eγ (MeV)

POLARIZABILITIES of the NUCLEON

Quick estimates

Electric polarizability:

αpE1 = 2

∑

n

|〈n|E1|0〉|2

En − E0

$2

3

(

e2

4π

)

〈r2E〉p∆E

∼ 10−3

fm3

Paramagnetic polarizability:

MAMI(2001)

Polarized Compton scattering

1 fm 10 fm 20km

1 fm 10 fm 20km

N

∆βN→∆M1 = 2

|〈∆|M1|N〉|2

M∆ − MN

βN→∆M1 !

e2

4π(M∆ − MN)

(

µV

MN

)

∼ 10−3

fm3

isovector magnetic moment µV =1

2(µp − µn) = 2.353

POLARIZABILITIES of the NUCLEON (contd.)

Chiral Effective Field Theory approach

nucleon = valence quark core + (multi)pion cloud

iriirr .rqr ur tsol sr ruJ s'0 --w l2t salBcs qfual le sJrs,(qd luBuodur leql .re^e,oq

:\,:'.qo l ll'slapou ed,1-auu,19 ur oseJ eql lcBJ ur sr srql .rBeddEstp plno,t\\tr:jii:rp srql 'oo <='w r4]ln r.uopso 3o saa.rEap uosetu Jolce^ .,u3zo,, Jo lrurl eqlr :rrt rtr)\'uorlnqrrlsrp reqr.unu uo,reg oql qlr^\ uoloqd rEIeJSosr oql slceuuoc:.Ut. .roi!Lu-ra Eurle8edord eql ol onp,ldrurs sr (!.r) pue s(];) uae,,n1eq ecueJogrpir::t:uir. aql'tuJ6t'O:s(?l) lecurdrue aql qlr,^d luaruea:3u pooE fra,L ul .ruJg.0

' ,u .r.rrctgo auo'(lepour lur.uturul) I olqel ur se ruJ9.g=.r,($.r):Hl qlr^\ ocueH

'ltu ' (o,r) + " "9

,q ,lrsuap Joqrunu uo,req orl Jo (gl): rl.l snrpur arunbs ueeu eql olr:ri:r \r trll) snrper eSreqc Jel€csosr orenbs ueotu eql teql ,(lalerpauur s,rolloJ 1I

' lt yog 1tb 1,t[-t tp " | ,!+?* -

' 'J i*uz

1t1,o1ftt',[rt tp "l+, : (,b)]c

, (r),,g8i: e)or(Iw _.L)

:(lopou letururtu :cgt.E) uorlenba plog uosaru_r,iur..1 '1egg'7) rope3 ruro3 e3:eqc releJsosr orl reprsuoc,srql elerlsnllr o1

'uolrlos eql eprsur raqlunu uo,:eq Jo uorlnqrrstp\.rlnieeru Hcrrrrr.r,(!.r):H/ sntper eql ueqlraE.rul ,(lgeraptsuoJ sr uoloqd eqt.q

'plau uosoru-o aqt .iq paunu:a1ep Xlrsuap-' 'jl\)l) rpleJsost aqt o1 saldnol uotoqd:e;ersost aql sealJq\\.llsuap:B.reqt uo,.rcq petrlp[!Jou. , I/ rl0H lapou leulultu oql JoJ (Itrun ot pazlleurou) suortnqlrlsrp a8:eq:r:elucsosl S.ilC

Ir!J]raL 0r 80 90 t0 z0 0

Z

[' u]l

\uoluos awtt$ls sD ''uoapnN I 'lD D /Duss!aI,! .c_.n

01rJ,4 UeC 3,4A

1

2

0.2 0.4 0.6 0.8 1.0 1.2 r [fm]

[fm−1]

4πr2

ρB(r)

4πr2

ρS(r)charge density

baryon density

isoscalar

0

recall:chiral (soliton) modelof the nucleon

compactbaryonic core

mesonic cloud

. . . treated properly in chiral EFT

N. Kaiser, U.-G. Meißner, W. W.Nucl. Phys. A 466 (1987) 685

〈r2〉1/2B

# 0.5 fm

〈r2〉1/2E,isoscalar # 0.8 fm

Leading order ChPT:V. Bernard, N. Kaiser, U.-G. Meißner

Phys. Rev. Lett. 67 (1991) 1515

αE1 = 10βM1 =

( e

4π

)2 5

24

(

gA

fπ

)2

! 12.6 · 10−4 fm3

Empirical (global fit): αE1 = [12.1 ± 0.3(stat.)∓0.4(syst.)±0.3(mod.)] 10−4

fm3

βM1 = [1.6 ± 0.4(stat.)∓0.4(syst.)±0.4(mod.)] 10−4

fm3V. Olmos de Leon et al.

Eur. Phys. J. A10 (2001) 207


State-of-the-art: Chiral Effective Field Theory to NNNNLO with inclusion of explicit degrees of freedom∆(1232)

FIG. 1: (Colour online) Tree diagrams that contribute to Compton scattering in the ε · v = 0 gauge,ordered by the typical size of their contributions in the two regimes ω ∼ mπ ∼ δ2 and ω ∼ ∆M ∼ δ,

respectively. The leading-order contribution in a particular regime is indicated by (LO). The vertices

are from: L(1)πN (no symbol), L(2)

πN (square), L(3)πN (triangle), L(4)

πN (diamond), L(4)ππ (disc). Permuted and

crossed diagrams not shown.

1. Regime I: ω ∼ mπ

We first discuss the graphs that contribute up to the order to which we work, Figs. 1 to 3,for photon energies ω ∼ mπ. References for the actual form of the resulting amplitudes aregiven in Appendix A.

The first class consists of the “tree” graphs of Fig. 1, For contributions without an explicit∆(1232), the only expansion scale in this regime is P ≡ δ2, and the power counting is that ofHBχPT, with only even powers of δ contributing [63]. Since we use the gauge v ·A = 0, directγN couplings do not occur at lowest order. Thus in regime I, the leading, O(e2δ0 ∼ e2P 0),contribution is the “seagull” diagram Fig. 1(i) which gives the Thomson amplitude; the restof the diagrams give the pion-pole contribution, and the nucleon Born amplitude expandedto O(1/M3

N). The 4th-order seagull graph, diagram (iii)(b), however, also contributes to thestructure amplitude in the form of the short-distance contributions δαE1, δβM1 to the scalarpolarisabilities of Eq. (3).

The second class of contributions comprises the pion-loop diagrams of Fig. 2. The leadingcontributions are O(e2δ2 ∼ e2P ) and give the polarisability contributions of Eq. (1). In dimen-sional regularisation the full amplitude contains only two divergences at order O(e2δ4), whichare cancelled by δαE1 and δβM1. The finite parts of these LECs encode contributions to thepolarisabilities from mechanisms other than soft-pion loops or Delta contributions, i.e. fromshort-distance effects. In contrast, the spin polarisabilities are still parameter-free predictionsat this order. Two- and higher-loop diagrams and contributions from higher-order Lagrangiansenter only at O(e2δ6), and are thus suppressed. (In Sec. IID we discuss the prescription whichputs the pion-production threshold at the kinematically correct position [56, 62].)

The diagrams in the third class contain a dynamical ∆(1232), as listed in Fig. 3. The∆(1232) propagator in these graphs is

S(0)∆ (ω∆ ∼ mπ) ∝

1

∆M ± ω∆, (10)

7

FIG. 2: (Colour online) Pion-nucleon loop diagrams. Notation as in Fig. 1. Permuted and crossed

diagrams not shown.

FIG. 3: (Colour online) ∆(1232) and ∆π loop diagrams. Notation as in Fig. 1, also double line:∆(1232); shaded blob: πN∆ couplings including vertex corrections, as detailed in Sect. II C and Fig. 3below. The given order for (i) of O(e2δ−1) for ω ∼ ∆M applies only to the leading coupling at the

vertices; other contributions will contribute at higher order. Permuted and crossed diagrams notshown. However, the s- and u-channel Delta-pole graphs in (i) occur at different orders in regime II,as detailed in the text.

where ω∆ is the kinetic energy of the Delta line, which is dominantly O(mπ) in regime I. Sinceω∆ " ∆M ∼ δ, the ∆(1232) propagator scales as δ−1, in contrast to the nucleon propagator,i/ωp, which scales as P−1 ∼ δ−2. Diagrams with a Delta are thus suppressed by one orderin δ relative to the corresponding nucleon diagrams, and hence start at O(e2δ3). Here, wemade the argument using the heavy-baryon propagator for the Delta, as it is more transparent,even though we actually choose to compute the pole graphs using a relativistic propagator, as

8

FIG. 2: (Colour online) Pion-nucleon loop diagrams. Notation as in Fig. 1. Permuted and crossed

diagrams not shown.

FIG. 3: (Colour online) ∆(1232) and ∆π loop diagrams. Notation as in Fig. 1, also double line:∆(1232); shaded blob: πN∆ couplings including vertex corrections, as detailed in Sect. II C and Fig. 3below. The given order for (i) of O(e2δ−1) for ω ∼ ∆M applies only to the leading coupling at the

vertices; other contributions will contribute at higher order. Permuted and crossed diagrams notshown. However, the s- and u-channel Delta-pole graphs in (i) occur at different orders in regime II,as detailed in the text.

where ω∆ is the kinetic energy of the Delta line, which is dominantly O(mπ) in regime I. Sinceω∆ " ∆M ∼ δ, the ∆(1232) propagator scales as δ−1, in contrast to the nucleon propagator,i/ωp, which scales as P−1 ∼ δ−2. Diagrams with a Delta are thus suppressed by one orderin δ relative to the corresponding nucleon diagrams, and hence start at O(e2δ3). Here, wemade the argument using the heavy-baryon propagator for the Delta, as it is more transparent,even though we actually choose to compute the pole graphs using a relativistic propagator, as

8

J.A. McGovern, D.R. Phillips, H.W. Grießhammer: Eur. Phys. J. A49 (2013) 12

Power counting issues: V. Pascalutsa, D.R. PhillipsPhys. Rev. C67 (2007) 055202

∆(1232)

40 60 80 100 120 140 1600

5

10

15

20

25

Θ lab"45°

0 50 100 150 200 250 300 3500

100

200

300

400

40 60 80 100 120 140 1600

5

10

15

20

25

Θ lab"60°

0 50 100 150 200 250 300 3500

50

100

150

200

250

300

40 60 80 100 120 140 1600

5

10

15

20

25

Θ lab"85°

0 50 100 150 200 250 300 3500

50

100

150

200

250

40 60 80 100 120 140 1600

5

10

15

20

25

Θ lab"110°

0 50 100 150 200 250 300 3500

50

100

150

200

250

40 60 80 100 120 140 16010

15

20

25

30

35

Θ lab"133°

0 50 100 150 200 250 300 3500

50

100

150

200

250

40 60 80 100 120 140 16010

15

20

25

30

35

Θ lab"155°

0 50 100 150 200 250 300 3500

50

100

150

200

250

Ωlab

dΣd%lab

FIG. 9: (Colour online) Results of fits to Compton scattering data, as described in the text. Thedifferential cross section in nb/sr is plotted as a function of lab energy in MeV, at fixed lab angle. Black

and black dot-dashed curves are fourth-order Baldin-constrained and unconstrained fits respectively(with a band showing the 1 σ errors on the former); green dashed (with band) and dot-dashed curvesare the corresponding third-order fits (see Sec. IIIC 4). Parameters are given in Eqs. (25), (24), (27),

(26) respectively. The insets show the low-energy fit region, and the grey area is beyond the high-energy fit region. The magenta/purple diamonds are MAMI data, principally Refs. [33, 38], while the

black squares are the data of Hallin [31], and the yellow stars are from Blandpied Ref. [40]. For thedefinition of other symbols see Ref. [1]. The data shown are within 5 of the nominal angle.

21


the proton’s scalar electric and magnetic dipole polarisabilities. The γN∆ couplings b1 andb2 can in principle be determined from the width and E2/M1 ratio of the ∆(1232) radiativedecay.

It is, however, useful to check the consistency of the values found for these parametersagainst other constraints. For example, the Baldin sum rule relates the sum of the scalar dipolepolarisabilities at zero energy to an integral of the weighted total dissociation cross sectionσγN→X starting at the production threshold ωthr via a dispersion relation. We use the valuefrom the recent global fit of [33]:

αE1 + βM1 =1

2π2

∞∫

ωthr

dωσγN→X

ω2= 13.8± 0.4, (22)

where the canonical units, 10−4 fm3, for scalar polarisabilities, are to be understood here andbelow. The “recommended value” of Ref. [3], 13.9± 0.3, which is based on an analysis of totalphoto-absorption cross sections, differs from the value (22) by less than the uncertainty in thesum-rule evaluation.

We also note that a recent reevaluation of the dispersion integral for γ0 [81] gives:

γ0 = −γE1E1 − γM1M1 − γE1M2 − γM1E2 = −0.90± 0.08(stat)± 0.11(sys), (23)

in the units, 10−4 fm4, we use throughout for spin polarisabilities. For further discussion, seeRef. [1].

III. RESULTS

The data sets we shall use to fit our free parameters are discussed extensively in our recentreview, Ref. [1]. For the reasons explained there, in the low-energy region we include data fromRefs. [5, 6, 8–10, 29–33], but not from Refs. [4, 7]. The Baranov 150 data [9, 10] were alsoexcluded. Above the photoproduction cusp, the data of Hallin [31] rises much more stronglythan that of Olmos de Leon [33]. Hence the former was excluded from the analysis we presentedin Ref. [1]. This leaves a gap in the database between ωlab = 164 MeV and ωlab = 198 MeV, soalthough we quote a cutoff of 170 MeV on the data set for our low-energy fits, in practice wecould have chosen this to be anywhere in the range 164–198 MeV without altering the results.(For further discussion see Sec. IIIC 2 below.) Below our cutoff two further points are excludedas clear outliers, namely Federspiel (135, 44 MeV) and Olmos de Leon (133, 108 MeV). Forthe other Olmos de Leon points we follow Wissmann [82] and add a point-to-point systematicerror of 5% in quadrature with the statistical error for the Olmos de Leon data set (not shownin plots). Lastly, although we remove the data of Hallin [31] above 150 MeV, the data fromthis experiment at or below the photoproduction cusp are completely consistent with the restof the database, and so, to maximise our statistical power, we include it in our fits.

We float the normalisation of each of these data sets within the quoted normalisation un-certainty, by using the standard augmented χ2 function as given in Eq. (4.19) of Ref. [1]; seeRef. [83] for more details. (The MacGibbon data [32] is counted as two data sets since thehigher-energy untagged data is treated separately).

16

B. Results with the O(e2δ4) low-energy amplitude

We fit both without and with the Baldin-sum-rule constraint αE1 + βM1 = 13.8 ± 0.4 (seeSec. II E). The statistical errors are obtained from the χ2

min+1 ellipsoid (ellipse). Since b1 is fitto an independent higher-energy data set, its small statistical error in principle feeds through toproduce an additional small error on the low-energy parameters, but this is negligible comparedto other uncertainties.

Our result then is, for the fit without the Baldin sum rule constraint:

αE1 = 11.7± 0.7(stat)± 0.6(theory), βM1 = 3.8± 0.55(stat)± 0.6(theory) (24)

with γM1M1 = 2.9± 0.65(stat) and b1 = 3.62± 0.02(stat), as well as a χ2 of 110.5 for 134 d.o.f.For the Baldin-constrained fit, we obtain αE1 − βM1 = 7.5± 0.7(stat)± 0.6(theory) or

αE1 =10.65± 0.35(stat)± 0.2(Baldin)± 0.3(theory),

βM1 =3.15∓ 0.35(stat)± 0.2(Baldin)∓ 0.3(theory),(25)

with γM1M1 = 2.2 ± 0.5(stat) and b1 = 3.61 ± 0.02(stat), with a χ2 of 113.2 for 135 d.o.f.If we instead take b1 = 3.8 from Ref. [78], the Baldin-constrained value of βM1 drops onlyslightly from 3.15 to 3.0, well within the statistical uncertainty. In both parameter sets we havealso included estimates of the individual residual theoretical uncertainties from higher-ordercorrections. Before we can justify them in Sec. IIIC 4, we need to explore the sensitivity of thefits to choices of the parameters and data sets, as well as convergence issues.

The statistical errors are calculated from the boundaries of the χ2min + 1 ellipsoid or ellipse.

In contrast, our 1σ regions in 3d-parameter space are calculated according to χ2 = χ2min + 3.5

(or χ2min+2.3 for 2d), and encompass 68% of the probability. The projections of such 1 σ regions

in αE1-βM1-γM1M1 parameter space onto the αE1-βM1 plane are shown in Fig. 8, together withthe Baldin-constraint band. We can see that the two extractions are fully compatible withone another, as we had required for an acceptable fit. We have also checked that the χ2 isnot unacceptably large for any individual data set, and that none of the normalisations floatsbeyond the quoted systematic error. In particular, the 65 Olmos de Leon points are fit witha χ2 of 69 (unconstrained) and 71 (constrained) with negligible floating of the normalisation.Our slightly restricted set of world low-energy data is clearly highly compatible, and indeed theerrors on some sets seem to be overstated.

In Fig. 9 the results are shown (along with third-order fits close to those of Ref. [1] whichwill be discussed later), together with all the world data, including those data sets that we didnot include in our fits. As expected, the differences between constrained and unconstrainedfits are small. Essentially no systematic deviation from the trend of the low-energy data isvisible. The trend of the high-energy Mainz data is well captured, except for a tendency to fallsomewhat low at forward angles.

For reference, with b1 = 3.61 and b2 = −0.34b1, the other three spin-polarisabilities take thevalues γE1E1 = −1.1, γE1M2 = −0.4 and γM1E2 = 1.9 (excluding the π0 pole contribution whichis -45.9 for γπ). See table 4.2 of the review [1] for values in other EFTs and in dispersion-relationextractions.

19

B. Results with the O(e2δ4) low-energy amplitude

We fit both without and with the Baldin-sum-rule constraint αE1 + βM1 = 13.8 ± 0.4 (seeSec. II E). The statistical errors are obtained from the χ2

min+1 ellipsoid (ellipse). Since b1 is fitto an independent higher-energy data set, its small statistical error in principle feeds through toproduce an additional small error on the low-energy parameters, but this is negligible comparedto other uncertainties.

Our result then is, for the fit without the Baldin sum rule constraint:

αE1 = 11.7± 0.7(stat)± 0.6(theory), βM1 = 3.8± 0.55(stat)± 0.6(theory) (24)

with γM1M1 = 2.9± 0.65(stat) and b1 = 3.62± 0.02(stat), as well as a χ2 of 110.5 for 134 d.o.f.For the Baldin-constrained fit, we obtain αE1 − βM1 = 7.5± 0.7(stat)± 0.6(theory) or

αE1 =10.65± 0.35(stat)± 0.2(Baldin)± 0.3(theory),

βM1 =3.15∓ 0.35(stat)± 0.2(Baldin)∓ 0.3(theory),(25)

with γM1M1 = 2.2 ± 0.5(stat) and b1 = 3.61 ± 0.02(stat), with a χ2 of 113.2 for 135 d.o.f.If we instead take b1 = 3.8 from Ref. [78], the Baldin-constrained value of βM1 drops onlyslightly from 3.15 to 3.0, well within the statistical uncertainty. In both parameter sets we havealso included estimates of the individual residual theoretical uncertainties from higher-ordercorrections. Before we can justify them in Sec. IIIC 4, we need to explore the sensitivity of thefits to choices of the parameters and data sets, as well as convergence issues.

The statistical errors are calculated from the boundaries of the χ2min + 1 ellipsoid or ellipse.

In contrast, our 1σ regions in 3d-parameter space are calculated according to χ2 = χ2min + 3.5

(or χ2min+2.3 for 2d), and encompass 68% of the probability. The projections of such 1 σ regions

in αE1-βM1-γM1M1 parameter space onto the αE1-βM1 plane are shown in Fig. 8, together withthe Baldin-constraint band. We can see that the two extractions are fully compatible withone another, as we had required for an acceptable fit. We have also checked that the χ2 isnot unacceptably large for any individual data set, and that none of the normalisations floatsbeyond the quoted systematic error. In particular, the 65 Olmos de Leon points are fit witha χ2 of 69 (unconstrained) and 71 (constrained) with negligible floating of the normalisation.Our slightly restricted set of world low-energy data is clearly highly compatible, and indeed theerrors on some sets seem to be overstated.

In Fig. 9 the results are shown (along with third-order fits close to those of Ref. [1] whichwill be discussed later), together with all the world data, including those data sets that we didnot include in our fits. As expected, the differences between constrained and unconstrainedfits are small. Essentially no systematic deviation from the trend of the low-energy data isvisible. The trend of the high-energy Mainz data is well captured, except for a tendency to fallsomewhat low at forward angles.

For reference, with b1 = 3.61 and b2 = −0.34b1, the other three spin-polarisabilities take thevalues γE1E1 = −1.1, γE1M2 = −0.4 and γM1E2 = 1.9 (excluding the π0 pole contribution whichis -45.9 for γπ). See table 4.2 of the review [1] for values in other EFTs and in dispersion-relationextractions.

19

J.A. McGovern, D.R. Phillips,

H.W. Grießhammer: Eur. Phys. J. A49 (2013) 12

Fits to differential Compton scatteringcross sections

Baldin sum rule incorporated

· 10−4

fm3


High-accuracy analysis of Compton scattering in Chiral Effective Field TheoryH.W. Grießhammer, D.R. Phillips, J.A. McGovern arXiv:1306.2200 [nucl-th]

Proton polarizabilities

Neutron polarizabilities

α(p)E1 = [10.7 ± 0.4stat ± 0.2Baldin ± 0.3theory] · 10−4 fm3

α(n)E1 = [11.1 ± 1.8stat ± 0.2Baldin ± 0.8theory] · 10−4 fm3

β(p)E1 = [3.1 ∓ 0.4stat ± 0.2Baldin ∓ 0.3theory] · 10−4 fm3

β(n)E1 = [4.2 ∓ 1.8stat ± 0.2Baldin ∓ 0.8theory] · 10−4 fm3

Figure 1: Tree diagrams at O(ε3). Solid, double and wiggly lines denote nucleons, deltas and photons,

in order. The filled circle is an insertion from L(2)πN∆.

(EFT) to include the delta based on the so-called covariant small scale expansion to O(ε3). We usethe explicit form of the spin-3/2 propagator from Ref. [14]. Also, we do not use infrared regularizationas done in Ref. [10] but rather utilize dimensional regularization. For the case at hand, this is aconsistent scheme as counter terms in the four-point function only show up at O(ε5). Therefore, nopower-counting violating contributions appear up-to-and-including O(ε4) and the corresponding loopcorrections to V2CS are all finite after mass and coupling constant renormalization. Note, however,that one has to deal with some LECs in the three-point functions that appear as parts of the fourthorder diagrams. In case of nucleon intermediate states, these are nothing but the anomalous magneticmoment of the proton and the neutron, see also Ref. [10]. In case of delta intermediate states, we have

in addition dimension two LECs from L(3)πN∆, that can be fixed from ∆ → N transition form factors.

The chiral Lagrangian for V2CS in the pion-nucleon sector is given in Ref. [10]. The pertinentnew Lagrangian structures related to the inclusion of the spin-3/2 fields read (for the constructionprinciples, see [13])

L(1)πN∆ = hA ψµ

i ωiµN + h.c. , (15)

L(1)π∆∆ = ψµ

i (i /Dijµν −m∆γµνδ

ij)ψνj , (16)

L(2)πN∆ =

1

2b1 ψ

µi if

i+µαγ

αγ5 N + h.c. , (17)

with

/Dijµν = γµνα Dα

ij , Dαij = (∂α + Γα) δij − iεijk〈τkΓα〉 ,

Γα =1

2[u†, ∂µu]−

i

2u†(vµ + aµ)u−

i

2u(vµ − aµ)u

†

f i+µα =

1

2〈τ if+µν〉 , ωi

µ =1

2〈τ iuµ〉 ,

γµνα =1

4

[γµ, γν ], γα

, γµν =1

2[γµ, γν ] . (18)

Here, ψµi is a conventional Rarita-Schwinger spinor for the spin-3/2 fields, N denotes the nucleon bi-

spinor (throughout, we work in the isospin limit), hA is the leading πN∆ axial-coupling (analogous togA in the pion-nucleon sector) and b1 is the leading photon-nucleon-delta coupling of chiral dimension

two (much like the nucleon magnetic moment that appears first in L(2)πN ). As usual, the pions are

collected in the matrix-valued field U(x) = u2(x). We only need the external vector source vµ = QAµ,with Aµ the photon field and Q = (1, 0)e the nucleon charge matrix. Therefore f+µν = Fµν(uQu† +u†Qu).

Based on this, we are now in the position to calculate V2CS at low energies in the covariantSSE. At order O(ε3), we have tree and the leading one-loop graphs involving the ∆-resonance, seeFigs. 1 and 2, respectively. The corresponding third order pion-nucleon loop graphs are e.g. displayed

5

Figure 2: Delta-loop diagrams at O(ε3). Solid, double, dashed and wiggly lines denote nucleons,deltas, pions and photons, in order. Crossed graphs are not shown.

in Fig. 1 of Ref. [10]. Note that since the delta propagator is of O(ε−1), the tree graphs with two

insertions from L(2)πN∆ appear first at third order in the SSE. Note further that at this order there are

no unknown low-energy constants (LECs), since the couplings hA and b1 can be determined from thedecays ∆ → Nπ and ∆ → Nγ, respectively. More precisely, the strong width of the ∆ is given interms of the LEC hA as

Γstr∆ = h2A

((m∆ −mN )2 −M2π)

3/2((m∆ +mN )2 −M2π)

5/2

192F 2π πm5

∆

= (118 ± 2) MeV , (19)

and similarly for the electromagnetic width in terms of b1

Γem∆ = e2b21

(m2∆ −m2

N )3(3m2∆ +m2)

576πm5∆

, (20)

with Γem∆ /(Γem

∆ + Γstr∆ ) = 0.55 . . . 0.65%. The predictions for the generalized spin polarizabilities and

other moments of the spin structure functions are thus parameter-free. It is also important to stressthat in the covariant scheme employed here one has more loop diagrams at leading order as comparedto the heavy baryon approach (cf. Fig. 2 in [8]). In that approach, the “missing” graphs only appearat fourth order due to the additional counting in the inverse baryon mass. Here, we have to deal with14 different topologies as shown in Fig. 2. None of them involves the leading, dimension-two ∆Nγ-vertices, such contributions only start at O(ε4). Still, the algebra to evaluate the diagrams shown inFig. 2 is non-trivial. In particular the box-type diagrams which three spin-3/2 propagators generatea large number of terms, as the spin-3/2 field propagator is given by

Sµν =p/+m∆

p2 −m2∆

(

−gµν +1

3γµγν +

1

3m∆(γµpν − γνpµ) +

2

3m2∆

pµpν)

. (21)

For example, the box diagram (right-most graph in the upper row of Fig. 2) has 53 = 125 times moreterms than the corresponding pion-nucleon box graph. Therefore, we have developed our own algebraicprogram that combines FORM [15] and Mathematica to calculate the tree and the loop diagrams. Thecode is able to reduce tensor integrals of any rank in the relativistic and the heavy baryon formalism.

6

SPIN POLARIZABILITIES of the NUCLEON

Covariant Chiral Effective Field Theory with spin-3/2 degrees of freedom

Strong sensitivity to ∆(1232) input parameters

-4

-2

0

0 0.05 0.1 0.15

! 0n (Q

2 )

Q2 [GeV2]

-4

-3

-2

-1

0

0 0.05 0.1 0.15

! 0p (Q

2 )

Q2 [GeV2]

Figure 3: Forward spin polarizability in units of 10−4 fm4 at finite photon virtuality for the neutron(left) and the proton (right). Neutron data: Ref. [3] and proton data from Ref. [17] (Q2 = 0) andRef. [4] (Q2 > 0). Only statistical errors are shown.

already found in Ref. [10], the corrections from tree-level delta graphs are large in the forward spin-polarizabilities whereas the delta loop corrections for γn,p0 are very small.#6 This is different for thetransverse-longitudinal polarizabilities, where the tree contributions are suppressed (as it was alsofound in the heavy baryon calculation of [8]). We note that the parameter-free prediction for γp0agrees within 1.5 σ with the empirical number, γp0 = −1.00± 0.08± 0.12 [17]. We note that the latternumber is obtained using the well-known sum rule for γ0 in terms of the measured difference of thephoton-proton cross sections with helicity 1/2 and 3/2 for photon energies between 200 and 1800 MeVcombined with the MAID2003 prediction for the region between the threshold and 200 MeV. Thisis a clear improvement as compared to earlier calculations employing either HBCHPT with explicitdeltas or the covariant O(q4) calculation adding tree-level ∆-contributions. Of course, before one canclaim success, one must consistently evaluate the fourth order contributions from nucleon and deltaintermediate states. We also note the marked difference in the delta-loop contribution to δp0 . Whilein the heavy baryon scheme this contribution is small, it is sizeable in our relativistic approach. Thiscan be largely traced back to the box diagram (the right-most diagram in the upper row of Fig. 2). AsmN and m∆ (with the splitting fixed) tend to infinity, the contribution from this diagram vanishes,whereas its value is 1.32 · 10−4 fm4 in our case. We have analyzed the 1/mN -expansion#7 of the boxdiagram. We find that it only starts to contribute at O(1/m2

N ) and that the convergence of the seriesis very slow. The effect is much more dramatic for the proton than for the neutron due to a muchlarger prefactor. This shows that the heavy baryon expansion does not provide a good approximationto the covariant result for this observable.

Next, we consider the various observables at finite photon virtuality. In Fig. 3, we show the neutron(left panel) and proton (right panel) forward spin polarizabilities for virtualities Q2 ≤ 0.15GeV2. Forthe neutron, there is only one data point at Q2 = 0.1GeV2, which lies slightly above the predictions.The trend of the proton data is not recovered, the discrepancy between the chiral prediction and thedata grows with increasing photon virtuality. This is also reflected in the deviations of the isoscalarand isovector combinations at Q2 = 0.1GeV2 given in Ref. [18]. To get more insights into these trends,we display the same decomposition for γp,n0 (Q2) at Q2 = 0.1GeV2 as given at the photon point in

#6In this work we use dimensional and not infrared regularization as in Ref. [10]. For this reason only qualitativecomparison is possible between our results and that of [10].

#7The 1/mN expansion with fixed ∆ = m∆ −mN leads to the usual heavy baryon SSE.

8

V. Bernard, E. Epelbaum, H. Krebs, U.-G. MeißnerPhys. Rev. D87 (2013) 054032

-2.4

-2

-1.6

-1.2

-0.8

0 0.05 0.1 0.15

I An (Q

2 )

Q2 [GeV2]

-3

-2.5

-2

-1.5

-1

0 0.05 0.1 0.15

I Ap (Q

2 )

Q2 [GeV2]

Figure 5: Generalized GDH integral IA for the neutron (left) and the proton (right). Neutron data:Ref. [2]. The two data points at the same value of Q2 refer to different extraction methods as describedin [2]. Only the dominant systematic errors are shown.

Finally, theQ2-dependence of the first moment Γ1(Q2) for the proton and the isovector combination

Γ(p−n)1 (Q2) are displayed in Fig. 6. While the O(ε3) contributions slightly improve the chiral prediction

for the proton, more curvature from the pion-nucleon and pion-delta loop graphs at fourth orderis required. This again points towards the necessity of performing such a complete fourth ordercalculations within the framework outlined here. However, we note that the third order calculation

already describes the admittedly relatively imprecise data for the isovector combination Γ(p−n)1 (Q2)

taken from Ref. [18]. Therefore, in this combination the fourth order corrections should largely cancel,which was found to be the case in the heavy baryon approach [12] but not in the infrared regularizedcovariant calculation [9].

-0.06

-0.04

-0.02

0

0 0.05 0.1 0.15

!1p (Q

2 )

Q2 [GeV2]

-0.01

0

0.01

0.02

0.03

0.04

0 0.05 0.1 0.15

!1(p

-n) (Q

2 )

Q2 [GeV2]

Figure 6: First moment of the integral I1(Q2) for the the proton (left) and the isovector nucleon (p−n)(right). The proton data are from Ref. [4] and the isovector data from Ref. [18]. Only statistical errorsare shown.

10

VCS forward spin polarizabilty 1st moment of spin structure fct.

. . . questions remaining

Explicit DEGREES of FREEDOM∆(1230)

Large spin-isospin polarizability of the Nucleon

β∆ =g2A

f2π(M∆ − MN)

∼ 5 fm3

M∆ − MN " 2 mπ << 4π fπ

(small scale)

N Nπ

π

∆

strong 3-body interaction

N

N

N

π

π

dominance of

Pionic Van der Waals - type intermediate range central potentialN. Kaiser, S. Fritsch, W. W. , NPA750 (2005) 259 N. Kaiser, S. Gerstendörfer, W. W. , NPA637 (1998) 395

Vc(r) = −

9g2A

32π2 f2π

β∆

e−2mπr

r6P(mπr)

J. Fujita, H. Miyazawa (1957)

Pieper, Pandharipande, Wiringa, Carlson (2001)

1 fm 10 fm 20km

1 fm 10 fm 20km

N

∆

Figure 4: Left: Total π+p cross section in the region of the ∆(1232) resonance (adapted from [37]).Right: Difference of polarized Compton scattering cross sections of the proton, σ3/2 and σ1/2, referringto channels with total angular momentum 3

2 and 12 , respectively. Data taken from [38]. Curves represent

dispersion relation and multipole analysis cited in [38].

4.3 Role of Explicit ∆(1232) Degrees of Freedom

The standard version of chiral meson-baryon effective field theory works with pions and nucleons only,both of which are stable particles with respect to the strong interaction. Effects of the ∆(1232) areencoded in low-energy constants such as c3 and c4 that are determined either by pion-nucleon scatteringdata or by fits to nucleon-nucleon phase shifts.

On the other hand, the mass difference between the ∆(1232) isobar and the nucleon is only about0.3GeV ! 2mπ, yet another “small” scale compared to the chiral symmetry breaking scale, 4πfπ ∼1GeV. This suggests incorporating the ∆(1232) as an additional explicit degree of freedom in theeffective field theory. In fact the ∆ isobar is by far the dominant feature in the excitation spectrum ofthe nucleon observed in pion and photon scattering measurements. An example is the strong spin-isospinexcitation observed in pion-nucleon scattering (see Fig.4, left). The ∆(1232) is also seen prominentlyin polarized Compton scattering on the proton (Fig.4, right).

In the low-energy expansion of the spin-independent πN scattering amplitude, the strong spin-isospinresponse of the nucleon manifests itself in a large “axial” polarizabilty:

α(∆)A =

g2Af 2π(M∆ −MN )

! 5 fm3 . (32)

Here the factor g2A/f2π comes from the axial vector coupling of the pion that drives the πN → ∆

transition. The mere magnitude of this polarizability, several times the volume of the nucleon itself,already illustrates the importance of this effect.

When implemented in the nucleon-nucleon interaction, the ∆ isobar plays an important role intwo-pion exchange processes such as the one shown in Fig.5 (left). This mechanism contributes a large

11

∆(1230) in pion-nucleon scattering

Explicit DEGREES of FREEDOM (contd.)∆(1230)

Kaiser et al. , Ordonez et al.

Krebs, Epelbaum, Meißner (2007)

Important physics of ∆(1230) promoted to NLO

Improved convergence

0 0.05 0.1 0.15 0.2

ρ [fm-3]-1

0

1

2

3

4

P [M

eV/fm

3 ]T=0MeV

T=5MeV

T=10MeV

T=15MeV

T=20MeVT=25MeV

T = 0

T = 25MeV 20

1510

5

NUCLEAR CHIRAL THERMODYNAMICS

π

π

N N

N N

+

Van der Waals + Pauli

contact terms

Liquid - Gas Transition atCritical Temperature T = 15 MeVc

c(empirical: T = 16 - 18 MeV)

baryon density

pressure

nuclear matter: equation of state

NUCLEAR CHIRAL (PION) DYNAMICS

N,∆

BINDING & SATURATION:

3-loop in-medium

ChEFT

+ 3-bodyforces

beam/E! = E!x0.4 0.5 0.6 0.7 0.8 0.9

RD /

MC

(sca

led

to 1

)

0.6

0.7

0.8

0.9

1

1.1

1.23 fm-4 10" 0.52 ± = 0.64 µ#

preliminary

Ni! -µ $ Ni -µCOMPASS 2009

beam/E! = E!x0.4 0.5 0.6 0.7 0.8 0.9

RD /

MC

(sca

led

to 1

)

0.6

0.7

0.8

0.9

1

1.1

1.23 fm-4 10" 0.7 ± = 1.9 #$

preliminary

Ni! -# % Ni -#COMPASS 2009

Pion polarizability signal measured at COMPASS

Muon control sample

POLARIZABILITIES of the PION

Chiral Perturbation Theory:prediction of forward and backward combinations

απ

+ + βπ

+ = (0.16 ± 0.10) · 10−4

fm3

απ

+ − βπ

+ = (5.7 ± 1.0) · 10−4

fm3

together with α

π+ = (2.64 ± 0.09) · 10

−4fm

3

J. Gasser, M.A. Ivanov, M.E. Sainio

Nucl. Phys. B745 (2006) 84

year of publication1980 1985 1990 1995 2000 2005 2010 2015

3 fm

-4 /

10!"

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0

10

20

30

40

50

Z$!%Z! Serpukhov Sigma

n+!$%p$ LebedevPACHRA

n+!$%p$ MAMI

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-!+!%$$ DM2, Mark IIPLUTO, DM1 Babusci

Mark IIDonoghue

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GIS '06

POLARIZABILITIES of the PION

J. M. Friedrich: Habilitationsschrift TU München 2012

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T39 Physics Group - HADRON POLARIZABILITIES · 2013. 12. 12. · the magnetic dipole excitation from the ground state 10) at energy E o into an excited state In> at exci- tation energy