Pion polarizabilities: Theory vs Experiment

The values of charged pion polarizabilities obtained in the framework of chiral perturbation theory at the level of two-loop accuracy are compared with the experimental result recently reported by COMPASS Collaboration. It is found that the calculated value for the dipole polarizabilities $(\alpha-\beta)_{\pi^\pm}= (5.7\pm 1.0)\times 10^{-4}\,{\rm fm}^3$ fits quite well the experimental result $(\alpha-\beta)_{\pi^\pm} = (4.0 \pm 1.2_{\rm stat} \pm 1.4_{\rm syst}) \times 10^{-4}\,{\rm fm}^3$.

The index "mod" denotes the uncertainty generated by the theoretical models used to analyze the data. The ChPT calculation was clearly in conflict with the MAMI result, see also 11 for a recent discussion.
The COMPASS collaboration at CERN has recently investigated pion Compton scattering by using the Primakoff effect. The pion polarizability has been determined This is an Open Access article published by World Scientific Publishing Company. It is distributed under the terms of the Creative Commons Attribution 3.0 (CC-BY) License. Further distribution of this work is permitted, provided the original work is properly cited.
to be 12 (α − β) π ± = (4.0 ± 1.2 stat ± 1.4 syst ) × 10 −4 fm 3 under the assumption (α+β) π ± = 0. This result is in agreement with the expectation from chiral perturbation theory. The concept of the polarizability of molecules, atoms and nuclei was applied for the first time to hadrons in Refs. 13,14,15 . By using the general properties of quantum field theory it was shown that an expansion of the Compton scattering amplitude for hadrons with spin one half in small photon energy up to the second order contains two structure parameters called the electric and magnetic hadron polarizabilities. The classical sum rule for these quantities has been derived in Ref. 16 . Further theoretical investigation of the pion polarizabilities has been pursued since the early 1970s. In the current algebra + PCAC approach of Terent'ev 17 , the fundamental low-energy theorem has been proven which allows one to relate the pion polarizability to the ratio γ = h A (0)/h V (0) of the vector and axial form factors in radiative pion decay π → eνγ. By using recent precise measurements of the pion weak form factors by the PIBETA collaboration 18 one finds α π ± = −β π ± = 2.78(10) × 10 −4 fm 3 .
There were many calculations of the pion polarizabilities by employing various models: the linear σ-model with quarks 19 , the chiral quark model 20 , the superconductor quark model 21 , some chiral models 22 and so on.
Almost all of them except Terent'ev approach predicted a value of the electric polarizability within the range which we call large-valued results. We note that models not based on a chiral Lagrangian, i.e., dispersion relations and finite-energy sum rules, also obtained the polarizability within this range of values 23,24 . The pion and kaon polarizabilities have been calculated in the quark confinement model 25 in which the emphasis is placed on quark confinement and the composite nature of hadrons. It was found for charged pions α π ± ∼ 3.6 × 10 −4 fm 3 which is smaller than the large-valued results but slightly larger than Terent'ev's prediction. The first correct calculation of the cross section γγ → ππ within chiral perturbation theory to next-to-leading order (one-loop accuracy) was performed in 26 . It was shown in 27 that chiral symmetry relates the low-energy constants (LECs) appearing in the γγ → ππ-amplitude with the axial form factor h A (0). Thus it was shown explicitly that Terent'ev's low-energy theorem follows from one-loop calculation of the γγ → ππ process within chiral perturbation theory.
Note that the axial form factor h A (0) can be expressed through the dispersion integral of the difference of the vector and axial spectral densities 28 . By using this sum rule the pion polarizability was estimated in 29 and found to be in perfect agreement with chiral perturbation theory.
An actual two-loop ChPT calculation of the γγ → ππ amplitude was done in 6 (neutral pions) and 7 (charged pions). Because the effective Lagrangian at order p 6 was not available at that time, the ultraviolet divergences were evaluated in the MS scheme, then dropped and replaced with a corresponding polynomial in the external momenta. The three new counterterms which enter at this order in the low-energy expansion were estimated with resonance saturation. Whereas such a procedure is legitimate from a technical point of view, it does not make use of the full information provided by chiral symmetry.
Later on, considerable progress has been made in this field, both in theory and experiment. As for theory, the Lagrangian at order p 6 has been constructed 30,31 , and its divergence structure has been determined 32 . This provides an important check on the above calculations: adding the counterterm contributions from the p 6 Lagrangian to the MS amplitude evaluated in 6 and in 7 must provide a scale independent result. Also in the theory, improved techniques to evaluate the twoloop diagrams that occur in these amplitudes have been developed 33 . The updated calculation of the γγ → ππ amplitude to two loops was then performed in 4 (neutral pions) and 5 (charged pions). The final results for the pion polarizabities were presented in a rather compact algebraic form. By using updated values for the LECs one obtains the values of the pion polarizabilities given in Eq. (1).
A comprehensive review of the modern status of this field maybe found in Refs. 34,35 . Finally, one has to mention that research on pion polarizabities using lattice simulation is currently conducted by several groups, see, for instance, Refs. 36,37,38,39 .

Definition of pion polarizabilities
The electric (α H ) and magnetic (β H ) polarizabilities characterize the response of hadron to two-photon interactions. These quantities must be considered as fundamental as the electromagnetic mean square radii, static magnetic moments, etc. They are defined by the expansion of the Compton scattering amplitude in small photon momenta and energies. Since our interest here is the pion polarizabilities, we plot in Fig. 1 the diagram describing the Compton scattering by charged pion. Expanding the Compton scattering amplitude in small photon momenta and ener- gies, one finds Born term It is convenient to use the linear combinations of the electric and magnetic polarizabilities: (α − β) π and (α + β) π which are obtained from the helicity flip and helicity non-flip amplitudes, respectively. As follows from the definition, the dipole pion polarizabilities are proportional to α π (β π ) ∼ α Mπ 1 Λ 2 ≈ 4 × 10 −4 fm 3 where the hadronic scale Λ ∼ 4πF π ∼ 1 GeV was used. Then a natural choice of units for the polarizabilities is 10 −4 fm 3 .

Effective Lagrangian
We consider an effective Lagrangian of QCD with two flavors in the isospin symmetry limit m u = m d =m. At next-to-next-to-leading order (NNLO), one has 2 The subscripts refer to the chiral order. The expression for L 2 is where e is the electric charge, and A µ denotes the electromagnetic field. The quantity F denotes the pion decay constant in the chiral limit, and M 2 is the leading term in the quark mass expansion of the pion (mass) 2 , M 2 π = M 2 (1 + O(m)). Further, the brackets . . . denote a trace in flavor space. In Eq. (7), we have retained only the terms relevant for the present application, i.e., we have dropped additional external fields. We choose the unitary 2 × 2 matrix U in the form The Lagrangian at NLO has the structure 2 where l i , h i denote low-energy couplings, not fixed by chiral symmetry. At NNLO, one has 31,32,30 As was shown in Ref. 40 the number of operators P i can be reduced by at least one from 57 to 56. For the explicit expressions of the polynomials K i ,K i and P i , we refer the reader to Refs. 2,31,32,30 . The vertices relevant for γγ → π + π − involve l 1 , . . . , l 6 from L 4 and several c i 's from L 6 , see below.
The couplings l i and c i absorb the divergences at order p 4 and p 6 , respectively, The physical couplings are l r i (µ, 4) and c r i (µ, 4), denoted by l r i , c r i in the following. The coefficients γ i are given in 2 , and γ (1,2,L) i are tabulated in 32 . We shall use the scale independent quantitiesl i introduced in 2 , where the chiral logarithm is l = ln(M 2 π /µ 2 ). We shall use 8 As follows from the resonance exchange model 7 The values of these constants were obtained in the ENJL model 42 (a r 1 , a r 2 , b r ) = (−8.7, 5.9, 0.38) One can see that only b r agrees in the two approaches. We shall use b r = 0.4 ± 0.4. The combinations (α ± β) π ± are independent of a r 2 and are determined precisely by the chiral expansion to two loops, once a r 1 is fixed. We will then simply display this quantity as a function of a r 1 -the result turns out to be rather independent of its exact value.

Evaluation of the diagrams
The lowest-order contributions to the scattering amplitude are described by treeand one-loop diagrams. These contributions were calculated in 26 . The two-loop diagrams are displayed in Figs. 2, 4 and 5. The two-loop diagrams in Fig. 2 may be generated according to the scheme indicated in Fig. 3, where the filled in blob denotes the d-dimensional elastic ππ-scattering amplitude at one-loop accuracy, with two pions off-shell.
The diagrams shown in Fig. 4 may be reduced to tree-diagrams by using Ward identities. They sum up to the expression where Z π is the pion renormalization constant. The function R(t) starts at order 1/F 4 π and can be obtained from the full pion propagator. Two further diagrams are displayed in Fig. 5. The first one -called "acnode" in the literature -may again be evaluated by use of a dispersion relation, see 4 . The second one is trivial to evaluate, because it is a product of one-loop diagrams. The remaining diagrams at order p 6 are shown in Fig. 6.
The evaluation of the diagrams was done in the manner described in 4,33 by invoking FORM 43 . In particular, we have verified that the counterterms from the Lagrangian L 6 32 remove all ultraviolet divergences, which is a very non-trivial check on our calculation. Furthermore, we have checked that the (ultra-violet finite) amplitude so obtained is scale independent. l1,2 ci (4) Fig. 6. The remaining diagrams at order p 6 : one-loop graphs generated by L 4 , and counterterm contributions from L 6 .

Chiral expansion for pion polarizabilities
Using the same notation as in 7 , we find for the dipole polarizabilities where with It would be interesting to numerically compare the values of ∆ ± given by Eq. (19) with those obtained in Refs. 7 . One has The results for the polarizabilities evaluated with the central values for the LECs in Eqs. (13)-(15) are shown in Table 1. The uncertainty in the prediction for Table 1. Central values of polarizabilities in units of 10 −4 fm 3 .
to one loop to two-loops (α − β) π + 6.0 5.7 (α + β) π + 0 0.16 the polarizability has two sources. First, the low-energy constants are not known precisely. Second, we are dealing here with an expansion in powers of the momenta and of the quark masses up to and including terms of order p 6 . The discussion of estimating uncertainties may be found in our paper 5 . It was shown that the value for the dipole polarizability (α − β) π ± is rather reliable -there is no sign of any large, uncontrolled correction to the two-loop result. The maximum deviation 1.0 from the central value 5.7 has been used as the final theoretical uncertainty for the dipole polarizability: The chiral expansion for the combination (α + β) π ± starts out at order p 6 so we have determined only its leading order term.

Experimental information
There are three types of experiments aiming to measure the pion polarizabilities: • The scattering of high energy pions off the Coulomb field of heavy nucleus using the Primakoff effect • Radiative pion photoproduction from the proton • Pion pair production in photon-photon collisions Schematically, they are shown in Fig. 7. The possibility to measure the pion polarizability via the Primakoff reaction was proposed in the early 1980s in 44 . The measurement of the pion-photon Compton scattering amplitude by using the Primakoff effect was performed in an experiment at Serpukhov 45 , but the small data sample led to only an imprecise value for the polarizability of α π = (6.8 ± 1.4 stat ±1.2 syst )×10 −4 fm 3 . Low statistics made it difficult to evaluate the systematic uncertainty.
COMPASS has now achieved a modern Primakoff experiment, using a 190 GeV pion beam from the Super Proton Synchrotron at CERN directed at a nickel target. It is important that COMPASS was also able to use a muon, which is point-like particle, to calibrate the experiment. The Compton π − γ → π − γ scattering is extracted from the reaction π − N i → π − γ N i by selecting events from the Coulomb peak at small momentum transfer Q 2 < 0.0015 GeV 2 . From the analysis of a sample of 63,000 events, the collaboration obtained a value of the pion electric polarizability of 12 α π = (2.0 ± 0.6 stat ± 0.7 syst ) × 10 −4 fm 3 under assumption (α + β) π = 0. The cross section for the radiative pion photoproduction γp → γπ + n has been measured at the Lebedev Institute 46 . By using an extrapolation to the pion pole in the unphysical region the value of the electric polarizability was obtained α π + = −β π + = (20 ± 12) × 10 −4 fm 3 . Similar experiment was performed at the Mainz Microtron MAMI 10 but the pion polarizability has been extracted by a comparison of the data with the predictions of two different models yielding the value (α − β) π + = (11.6 ± 1.5 stat ± 3.0 syst ± 0.5 mod ) × 10 −4 fm 3 .
Another possibility to obtain the value for the pion polarizability is to extrapolate the data from the pion pair production in photon-photon collisions γγ → ππ to the region of the Compton scattering threshold by using crossing symmetry and analyticity. Normally, the procedure involves the construction of the dispersion relations with one or two subtractions. The most recent analysis preformed in 24 produced the value (α − β) π ± = 13.0 +2.6 −1.9 × 10 −4 fm 3 which is close to the MAMI data. There are plenty of previous studies in this direction which give quite a broad region for the value of the pion polarizability. The available experimental information is shown in Table 2. Table 2. Experimental information on (α − β) π ± , in units of 10 −4 fm 3 . We indicate the reaction and data used. In 52 , 47 and 12 απ was determined, using as a constraint απ = −βπ. To obtain (α − β) π ± , we multiplied the results by a factor of 2.