1 Motivation and overview


Facets of chiral perturbation theory***To appear in the Proceedings of Hadron Structure ’13, July 2013, Tatranske Matliare, Slovakia


Gerhard Ecker

[.6cm] University of Vienna, Faculty of Physics

Boltzmanngasse 5, A-1090 Wien, Austria

Chiral perturbation theory is the effective field theory of the Standard Model at low energies. After a short introduction and overview, I discuss three topics where the chiral approach leads to a deeper understanding of low-energy hadron physics: radiative kaon decays, carbogenesis in stellar nucleosynthesis and the interplay of chiral perturbation theory and lattice QCD.

1 Motivation and overview

For a systematic and quantitative treatment of the Standard Model (SM) at low energies ( GeV), two approaches have survived the scrutiny of time:

  • Effective Field Theory (EFT)

  • Lattice Field Theory

The main objectives are to understand the physics of the SM in the hadron sector at low energies and to look for evidence of new physics.

The low-energy region is not accessible in standard perturbation theory because it is the strong-coupling regime of QCD. The key concept for the EFT approach is the approximate chiral symmetry of QCD:


For massless quarks (), the chiral components can be rotated independently, leading to the chiral symmetry of QCD with massless quarks .

Although is a very good approximation for ( quarks) and a reasonable one for (), there is no sign of chiral symmetry in the hadron spectrum. There are many additional arguments pointing to the spontaneous breakdown of chiral symmetry,


where the diagonal subgroup is either isospin () or flavour (). As a consequence, the spectrum of the theory contains massless Goldstone bosons. The associated fields parametrize the coset space :

Goldstone bosons
2 3
3 8

Even in the real world with nonvanishing quark masses, pseudoscalar meson exchange dominates amplitudes at low energies. For an EFT of pseudo-Goldstone bosons only, chiral symmetry is realized nonlinearly and the associated effective Lagrangian is necessarily nonpolynomial. The EFT of the SM at low energies is called Chiral Perturbation Theory (CHPT) [1, 2, 3] and it is a nonrenormalizable quantum field theory. Nevertheless, CHPT is a fully renormalized QFT (in practice up to NNLO) and therefore independent of the regularization procedure.

Another important consequence of Goldstone’s theorem is at the basis of the systematic low-energy expansion of CHPT: pseudo-Goldstone bosons decouple for vanishing meson momenta and masses. The systematic CHPT approach for low-energy hadron physics (for reviews, see Refs. [4, 5, 6, 7, 8]) is

  • most advanced in the meson sector (up to two loops, Table 1);

  • it is also well developed for single-baryon and few-nucleon systems;

  • electroweak interactions can be and have been included.

           ( of LECs) loop  order
 +   +   + 
 +   + 
 +   +   +   + 
 +   +   + 
Table 1: Effective chiral Lagrangian in the meson sector for chiral . In brackets, the number of coupling constants (LECs) of CHPT.

For this talk, I have chosen three topics where the main emphasis is on obtaining a better understanding of hadronic interactions at low energies rather than on high-precision studies with the potential to look for evidence of new physics (e.g., in semileptonic kaon decays). Theoretical and experimental investigations of the radiative kaon decays and span a period of more than a quarter century, from the second half of the 80s of last century where CHPT was just one of many “hadronic models” to fairly recent times where CHPT predictions have been verified experimentally. An interesting application of chiral EFTs in nuclear physics is the recent attempt to quantify the sensitivity of the so-called Hoyle state to fundamental parameters of the SM, the light quark mass and the electromagnetic fine-structure constant. The results add a new touch to the understanding of the abundance of carbon and oxygen in the universe in terms of the anthropic principle. Finally, to illustrate the fruitful collaboration between the two main players in low-energy hadron physics, CHPT and lattice QCD, I discuss ongoing attempts to extract information on some low-energy constants (LECs) from lattice simulations. I present preliminary results of an approach making use of an analytic approximation of two-loop amplitudes in chiral .

2 Nonleptonic kaon decays

Kaon decays are a fertile field for CHPT (for a general review, see Ref. [9]) . While in some semileptonic decays the precision provided by CHPT allows to search for evidence of new physics, the situation is much more complicated in nonleptonic decays. Nevertheless, a comprehensive picture has emerged over the years through the collaboration between theory and experiment. In this section, I briefly review the status of a specific subclass of radiative kaon decays.

The two basic couplings of the leading-order nonleptonic chiral Lagrangian , usually called , are by now well established from studies of the dominant nonleptonic kaon decays up to NLO, including isospin-violating and radiative corrections [10, 11, 12, 13, 14, 15].

Figure 1: One-loop diagrams for [16, 17]. For , replace by and drop the line [18, 19].

All other nonleptonic kaon decays start at NLO, , only. As indicated in Table 1, there are 22 (octet) plus 28 (27-plet) new LECs entering at NLO. Therefore, the radiative decays , and have been especially popular among CHPT theorists: none of the NLO LECs contributes! Therefore, at NLO the decay amplitudes are given by finite one-loop contributions in terms of the known LO couplings only. The channel is not only interesting in its own right because it generates a CP-conserving contribution via the two-photon cut to the dominantly CP-violating decays [20, 21, 22].

In the remainder of this section, I review the status of the decays and (the decay [23] has not been observed yet). At , the relevant diagrams are shown in Fig. 1. Note that each one of the diagrams is quadratically divergent: chiral symmetry ensures that the sum is finite.

As predicted by CHPT, already the first observation of [24] demonstrated that the two-photon spectrum is dominated by the pion-loop contribution, in contrast to the previously assumed vector meson dominance. However, it also became clear that the rate was underestimated. Higher-order corrections needed to be taken into account, starting at .

Figure 2: Experimental results for the branching ratio in comparison with the chiral prediction.
  1. Rescattering (unitarity) corrections turned out to be small for [25], but they are sizable in the case of [26, 27].

  2. Resonance contributions were estimated to be small for the decay mainly because vector mesons cannot contribute. This is again different for the decay: although the vector meson contribution is model dependent, it is to a good approximation parametrized by a single parameter [27, 28].

Figure 3: Two-photon spectra for from NA48 [29] (top) and KTeV [30] (bottom).

It therefore came as a surprise when NA48 [31] announced a rate for substantially bigger than the chiral prediction (see Fig. 2). Fortunately, the more recent result of KLOE [32], , is again in perfect agreement with expectations. The decision by the Particle Data Group [33] to average the results of NA48 and KLOE does not appear very illuminating: another experiment is needed to clarify the issue.

After several years of discrepancies, the experimental situation for has now been clarified [29, 30]. Both the two-photon spectra shown in Fig. 3 and the branching ratios agree among each other and with CHPT [33]:


As an important by-product of this result, the CP-conserving contribution is indeed negligible in comparison with the CP-violating amplitudes.

3 Carbogenesis

Figure 4: Triple- process for C.

Almost all carbon in the universe is produced in stellar nucleosynthesis via the triple- process shown in Fig. 4. In order to explain the observed carbon abundance, Hoyle [34] postulated the existence of an excited state of C near the Be- threshold that was observed soon afterwards. Two important characteristics of this resonance are the energy above the threshold and its radiative width :


Since the triple- rate is proportional to , the rate is most sensitive to . This sensitivity has often been considered a prime example of the anthropic principle, but later investigations showed that a difference keV could be tolerated to explain the abundance of C and O [35, 36].

Although this range cannot be considered extreme fine-tuning, the more interesting issue is the dependence of on fundamental parameters of strong and electromagnetic interactions. Using a one-parameter nuclear cluster model, Schlattl et al. [37] found that the tolerances for the strength parameter and the Coulomb force were indeed small:


However, the quantities and in the model of Ref. [37] are difficult to relate to fundamental parameters of QCD and QED. CHPT can provide at least a partial solution of this problem.

The chiral EFT of nuclear forces put forward by Weinberg [38] has proven to be very successful for small nuclei (). In a more recent development, the nuclear chiral EFT has been put on the lattice (for a review of nuclear lattice simulations, see Ref. [39]). The important difference to lattice QCD (see Sec. 4) is that instead of quarks and gluons the lattice degrees of freedom are now nucleons and pions. The nuclear simulations have been quite successful in calculating energy spectra of light nuclei. As an example, I reproduce in Table 2 the low-lying even-parity spectrum of C calculated by Epelbaum et al. [40].

Table 2: Energies of low-lying even-parity states of C (in MeV) from Ref. [40]. is the Hoyle resonance.

With nuclear CHPT one cannot study the influence of the strong coupling (hidden in nucleons and pions), but the impact of the light quark mass in the isospin limit (via at lowest order CHPT) and of the fine-structure constant can be investigated. The final conclusion obtained in Ref. [41] is that the necessary fine-tuning of and is much more severe than for the energy difference in Eq. (4):


While the constraint on the fine-structure constant is in accordance with the previous result in Eq. (5), the allowed range for the light quark mass adds another touch to the interpretation of the anthropic principle.

4 Low-energy constants and lattice QCD

In recent years, the collaboration between the two major players in low-energy hadron physics, CHPT and lattice QCD, has intensified considerably.

  • Extrapolation to the physical quark (and meson) masses provided by CHPT is still useful for lattice simulations, but because of more powerful computers less so than some five years ago. On the other hand, finite-volume corrections accessible in CHPT are still needed for a reliable estimate of lattice uncertainties.

  • On the other hand, the determination of LECs from lattice studies has become more important over the years. This input is especially welcome for those LECs that modulate quark mass terms: unlike in standard phenomenological analysis, the lattice physicist can tune quark (and therefore meson) masses.

The present situation can be characterized by the following motto [42], modeled after a famous quote: “Ask not what CHPT can do for the lattice, but ask what the lattice can do for CHPT”.

As an illustrative example, consider one of the two leading-order LECs, the meson decay constant in the chiral limit. The chiral LEC is well known, mainly from a combined analysis of lattice data by the FLAG Collaboration [43, 44]:


The situation is different in the case. The lattice results for cover a much wider range, from about 66 MeV to 84 MeV [44]. Consequently, the FLAG group refrains from performing an average. A similar range is covered in the phenomenological fits of Bijnens and Jemos [45] as shown in Fig. 5.

Figure 5: Results for and from various fits in Ref. [45]. is not fitted but calculated a posteriori from the fitted LECs.

The low-energy expansion in chiral is characterized by the ratio where stands for a generic meson momentum or mass. The LEC thus sets the scale for the chiral expansion. In practical work, is usually traded for at successive orders of the chiral expansion. Nevertheless, sets the scale of “convergence” of the chiral expansion: a smaller tends to produce bigger fluctuations at higher orders. It is therefore disturbing that its value is less known than for many higher-order LECs.

One clue to the difficulty of extracting is the apparent anti-correlation with the NLO LEC in the fits of Ref. [45]: the bigger , the smaller , and vice versa (see Fig. 5). The large- suppression of is not manifest in the fits with small .

This anti-correlation can be understood to some extent from the structure of the chiral Lagrangian up to and including NLO (see Table 1):


where  meson fields, ( quark condensate, is the quark mass matrix), stands for the flavour trace and denotes the lowest-order meson masses. The dots refer to the remainder of the NLO Lagrangian in the first line and to terms of higher order in the meson fields in the second line. Therefore, a LO tree-level contribution is always accompanied by an contribution in the combination


Of course, there will in general be additional contributions involving at NLO, especially in higher-point functions (e.g., in meson meson scattering). Nevertheless, the observed anti-correlation between and is clearly related to the structure of the chiral Lagrangian. Note that is the typical size of a NLO LEC. Although of different chiral order, the two terms in could a priori be of the same order of magnitude.

Independent information on comes from comparing the and expressions for . To in chiral , is given by [2]


where is a NLO LEC and is a one-loop function. Expressing in terms of , and a kaon loop contribution [3] and equating Eq. (10) with the result for [3], one arrives at the following relation:


Assuming the “paramagnetic” inequality [46] to hold already at , one gets a lower bound for ,


well compatible with existing estimates.

lattice data for seem well suited for a determination of and . For a quantitative analysis, the use of CHPT to NNLO, , is essential. In many analyses of lattice data, the complete NNLO result for in chiral [47], which is available in numerical form only, has not been employed so far. Some time ago, we proposed a large- motivated approximation for NNLO calculations in chiral where the loop amplitudes are given in analytic form [48]. In the remainder of this section, I report on a preliminary analysis of within this framework to extract the LECs , [49].

The following input is needed. The two-loop contribution depends on a single additional parameter , the scale of double logs. Comparing with a numerical analysis [50], we have convinced ourselves that as expected, at least for and for itself. In addition, some knowledge of the only other LEC appearing at NLO and of the other LECs entering at is required. The following (preliminary) results [49] take the uncertainties of and the LECs involved into account, adding errors in quadrature to the lattice errors. We use lattice data for from the RBC/UKQCD Collaboration [51, 52].

Figure 6: Fitted values of , using RBC/UKQCD data [51, 52] with MeV, with (blue ellipse) and without (green ellipse) including . The red band results from the comparison of between and as expressed by Eq. (11), taking from the FLAG compilation [44]. The horizontal line denotes MeV.

The extracted values of and are shown in Fig. 6. The ellipses describe two options, depending on whether the physical value of is included in the fit (blue ellipse) or not (green ellipse). The red band originates from the comparison between chiral and as expressed by Eq. (11). Referring to Ref. [49] for a more complete discussion, I list the values of and corresponding to the blue ellipse ( included):

Figure 7: A compilation of lattice determinations of [44].

The two ellipses are roughly compatible with each other. The green ellipse is a little lower because from the RBC/UKQCD data alone the fitted value of is slightly smaller than the experimental value. The value for is consistent with large and with available lattice results [44] shown in Fig. 7. The result for is more precise than both phenomenological (cp. Fig. 5) and existing lattice determinations [43, 44]. It is a little bigger than expected [46], approximately of the same size as the LEC in Eq. (7). Moreover, in Eq. (13) does not match with the comparison between and as indicated by the red band in Fig. 6.

Aside from possibly underestimated uncertainties, this discrepancy may be due to the fact that the red band in Fig. 6 is based on calculations whereas the fit values in Eq. (13) result from an (albeit approximate) calculation to . Note also that the value for in Eq. (7) is an average over all existing lattice results; the most precise determinations with active flavours produce a slightly bigger average MeV [44]. Nevertheless, the discrepancy between the direct fit (13) and the constraint (11) would remain.

The strong anti-correlation between and persists because the kaon masses in the RBC/UKQCD data are all close to the physical kaon mass. Simulations with smaller kaon masses [53] would not only be welcome from the point of view of convergence of the chiral series, but they could also provide a better lever arm for reducing the anti-correlation and the fit errors of and . This expectation is justified because the quantity defined in Eq. (9) is much better determined than .

5 Conclusions

We started out with stating the main objectives of CHPT: to understand the structure of the SM at low energies and to look for possible evidence of new physics. Have these objectives been accomplished?

We have certainly come some way with CHPT in understanding hadronic interactions at low energies. I discussed three examples where the CHPT approach has given rise to significant new insights. The investigation of the radiative nonleptonic kaon decays and during the past 25 years has led to an overall agreement between theory and experiment, with a minor discrepancy between two experiments for the decay still pending. Nuclear physics for light nuclei has made impressive progress with the help of chiral EFTs. A recent formulation on the lattice (with nucleons and pions) seems very promising. As an example, the impact of the light quark masses and of the fine-structure constant on the Hoyle resonance in C was studied with such an approach. Finally, the interaction between CHPT and lattice QCD is prospering. Many of the CHPT couplings that are difficult to obtain from phenomenology are now becoming accessible on the lattice.

Concerning the second objective mentioned in the introduction, we have not found any evidence for new physics with CHPT. But neither has the LHC!


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