Chiral anomaly and radiative coupling

Finally, we can determine the anomaly 𝐹_3𝜋 and the radiative coupling at the physical point via Eq. (7.87) and Eq. (7.34), respectively. The values are listed for the different fit strategies in Table 9.3.

Since the outcomes of the different fit variants are highly correlated with only minor differences in fit quality and very similar statistical errors, we do not compute weighted averages, but instead only perform plain averages to determine the central values. Doing so for the acceptable fits without a

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

√𝑠/GeV 0

5 10 15 20

𝜎/𝜇b

strategy II strategy IIP

Figure 9.5: The cross section at the physical point for two different fit strategies. The error bands are as in Fig. 9.4.

pole factor, i.e., averaging over strategy II and III, results in 𝐹_3𝜋 = 24(13)(1)GeV⁻³,

𝑔_{𝜌𝛾 𝜋} = [0.51(6)(4) + 𝑖0.03(13)(2)] GeV⁻¹,

|𝑔_{𝜌𝛾 𝜋}| = 0.51^+0.08_−0.05(4)GeV⁻¹,

(9.4)

with errors as in Table 9.3, while the strategies including an𝜔pole, i.e., IP, IIP, and IIIP, yield 𝐹_3𝜋 = 47(18)(1)GeV⁻³,

𝑔_{𝜌𝛾 𝜋} = [0.60(8)(4) + 𝑖0.26(18)(3)] GeV⁻¹,

|𝑔_{𝜌𝛾 𝜋}| = 0.66^+0.15_−0.12(3)GeV⁻¹.

(9.5)

Both values of 𝐹_3𝜋 are compatible with the prediction (7.30), albeit only due to their large errors.

Fits including the pole ansatz do display a better fit quality, but not at a level that would conclu-sively demonstrate the necessity of the pole. Since, further, both fit variants agree within statistical uncertainties, we conclude that the current lattice data cannot discriminate between Eq. (9.4) and Eq. (9.5) and quote the resulting spread as an additional systematic error. This error also arises due to the absence of lattice data at several different pion masses by one collaboration, forcing us to fit our representation to two data sets by two different collaborations at only two different pion masses, which makes it impossible to fix the pion-mass dependence of the subtraction functions beyond the simple ansatz (7.86). Averaging over all fit results except for strategy I, we finally quote

𝐹_3𝜋 = 38(16)(1)(11)GeV⁻³,

𝑔_{𝜌𝛾 𝜋} = [0.57(7)(4)(4) + 𝑖0.17(16)(3)(12)] GeV⁻¹,

|𝑔_{𝜌𝛾 𝜋}| = 0.60^+0.12_−0.09(3)(7)GeV⁻¹,

(9.6)

where the last error is our estimate of the systematic uncertainty associated with the parameteri-zation of the subtraction functions.

𝐹_3𝜋×GeV³ Re(𝑔_{𝜌𝛾 𝜋}) ×GeV Im(𝑔_{𝜌𝛾 𝜋}) ×GeV I 13(11)(0) 0.50(6)(4) 0.09(11)(2) IP 46(18)(1) 0.59(8)(4) 0.26(18)(3) II 23(13)(1) 0.51(6)(4) 0.02(13)(2) IIP 48(18)(1) 0.61(8)(4) 0.27(18)(3) III 26(13)(1) 0.52(6)(4) 0.05(13)(2) IIIP 48(18)(1) 0.61(8)(4) 0.27(18)(3)

Table 9.3: The anomaly and the radiative coupling at the physical point. The fit uncertainty gives the first error, the second error corresponds to the error of the lattice spacings.

The resulting value of 𝐹_3𝜋 is perfectly consistent with the chiral prediction (7.30), but carries a large uncertainty. This is the first extraction of this low-energy parameter from lattice-QCD calculations, and will improve accordingly once better data become available. The residue 𝑔_{𝜌𝛾 𝜋} is currently not known better than from an 𝑆𝑈 (3) VMD estimate [211], which suggests|𝑔_{𝜌𝛾 𝜋}| = 0.79(8)GeV⁻¹ [97], again compatible with Eq. (9.6) (within1.2𝜎).¹ The difference to the VMD es-timate increases to2.3𝜎for Eq. (9.4), while there is full agreement with Eq. (9.5). This provides a-posteriori evidence for the presence of an𝜔pole in the subtraction functions, as does the final result for the cross section shown in Fig. 9.5 when compared to the expected peak cross section around 20 𝜇b [106]. The radiative coupling has also been extracted in Ref. [108] under the assumption that the pion-mass dependence of|𝐺_{𝜌𝛾 𝜋}| = |𝑔_{𝜌𝛾 𝜋}|𝑀_𝜋/2is weak, leading to|𝑔_{𝜌𝛾 𝜋}|_[108] = 1.15(5)(3)GeV⁻¹. This value differs from the VMD estimate by3.6𝜎, a discrepancy that went unnoticed in Ref. [108]

because it is mitigated by a missing factor 2 in Eq. (17) forΓ(𝜌 → 𝜋𝛾 ) therein [217]. Moreover, our analysis shows that the uncertainties especially from the chiral extrapolations are substan-tially larger. In particular, a pion-mass independent |𝐺_{𝜌𝛾 𝜋}| renders the residue divergent in the chiral limit, while at 𝑀_𝜋 = 317MeV one has |𝑔_{𝜌𝛾 𝜋}|^𝑀_[108]^𝜋=317MeV = 0.507(20)(13)GeV⁻¹ as well as

|𝑔_{𝜌𝛾 𝜋}|^{𝑀𝜋=317MeV} = 0.552(18)(18)(0), the latter being the average (9.6) at this pion mass. We con-clude that |𝑔_{𝜌𝛾 𝜋}|instead of|𝐺_{𝜌𝛾 𝜋}| is approximately pion-mass independent, thus avoiding the di-vergence in the chiral limit.

1The branching fractions cited in Ref. [9] imply |𝑔_{𝜌𝛾 𝜋}| = 0.72(4)GeV⁻¹ for the charged channel and |𝑔_{𝜌𝛾 𝜋}| = 0.73(6)GeV⁻¹for the neutral one. However, these values derive from high-energy Primakoff measurements [212–

214] and VMD fits to𝑒⁺𝑒⁻→ 𝜋⁰𝛾data [215, 216], respectively, and thus involve a substantial model dependence.

Rescattering effects in 𝟑𝝅 decays

Rescattering of pions

We have already encountered rescattering of pions in several places. In Sec. 4.2 we mentioned that the unitarization procedure via the IAM accounts, in fact, for rescattering of pions in the 𝑠 channel. Furthermore, the basis functions of the KT equations encode rescattering, too, as stated in Sec. 7.2.2. It is now the time to make this notion of rescattering more precise, in particular, to distinguish more carefully between different kinds of rescattering. We stress that in this chapter we care mainly about qualitative insights into the nature of rescattering, so contrarily to the rest of this thesis we are not too concerned with correct bookkeeping and instead focus on the key concepts only.

10.1 Rescattering and unitarity

Consider 𝑇_𝑠, the part of the 𝜋𝜋 → 𝜋𝜋 amplitude that contains all 𝑠-channel loops as depicted in Fig. 10.1. This sum of iterated loops is precisely what we mean by𝑠-channel rescattering of pions.

For simplicity, we work in the CM frame. Denoting the vertex by𝑉and the loop by𝐿, formally we can write

𝑇_𝑠 = 𝑉 + 𝑉 𝐿𝑉 + 𝑉 𝐿𝑉 𝐿𝑉 + …

= 𝑉

∞

∑

𝑘=0

(𝐿𝑉)^𝑘

= 𝑉

1 − 𝐿𝑉

= 1

1 𝑉 − 𝐿,

(10.1)

where we used the geometric series. The loop𝐿is a special case of𝐼_IV(𝑃)as defined in Eq. (3.24), with𝑃 = (𝑠, 𝟎)and𝑓 (𝑘) = −𝑖.^1,2 Its discontinuity as defined in Eq. (7.39) can be straightforwardly

1The factor−1arises from two powers of𝑖stemming from the numerators of the propagators, while the factor 𝑖corresponds to the global factor𝑖ℳin the definition of the scattering amplitudeℳ, see Eq. (2.1), and needs to be included if we want to use the Cutkosky rules in the form of Eq. (10.2).

2Of course, the propagators are to be replaced by the full 1PI contribution. However, the only aspect that matters here is the imaginary part of the loop𝐿, which is computed by setting the propagators on-shell, see Eq. (10.2). Picking the renormalization conditions (3.59), the argument remains unaffected by this subtlety.

𝑇_𝑠≔ + + + + …

≕ +

Figure 10.1: Pion–pion rescattering in the𝑠channel. Double lines denotes the full pion propagator Δas defined in Fig. 3.1, while the thick line denotes the two-pion propagator that is defined as the sum of all iterated pion loops.

computed via the Cutkosky rules [189, 218], i.e., by performing the replacements 1

𝑘²− 𝑀_𝜋²+ 𝑖𝜖 → −2𝜋𝑖 Θ (𝑘⁰) 𝛿 (𝑘²− 𝑀_𝜋²) , 1

(𝑃 − 𝑘)²− 𝑀_𝜋²+ 𝑖𝜖 → −2𝜋𝑖 Θ ((𝑃 − 𝑘)⁰) 𝛿 ((𝑃 − 𝑘)²− 𝑀_𝜋²) , (10.2) in Eq. (3.24), with Θ defined in Eq. (3.83). Since the loop obeys the Schwarz reflection princi-ple (2.7),disc_𝑠[𝐿(𝑠)] = 2𝑖Im[𝐿(𝑠)]holds. Altogether, this results in

Im[𝐿 (𝑠)] = 𝜎 (𝑠)

16𝜋, (10.3)

with𝜎being the phase space as introduced in Eq. (2.29). By inserting Eq. (10.3) into Eq. (10.1) we finally obtain

𝑇_𝑠 = 1

1

𝑉 −Re[𝐿] −_16𝜋^𝑖 𝜎. (10.4)

Note that the vertex𝑉is real, for the Hamiltonian needs to be Hermitian, and thus the couplings in the underlying Lagrangian must be real.

The comparison of Eq. (10.4) with the single-channel 𝐾-matrix representation (2.17) shows a remarkable similarity. In fact, the factor1/(16𝜋)that spoils perfect agreement is an artifact of our sloppy bookkeeping, which ignored the partial-wave projection, see Eq. (2.3) and Eq. (2.5). Thus, Eq. (10.4) is a special case of the𝐾-matrix representation, with

𝐾 ∼ 1

𝑉 −Re[𝐿] . (10.5)

Recalling that elastic unitarity (2.16) is equivalent to the amplitude taking on the form of a 𝐾-matrix representation as shown in Sec. 2.1, we conclude that𝑠-channel rescattering generates an amplitude that fulfills elastic unitarity. By inverting the line of thought, we see that each𝐾-matrix representation at least correctly accounts for the imaginary part generated by 𝑠-channel rescat-tering, although of course to obtain the corresponding real part the correct𝐾 matrix needs to be picked. It is this argument that makes us identify pion rescattering in the 𝑠channel with elastic unitarity in the very same channel.

Figure 10.2: The two-pion propagator in the𝑡channel.

Accordingly—keeping in mind its derivation in Sec. 4.2—the IAM accounts for pion rescattering in the𝑠channel, where the 𝐾matrix has been approximated by ChPT truncated at either NLO or NNLO. In this regard, the unitarization can be interpreted as an approximated resummation of the perturbative expansion.

It is instructive to take a look at the crossed version of the two-pion propagator, see Fig. 10.2 for the𝑡-channel one. This equals the Bethe–Salpeter kernel 𝐵introduced in Sec. 3.2.5, if con-tributions to the latter of the 𝑢 channel and of higher-energetic intermediate states are ignored.

If we replace the vertex𝑉by the kernel𝐵in Eq. (10.1), we will obtain the exact representation of the scattering amplitude𝑖ℳas depicted in Fig. 3.2. The derivation of Eq. (10.4) still works out, in particular, 𝐵is real inside the elastic𝑠-channel scattering region, but the real part, and thus the𝐾 matrix, becomes much more convoluted, because it describes more physics.

Im Dokument Quark-mass dependence of pion scattering amplitudes (Seite 112-118)