First direct lattice calculation of the chiral perturbation theory low-energy constant $\ell_7$

We evaluate by means of lattice QCD calculations the low-energy constant $\ell_{7}$ which parametrizes strong isospin effects at NLO in $\rm{SU}(2)$ chiral perturbation theory. Among all low-energy constants at NLO, $\ell_{7}$ is the one known less precisely, and its uncertainty is currently larger than $50\%$. Our strategy is based on the RM123 approach in which the lattice path-integral is expanded in powers of the isospin breaking parameter $\Delta m= (m_{d}-m_{u})/2$. In order to evaluate the relevant lattice correlators we make use of the recently proposed rotated twisted-mass (RTM) scheme. Within the RM123 approach, it is possible to cleanly extract the value of $\ell_{7}$ from either the pion mass splitting $M_{\pi^{+}}-M_{\pi^{0}}$ induced by strong isospin breaking at order $\mathcal{O}\left((\Delta m)^{2}\right)$ (mass method), or from the coupling of the neutral pion $\pi^{0}$ to the isoscalar operator $\left(\bar{u}\gamma_{5}u + \bar{d}\gamma_{5} d\right)/\sqrt{2}$ at order $\mathcal{O}(\Delta m)$ (matrix element method). In this pilot study we limit the analysis to a single ensemble generated by the Extended Twisted Mass Collaboration (ETMC) with $N_{f}=2+1+1$ dynamical quark flavours, which corresponds to a lattice spacing $a\simeq 0.095~{\rm fm}$ and to a pion mass $M_{\pi}\simeq 260~{\rm MeV}$. We find that the matrix element method outperforms the mass method in terms of resulting statistical accuracy. Our determination, $\ell_{7} = 2.5(1.4)\times 10^{-3}$, is in agreement and improves previous calculations.


I. INTRODUCTION
Chiral Perturbation Theory (ChPT) represents a powerful theoretical framework to describe the low-energy dynamics of QCD taking full advantage of the consequences of spontaneous chiral symmetry breaking. The ChPT Lagrangian is organized as a power expansion in terms of the external momenta and quark masses and, depending on whether only light quarks are considered or the strange quark is included, one has SU (2) or SU (3) ChPT. The chiral expansion is then written in terms of low-energy constants (LECs) whose values are fixed by matching a number of observables to the predictions of fundamental QCD or to their experimental determination. The ChPT Lagrangian at LO contains only two LECs: the pion decay constant f π ∼ 132 MeV and the parameter B 0 proportional to the chiral condensate ψ ψ , while at NLO the SU(2) ChPT Lagrangian is parametrized, apart from contact terms, by the seven LECs { i } i=1,...,7 [1]. Strong isospin-breaking (IB) effects at NLO are all parametrized by 7 , which corresponds to the only LEC that couples to an operator function of the up-down quark mass difference m u − m d . In the seminal papers by Gasser and Leutwyler [1,2], a phenomenological estimate of the values of the NLO LECs { i } i=1,...,7 , based on the available experimental informations, was given. 7 enters any ChPT-based calculations where IB effects play an important role, and for this reason a first principle and high precision evaluation of its value is important for several phenomenological analyses. To give some example, according to Ref. [3] the prediction at NLO of the axion mass m a = 5.70(6)(4) µeV 10 12 GeV/f a , where f a is the axion decay constant, has an overall uncertainty of order O(1%) where the first source of error comes from the uncertainty on the up-down quark mass ratio m u /m d , while the second one is due to the uncertainty on the value of 7 . A precise determination of 7 turns out to be even more crucial in the evaluation at NLO of the axion quartic self-coupling λ a = −0.346(22) m 2 a /f 2 a , where the resulting 6% error is completely dominated by the uncertainty on 7 . More recently [4], the axion-pion scattering process aπ → ππ has been computed in SU (2) ChPT at NLO in order to probe the convergence of the chiral expansion of the axion-pion thermalization rate Γ a (T ), from which it is possible to put the so-called hot dark matter (HDM) bounds on the axion mass. In this case, the uncertainty on the value of 7 produces a 15 − 20% uncertainty in the amplitude for aπ → ππ at NLO.
Among the LECs 7 turns out to be one affected by the largest uncertainty. A first estimate of its value was obtained by matching the charged/neutral pion mass splitting M π + − M π 0 parametrized by 7 in SU(2) ChPT at NLO to the value predicted by SU (3) ChPT at LO and due to η − π 0 mixing [1]. This gives, for this constant which is scheme and scale independent at NLO, the relation Higher order corrections to the SU(3) tree level prediction, which depend on the knowledge of the SU(3) LECs at NLO, however, turn out to be numerically of the same size as the LO prediction of Eq. (1), giving rise to a large systematic uncertainty, namely 7 = 7(4) × 10 −3 [3]. A first attempt to determine the value of 7 from lattice QCD simulations has been made by the RBC-UKQCD Collaboration [5], where the NLO and NNLO partially quenched SU(2) LECs have been extracted by means of a global fit to various pseudoscalar masses and decay constants. However, the value 7 = 6.5(3.8) × 10 −3 that has been reported, being obtained from a NNLO ChPT global fit, is strongly correlated to the other LECs which contribute at NLO to the dependence of the fitted meson masses and decay constants on the light quark masses.
In this paper we propose to determine 7 directly from first principle lattice QCD simulations, evaluating IB effects within the RM123 approach [6,7], in which the path-integral is expanded around the isosymmetric point m d = m u in powers of ∆m = (m d − m u ) /2. We use the rotated twisted-mass (RTM) scheme [8], which have been shown to reduce the statistical noise of some mesonic lattice correlation functions. We employ two different strategies to determine 7 : the first one (in the following mass method ) is based on the computation of the charged/neutral pion mass splitting M π + − M π 0 at order O (∆m) 2 , while the second one (in the following matrix element method ) consists in extracting 7 from the coupling Z P 0 π 0 = 0|P 0 |π 0 of the neutral pion to the isoscalar operator at leading order O(∆m). The clear advantage of our strategy is represented by the fact that within the RM123 approach one evaluates directly the derivatives in ∆m of both M π + − M π 0 and Z P 0 π 0 which, being proportional to 7 , allow for a clean extraction of its value. This is different from what happens in the global fit procedure, where the value of 7 must be evaluated together with the other LECs which enter at the same and lower order in ChPT, by fitting the combined quark mass dependence of several meson masses and decay constants.
We make use of the gauge configurations produced with Wilson-clover TM fermions by the Extended Twisted Mass Collaboration (ETMC) [9,10]. For this feasibility study, we limit our simulations to a single value of the lattice spacing a 0.095 fm and to an higher-than-physical pion mass M π 260 MeV, postponing the extrapolation to the continuum and chiral limit to a future work. We also note, however, that any residual pion mass dependence which is left in our present estimate of 7 represents a NNLO (or higher order) effect in ChPT. This is only relevant for phenomenological applications which aim to an accuracy beyond NLO in the chiral expansion, where many other unknown LECs are involved in any case.
The remaining of the paper is organized as follows: in Sec.
[II] we briefly introduce the mass method and the matrix element method. In Sec.
[III] we derive the diagrammatic expansion of the relevant lattice correlators in the RTM scheme, describing the procedure we used to relate them to the pion mass splitting M π + − M π 0 and to the matrix element Z P 0 π 0 . In Sec.
[IV] we present our numerical results, and finally in Sec.
[V] we draw our conclusions.

II. THE MASS METHOD AND THE MATRIX ELEMENT METHOD
In this section we describe the two methods that, following Gasser and Leutwyler [1], we considered for a direct determination of 7 . The mass method relies on the fact that 7 parametrizes the charged/neutral pion mass difference induced by QCD IB through where f π is the pion decay constant normalized as f π 132 MeV. Expanding the l.h.s. of the previous equation using and by noticing that at LO in the chiral expansion M 2 This allows to compute 7 through where we emphasized that the difference (M π + − M π 0 ) QCD in Eq. (6) indicates only the pure QCD contribution to the pion mass splitting, which is subdominant with respect to the leading QED contribution of O(α em ).
The matrix element method relies instead on the fact that, away from the isosymmetric limit, i.e. for different up and down quark masses, the neutral pion has a non vanishing iso-singlet component the size of which is quantified by the matrix element The matrix element Z P 0 π 0 is directly proportional to 7 through [1] which allows to determine 7 via Eqs. (6) and (9) show that in a mass independent scheme 7 is a dimensionless RGI quantity. Therefore it can be expressed equivalently in terms of the bare lattice quantities or the renormalized ones.
To evaluate the pion mass splitting at order O (m u − m d ) 2 and the matrix element we adopt the RM123 method which is based on the Taylor expansion of the QCD path-integral around the isosymmetric point [6,7]. In order to reduce the statistical noise of the correlators involved in the calculation, we will make use of the RTM scheme introduced in Ref. [8].
For completeness this scheme will be briefly introduced in the next section, the interested reader is referred to Ref. [8] for more details.

III. RM123 EXPANSION IN THE RTM SCHEME
The lattice QCD RTM Lagrangian of the light doublet ψ = (u , d ), is given by [8] where ∆m = 1 2 (m d − m u ), ∇ µ is the lattice symmetric covariant derivative, written in terms of the forward (∇ µ ) and backward (∇ * µ ) covariant derivatives, and W (m cr ) is the critical Wilson term, which includes the mass and is globally odd under r → −r, The quark fields u , d appearing in the RTM Lagrangian, which are regularized in Eq. (10) with opposite values of the Wilson parameter r = ±1, are not the physical ones. They are related to the physical up and down quark fields u and d, through The RTM Lagrangian is not equivalent to the standard twisted mass Lagrangian. It can be shown [8] that rotating back to the physical doublet ψ = (u, d), the RTM Lagrangian coincides with the regularization proposed in Ref. [11] and adopted by the ETMC to discretize the Lagrangian of the heavy doublet ψ h = (c, s). From this observation, it follows that the Lagrangian of Eq. (10) inherits all the benefits of the standard twisted mass regularization, including the O(a 2 ) improvement of parity even observables at maximal twist.
In order to determine the diagrammatic expansion (in the RTM basis) of M π + − M π 0 at order O(∆m 2 ) and the matrix element Z P 0 π 0 at order O(∆m) it is necessary to establish the relation between the physical correlators C π + π + (t) − C π 0 π 0 (t) and written in terms of the physical quark fields u and d, and the correlators written in the rotated basis. Using Eq. (14), it is straightforward to show that such relations are given by where We now discuss the RM123 expansion of the correlators appearing in the r.h.s. of Eqs. (15) and (16), respectively at second and first order in ∆m, postponing to the next subsection the description of the relations between the correlators and the physical observables that we want to extract.
In full generality, the expansion of the VEV of a given observable O up to second order in ∆m, can be written as where · 0 denotes the average in isosymmetric QCD. For the correlators C π + π − (t), C P 0 π + (t) and , the expansion reads (the latter two must be expanded at leading order only) where P P (x) is an interpolator of the meson P . In the previous expression it is implied that the interpolating fields are projected to zero three momentum p = 0. In all cases, the leading zeroth order term vanishes because in the isosymmetric limit there is no mixing between the rotated pions, and they also do not possess an isoscalar component. Moreover, the order O(∆m) term in the expansion of C π + π − also vanishes (as expected), because at least two insertions of the perturbation L IB are needed in order to convertū ↔d and have a mixing between π + and π − .
In the physical basis, this corresponds to the fact that the pion correlators are symmetric with respect to u ↔ d and can receive, therefore, only corrections proportional to even powers of ∆m.
Performing the corresponding Wick contractions and then taking the isosymmetric limit, one obtains the following diagrammatic expansion for the previous correlators 1 The

A. From correlators to the physical observables
We now discuss how to relate the correlators defined in the previous section to the charged/neutral pion mass difference (mass method) and to the coupling Z P 0 π 0 of the neutral pion to the isoscalar density (matrix element method) induced by strong IB.
Let us start from the mass method, and consider the correlators of the charged and neutral pion in the physical basis. In the complete theory, where the perturbation ∆mL IB is treated to all orders, one has where T is the temporal extent of the lattice, the dots indicate excited-state contributions which from now on will be neglected assuming the ground state dominance, and for a meson P the amplitude A P is related to the matrix element Z P P = P |P † P |0 , through At fixed value of m , the masses M π + , M π 0 and the amplitudes A π + , A π 0 are implicitly functions of the quark mass difference. Hence we can express the derivatives of the correlators w.r.t. ∆m in terms of the derivatives of amplitudes and masses. For the generic correlator C P P (t) the first derivative in ∆m is given by (Q = dQ/d(∆m)) while for the second derivative one gets The first and second derivatives must be evaluated at ∆m = 0, where several simplifications occur: the first derivative of the pion masses and amplitudes at ∆m = 0 vanishes and one also has This implies that The isosymmetric pion mass, M π , and amplitude, A π , can be computed from the ground state of the isosymmetric pion correlator C isoQCD Given that at second order in ∆m the charged/neutral pion mass difference at order O((∆m) 2 ) can be extracted through a standard "effective slope" analysis in time of the diagrams displayed in Eq. (22), as it will be explained in the next section.
Let us now discuss the matrix element method and consider first the correlator C P 0 π 0 (t) in the physical basis. Inserting a complete set of eigenstates between the pion source and the isoscalar operator, one gets where the dots correspond to terms that are exponentially suppressed w.r.t. the neutral pion contribution for large time separations t a, T − t a. Assuming the ground state dominance we have where As in the previous case, the matrix elements Z P 0 π 0 , Z π 0 π 0 and the neutral pion mass M π 0 are implicitly function of ∆m. Therefore we can expand C P 0 π 0 at first order in ∆m, in terms of the derivatives of Z P 0 π 0 , Z π 0 π 0 and M π 0 . In this case, however, Z P 0 π 0 vanishes in the isosymmetric limit, because the neutral pion does not have an isoscalar component for ∆m = 0, and one gets where and In order to isolate the quantity we are interested in, namely Z P 0 π 0 , it is useful to normalize again over the isosymmetric charged pion correlator In this way we get By performing a constant fit to the ratio of correlators in Eq. (42), and by extracting Z ππ from C isoQCD ππ (t), it is possible to determine Z P 0 π 0 , and thus Z P 0 π 0 at O(∆m).

IV. NUMERICAL RESULTS
We performed simulations on the cA211.30.32 ensemble generated by the ETMC [9], which has a spatial extent L = 32 and aspect ratio T /L = 2. The number of gauge configurations that have been analyzed is N cf g = 1232. The ensemble corresponds to an higher-than-physical pion Relying on Eqs. (6) and (9), which are the basis for our mass and matrix element methods, we have built the following estimators to extract 7 from the diagrams in Eqs. (22) and (23): In our twisted mass setup, the pion decay constant can be obtained fromẐ ππ , which is the bare matrix element π 0 |P † π 0 |0 extracted from C isoQCD ππ , using 2 The operator −∂ t in Eq. (43), corresponds to the evaluation of the so-called effective slope δm ef f (t) from the ratio of correlators δC/C, which is defined through where in our case In Eq. (46), M is the ground state mass extracted from the correlator C(t), T is the temporal extent of the lattice and F (x, M ) is an analytical factor given by M M (t) contributing to C π + π + (t) − C π 0 π 0 (t) and normalized over the isosymmetric charged pion correlator C isoQCD ππ (t). M EM (t) and the disconnected diagram C disc. M EM (t) contributing to C P 0 π 0 (t).
In the large time limit t a, (T − t) a, both estimators in Eqs. (43) and (44) tend to 7 .
In Tab. I we collected the values of the input parameters that have been used for the determination of 7 on the cA211.30.32 ensemble. The ratio between the RCs Z S and Z P has been computed using whereẐ OS ππ is the bare matrix element π 0 |P † π 0 |0 extracted from the Osterwalder-Seiler pion correlator, i.e. from the single connected diagram in which the quark and antiquark propagators are computed with the same value of the Wilson parameter r. In Fig. 1 we show our determination of the diagrams C conn.
M M (t) and C disc. M M (t) normalized over the isosymmetric charged pion correlator C isoQCD ππ (t), while in Fig. 2 we show our determination of C conn.
M EM (t) and C disc. M EM (t). As the figures show, for both mass and matrix element methods, the signal of 7 comes from a large cancellation between the connected and the disconnected contributions. This makes the evaluation of 7 a nontrivial task, since a very good precision on both diagrams is needed to ensure that the difference is not dominated by the statistical noise. In this respect, the use of the RTM scheme turns out to be crucial (see Ref. [8] for more details on this improvement). In Fig. 3, we show our determination of¯ 7 (t) as obtained from both mass and matrix element methods. As can be seen, the matrix element method seems to perform better, giving a more precise result w.r.t. the mass method. In both cases, the signal disappears after a time separation t/a ∼ 15, and 7 can be extracted only at smaller times. We decided to fit both estimators in the time interval [5,13]. We obtain in this way Even if our analysis is limited to a single value of the lattice spacing and to a single pion mass Our determination can be compared with the phenomenological estimate given in Ref. [3] ph.
and with the global-fit based result obtained by the RBC-UKQCD Collaboration 7 = 6.5(3.8) × 10 −3 [5]. Our result is in agreement but significantly improves both estimates, and shows the effectiveness of the RM123 approach to determine 7 .

V. CONCLUSIONS
In this paper we showed that it is possible to determine directly from lattice QCD calculations the SU(2) ChPT low-energy costant 7 which parametrizes QCD isospin-breaking effects in the chiral Lagrangian at NLO and that is crucial for several phenomenological analyses. Our strategy is based on the the RM123 approach [6,7], which allows to evaluate isospin-breaking effects perturbatively in the up-down quark mass difference m u − m d . In addition, in order to increase the precision of the lattice correlators involved in the calculation we made use of the recently proposed rotated twisted-mass (RTM) scheme [8].
To determine 7 we explored two strategies. The first one, the mass method, is based on the computation of the charged/neutral pion mass difference M π + − M π 0 , whose second derivative in m u − m d , evaluated at the isosymmetric point m u = m d , is directly proportional to 7 . The second strategy, the matrix element method, allows to compute 7 from the slope in m u − m d of the coupling Z P 0 π 0 between the neutral pion and the isoscalar operator P 0 , which is a pure isospin breaking effect. The two methods give rise to consistent results, but we find that the matrix element method displays an higher statistical accuracy w.r.t. the mass method. The comparison of our determination with existing results reveals substantial agreement, although the difference is slightly larger than one standard deviation. It is however important to remind that our results are obtained at a single value of the lattice spacing and with a pion mass M π 260 MeV. Therefore, our result should be understood as a proof-of-principle calculation, showing the feasibility of a direct determination of 7 from lattice QCD calculations. In the future we plan to perform simulations at different values of the lattice spacing and explore different light quark masses in order to perform a reliable extrapolation towards the continuum and chiral limit. A possibility would be to extend the calculation to the other ensembles produced by the ETM Collaboration with Wilson-clover TM fermions, or alternatively, one could consider other lattice discretizations.

VI. ACKNOWLEDGEMENT
We thank L. Di Luzio, G. Piazza and C. Tarantino for useful discussions, and all members of ETMC for the most enjoyable collaboration. We acknowledge CINECA for the provision of CPU