Anatomy of the $\rho$ resonance from lattice QCD at the physical point

We propose a strategy to access the $q\bar{q}$ component of the $\rho$ resonance in lattice QCD. Through a mixed action formalism (overlap valence on domain wall sea), the energy of the $q\bar{q}$ component is derived at different valence quark masses, and shows a linear dependence on $m_\pi^2$. The slope is determined to be $c_1=0.505(3)\,{\rm GeV}^{-1}$, from which the valence $\pi \rho$ sigma term is extracted to be $\sigma_{\pi \rho}^{(\rm val)}=9.82(6)$ MeV using the Feynman-Hellman theorem. At the physical pion mass, the mass of the $q\bar{q}$ component is interpolated to be $m_\rho=775.9\pm 6.0\pm 1.8$ MeV, which is close to the $\rho$ resonance mass. We also obtain the leptonic decay constant of the $q\bar{q}$ component to be $f_{\rho^-}=208.5\pm 5.5\pm 0.9$ MeV, which can be compared with the experimental value $f_{\rho}^{\rm exp}\approx 221$ MeV through the relation $f_{\rho}^{\rm exp}=\sqrt{Z_\rho}f_{\rho^\pm} $ with $Z_\rho\approx 1.13$ being the on-shell wavefunction renormalization of $\rho$ owing to the $\rho-\pi$ interaction. We emphasize that $m_\rho$ and $f_\rho$ of the $q\bar{q}$ component, which are obtained for the first time from QCD, can be taken as the input parameters of $\rho$ in effective field theory studies where $\rho$ acts as a fundamental degree of freedom.

The standard quantities that are used to test lattice QCD against experiments include hadron masses, pseudoscalar meson decay constants f π and f K , and B K [1]. On the other hand, the lattice results are getting precise enough, such as f Ds [2,3], so that any significant deviation from experiments could be taken as a signal for physics beyond the standard model. Now that lattice configurations with N f = 2 + 1 flavors are available for physical pion mass and large volume, the list of physical observables should be expanded to have a broader examination of lattice calculations against experiments. The obvious choices include the vector meson decay constants for ρ, K * , D * and D * s and the vector to pseudoscalar radiative decays V → P γ, etc. In this work we shall report our analysis of the leptonic decay constant f ρ ± of the charged ρ meson, which describes the coupling of ρ ± and lepton pair lν through the charged weak current interaction.
Gauge configurations of N f = 2 + 1 domain-wall fermions with large spatial volume and physical pion mass have been generated by the RBC & UKQCD Collaborations [1]. This work is based on the 48I gauge ensemble with lattice size L 3 × T = 48 3 × 96 [1]. The lattice spacing has been determined to be a −1 = 1.730 (4) GeV, such that the spatial extension of the lattice is approximately La ∼ 5.5 fm. The light sea quark mass is set to give the pion mass m (sea) π = 139.2 (4) MeV. For the valence quarks, we adopt the overlap fermion action, which is another realization of the chiral fermions on the lattice. The low-energy constant ∆ mix , which measures the mismatch of the mixed valence and sea pion masses between the domain-wall fermion and the overlap fermion, is shown to be very small [4]. Since overlap fermion accommodates the multi-mass algorithm and the eigenvectors are the same for different quark masses, we use 1000 pairs of eigenvectors plus the * cheny@ihep.ac.cn  (2) 135(2) 149(2) 182(2) 208(2) 371(1) fπ(MeV) 130.3(9) 131.0(9) 131.6(8)) ... ... ... mρ(MeV) 773 (7) 775(6) 779(6) 784(5) 789(5) 836(3) zero modes for deflation in calculating quark propagators for several masses on 45 configurations (see Ref [5] for details). The bare mass parameters are chosen as am (val) q = 0.00170, 0.00240, 0.00300, 0.00455, 0.00600 and 0.02030, which give the pion mass ranging from 114 to 371 MeV. In this way we can discern the chiral behaviors of the mass and the leptonic decay width of the ρ meson.
We first extract the decay constant of pion according to the partially conserved axial current relation For overlap fermions, the quark mass renormalization constant Z m and the renormalization constant Z P of the psuedoscalar densityūγ 5 d satisfy the relation Z m Z P = 1.
Thus the calculation of f π can be carried out straightforwardly by taking m u = m d = m (val) q after the matrix element in Eq. (1) is obtained from the A 4 P and P P correlation functions. The pion masses and decay constants are listed in Table I. Since the pion masses are very close to the physical pion mass, we perform simply a linear interpolation in m 2 π and get f π = 131.3(6) MeV at the physical pion mass m π = 139.5 MeV, which agrees with RBC&UKQCD's result f π = 131.1(3) MeV on the same lattice and their final theoretical prediction f π = 130.2(9) MeV [1]. RBC&UKQCD also calculates f π on a larger lattice with a smaller lattice spacing, a −1 = 2.359(7) GeV, with a result f π = 130.9(4) MeV. Their f π s on the two lattices imply very small finite a artifacts. This comparison can be taken as a calibration of our formalism. When calculating the two-point functions in the vector channel, we first perform Coulomb gauge fixing on the 45 gauge configurations. The quark propagators are computed using a wall source, thus the source operators of the vector can be expressed as For the sink operators of the vector, we use the spatially extended operators O V,i (x, t; r) by splitting the quark and antiquark field operator with different spatial displacements r, namely, Subsequently the two-point functions C(t, r) with different spatial separation r are calculated as where N r is the number of rs that satisfy |r| = r. The effective mass plateaus of C(r, t) with r = 0 (blue points) and r = √ 20a = 4.58a(black points) at m π = 208(2) MeV are plotted in Fig. 1. It is seen that the plateaus are converged to a uniform effective mass in large time range. The difference of the plateaus in the short time range shows the r-dependence of the contamination from higher states. In order to reduce the higher state contamination, we linearly combine the two correlation functions as C mix (t) = C(0, t) + ωC(4.58a, t) with an optimal mixing parameter ω = 10, by which we can get a very flat effective mass plateau starting from t/a = 3, as shown in the figure (red points). We fit C mix (t) using a singleexponential form in the time range t/a ∈ [3,13] and get m V a = 0.456(4) (plotted as a red band), where the error is statistical and is obtained through a jackknife analysis.
Through the large N c analysis, it is suggested by Jaffe [6] that the ρ meson is an ordinary meson like the Feshbach resonance which exists as a confined qq state in the continuum with zero width in the large N c limit. Its Breit-Wigner width manifests the coupling of the bound state and the scattering ππ states, which vanishes as N c goes to infinity, while its pole mass is insensitive to the strength of the coupling [7]. In the continuum, the resonance parameters of ρ, m ρ = 775 MeV and Γ ρ ≈ 150 MeV, have been extracted in the I = 1 ππ scattering channel. On a finite lattice, the eigenstates of the QCD Hamiltonian in the ρ channel are actually the superpositions of the discretized interacting ππ scattering states and the would-be qq confined state, and the energy shifts of these eigenstates from the interacting ππ states or the qq state encode the resonance properties. Many lattice calculations [8][9][10][11] have been carried out to extract the resonance parameters of ρ according to the Lüscher method [12]. It is known that the lattice spectrum in the ρ channel should show an avoided level crossing behavior when the energy of ππ states approaches to the mass of the resonance by varying the lattice size or the relative momentum of the ππ states, and the mixing of the would-be qq confined state and the nearby scattering state gives two eigenstates. For the spatial extension La ∼ 5.5 fm of our lattice and in the case of m π = 208 MeV, the two lowest energy thresholds of the P − wave ππ states are 2E π (001) = 614 MeV (shown in Fig. 1 as a blue line) and 2E π (011) = 761 MeV (shown in Fig. 1 as a black line) with the relative momenta pa = 2π/L(0, 0, ±1) and pa = 2π/L(0, ±1, ±1), respectively. Since 2E π (001) is far from the expected ρ mass, the corresponding ππ state should mix little with ρ and therefore have an energy close to 2E π (001), but we do not observe this state in the vector correlation function. The reason for its absence might be due to the implementation of the Coulomb wall-source operator which suppresses the scattering states with non-zero relative momenta (we will discuss this in depth in another publication). However, 2E π (011) is very close to the expected ρ mass, so the mixing of the corresponding ππ state and the qq state can be sizable. According to the feature of the avoided level crossing, the mixing gives two states with nearly degenerate energies around m ρ , which, however, cannot be resolved in our analysis. In this sense, we expect that the plateau in Fig. 1 illustrates the effective energy of the two adjacent states, and will take the fitted energy value m V a = 0.456(4) as an approximation of the ρ mass in the lattice unit.
We take a similar analysis procedure for the correlation functions at other pion masses, and the extracted masses of ρ are listed in Table I and are also plotted in Fig. 2 with respect to m 2 π , by which the chiral behavior of m ρ can be investigated. A new chiral extrapolation formula based on a modified M S regulator and a power counting scheme [13] gives m ρ (m π ) = m ρ (0) + c 1 m 2 π + c 2 m 3 π +  , where the log term is due to self energy. However, from Fig. 2 it is seen that m ρ is very linear in m 2 π for m π ranging from 114 MeV to 371 MeV. A correlated jackknife analysis using the form gives m ρ (0) = 766 (7) MeV and c 1 = 0.505(3) GeV −1 with χ 2 /d.o.f = 0.13. The ρ mass at the physical m π is m ρ = 775.9 ± 6.0 ± 1.8 MeV, where the second error is due to the 0.23% uncertainty of the lattice spacing. Our data cannot discern higher order terms in m π . We would like to point out that our study is carried out for the first time in the chiral region around the physical point and with chiral fermions, although there have been many lattice studies on this topic [14][15][16][17]. We note that c 1 is precisely determined, and serves potentially as a constraint on the chiral perturbation study of ρ. Furthermore, c 1 is exactly the valence or connected insertion part of the πρ sigma term from the Feynman-Hellman theorem σ (val) πρ = m 2 π dm ρ /dm 2 π = c 1 m 2 π , since the sea is fixed in our partially quenched calculation of m ρ . One can determine the disconnected part from a direct calculation of the mψψ matrix element in the disconnected three-point correlator. From the fitted c 1 in Eq. (4), we find σ (val) πρ = 9.82(6) MeV. The leptonic decay constant f ρ is also an important characteristic quantity of ρ. Most of previous lattice studies on f ρ were performed at relatively heavy pion masses [18][19][20][21][22]. Here we would like to extract the decay constant of charged ρ at the physical point. For charged ρ, for example, ρ − , f ρ − is defined by where J V,i |V ( p, ζ) . Usually this matrix element can be derived by calculating the wall-wall correlation function where the last equation uses the definition of C(r, t) in Eq. (3). However, a very large statistics is required to obtain a satisfactory signal-to-noise ratio for this kind of correlation function. The reason for noisy C (w) (t) is analyzed as follows. Using the spectral expression C(r, t) = i Φ n (r)e −Ent , when t → ∞ one has In practice, we calculate C(r, t) for r ranging from 0 to 10a and observe the profile of Φ 1 (r) for t = 7a where all C(r, t) are almost saturated by the ground state. The Φ 1 (r) at m π = 208(2) MeV (normalized as Φ 1 (0) = 1) is plotted in Fig. 3 as red points. It is seen that Φ 1 (r) damps rapidly with r and can be parameterized as with the parameters α = 1.60 and r 0 = 5.88a. The curve illustrates this parameterization in the figure. We check  (7) 208(7) 211(6) 215(5) 217(5) 223(3) this at other pion masses and find Φ 1 (r) is similar for all the cases and is very insensitive to m π . This means that when calculating the wall-to-wall correlation function C (w) (t), the C(r, t)s (see in Eq. (6)) with very large r contribute only noise and make C (w) (t) very noisy. In order to circumvent this difficulty, we introduce a cutoff r c to exclude the contribute of C(r, t)s with r > r c from C (w) (t) and use the correlation function to approximate C (w) (t). Letting I 1 (r ′ ) = r ′ 0 drr 2 Φ 1 (r), one can see that the ratio C (w) (r c , t)/C (w) (t) can be depicted by the ratio I 1 (r c )/I 1 (∞) at large t. The ratio I 1 (r c )/I 1 (∞) using the parameterization above is also plotted in Fig. 3. It approaches to 1 beyond r c = 15a and is equal to 0.995 at r c = 20a, whose deviation from one is already much smaller than the statistical error. So we take C (w) (20a, t) as a satisfactory approximation of C (w) throughout this work.
With this prescription, we carry out the joint fit to the correlation functions where Z (w) n is the matrix element of the wall source operator between the vacuum and the n-th state, and f n is the bare decay constant of the n-th state according to the definition in Eq. (5). In practice, two exponentials are used in the fit and f 1 is taken as the bare decay constant of ρ (the second term is introduced to account for the contamination of higher states). Since f 1 is sensitive to the value of m 1 , we adopt the single-elimination jackknife analysis procedure as follows. On each jackknife re-sampled ensemble, we first obtain the mass parameter m 1 from C mix (t) defined previously, and then extract f 1 through a joint fit to Eq. (10) with m 1 fixed. After that, we quote the jackknife error of f 1 as the statistical error.
For the overlap fermion, the chiral symmetry dictates that Z V = Z A for the local currents and it has been verified in the non-perturbative renormalization [23]. We have calculated Z A from the Ward identity for a few bare quark masses and they are listed in Table II, where one can see that the quark mass dependence of Z A is mild and the chiral limit value is Z A = 1.1045 (8). π . The f (exp) ρ ± extracted from the τ leptonic decay is also plotted as an asterisk.
From Z V = Z A and the bare decay constant f 1 , the renormalized decay constant f ρ is obtained at different m π , as illustrated in Fig. 4, where one can see that f ρ is not linear in m 2 π throughout the pion mass range. Fortunately, we have several m π s very close to the physical pion mass, which facilitate us to do an linear interpolation in m 2 π in the neighborhood of the physical pion mass. The final result of f ρ at the physical point is f ρ ± = 208.5 ± 5.5 ± 0.9 MeV, (11) where the first error is statistical and the second is the combined uncertainty of Z V , the scale parameter a −1 , and the approximated wall-wall correlation function. Experimentally, f ρ ± can be derived from the branching ratio of τ lepton decaying into ρ − ν τ , (12) The branching ratio Br(τ → π 0 π − ν τ ) is measured to be 25.52(9)% [24]. After subtracting the 0.30(32)% nonρ branching fraction, one has the branching fraction Br(τ → ρ − ν τ ) = 25.22(33)%. Using this value along with the latest PDG2014 values of G F , |V ud |, m τ , τ τ , and m ρ , one can get the experimental value f ±,exp ρ = 209.4 ± 1.5 MeV, where the error is predominantly given by the uncertainty of the above branching fraction. Our result Eq. (11) and f ±,exp ρ are in good agreement. To summarize, we use overlap valence fermion for several quark masses on the N f = 2+1 domain-wall fermion configurations generated by the RBC&UKQCD Collaboration. The light sea quark is at the physical point and the spatial extension is 5.5 fm. As a benchmark, we first calculate the decay constant of pion and obtain a value f π = 131.3(6) MeV at the physical pion mass, which is consistent with RBC & UKQCD's resultf π = 130.2 (9) MeV. We extract the spectrum of the I = 1 vector channel from the Coulomb gauge fixed wall-source correlation functions. As such the ρ mass is precisely determined to be m ρ = 775.9 ± 6.0 ± 1.8 MeV at the physical pion mass. We also propose a strategy to reduce the noise of the wall-wall correlation functions of hadrons, through which the leptonic decay constant of ρ, f ρ , is determined to be 208.5 ± 5.5 ± 0.9 MeV at the physical m π , which agrees well with the value f ρ = 209.4 ± 1.5 MeV derived from the process τ → ρν τ . From the slope of m ρ vs. m 2 π , we obtain the valence πρ sigma term from the Feynman-Hellman theorem which gives σ (val) πρ = 9.82(6) MeV. Future results at different lattice spacings are needed for the continuum extrapolation.