Large \(N_c\) scaling of meson masses and decay constants
Abstract
We perform an ab initio calculation of the \(N_c\) scaling of the low-energy couplings of the chiral Lagrangian of low-energy strong interactions, extracted from the mass dependence of meson masses and decay constants. We compute these observables on the lattice with four degenerate fermions, \(N_f=4\), and varying number of colours, \(N_c=3\)–6, at a lattice spacing of \(a\simeq 0.075\) fm. We find good agreement with the expected \(N_c\) scaling and measure the coefficients of the leading and subleading terms in the large \(N_c\) expansion. From the subleading \(N_c\) corrections, we can also infer the \(N_f\) dependence, that we use to extract the value of the low-energy couplings for different values of \(N_f\). We find agreement with previous determinations at \(N_c=3\) and \(N_f=2, 3\) and also, our results support a strong paramagnetic suppression of the chiral condensate in moving from \(N_f=2\) to \(N_f=3\).
1 Introduction
The ’t Hooft limit of QCD [1] is well known to capture correctly most of its non-perturbative features, such as confinement and chiral symmetry breaking. Large \(N_c\) inspired approximations are often employed in phenomenological approaches to hadron physics [2, 3, 4, 5, 6, 7, 8, 9, 10, 11], but systematic errors from subleading \(N_c\) corrections are only naively estimated.
Lattice Field Theory offers the possibility of ab initio explorations of the large \(N_c\) limit of QCD, by simulating at different values of \(N_c\) [12, 13]. Several studies have already been performed. In Ref. [13] a thorough study of mesonic two-point functions was carried out in the quenched approximation, a limit that captures correctly the leading order terms in \(N_c\), but modifies subleading corrections in an uncontrolled way. Furthermore, in Ref. [14] a similar study was performed for \(N_c= 2\)–5 using \(N_f=2\) dynamical fermions at rather high pion masses.
In addition to the standard approach, the study of QCD in the large \(N_c\) limit can also be achieved using reduced models (see [15] for a review). In this context, there has been significant progress regarding the properties of mesons [16, 17, 18, 19, 20].
Besides, lattice simulations have been used to perform studies of various observables in theories with different number of colours, flavours or fermion representations in the context of Beyond-the-Standard-Model theories. Some recent results can be found in [21, 22, 23, 24, 25, 26] and for recent reviews see [27, 28].
In this work, we use previously generated lattice configurations with \(N_c=3\)–6 and four dynamical fermions. Our particular choice of \(N_f\) has also advantages for weak matrix elements [29]. On these ensembles, we compute meson masses and decay constants as a function of the quark mass at the different values of \(N_c\). We fit these to chiral perturbation theory (ChPT) in order to extract the leading order and next-to-leading order low-energy chiral couplings (LECs). We then study their \(N_c\) scaling and extract the first two terms in the ’t Hooft series. Our study builds on previous lattice determinations of the LECs for \(N_c=3\) [30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44], whose main results are summarized in [45].
Interestingly, within the large \(N_c\) expansion, the \(1/N_c\) corrections have a well-defined linear dependence on \(N_f\), while the ’t Hooft limit is independent on \(N_f\). Using this fact, we can predict the low-energy couplings at different values of \(N_f\) up to higher orders in \(N_c\). This allows us to compare with previous determinations, and check the prediction of paramagnetic suppression at large \(N_f\) of Refs. [46, 47].
This paper is organized as follows. First, we describe chiral perturbation theory predictions and the relation to the large \(N_c\) limit in Sect. 2. In Sect. 3, we present the lattice setup that involves a mixed-action formulation. Next, we explain our scale setting procedure at different \(N_c\) consistent with ’t Hooft scaling in Sect. 4. In Sect. 5 we present the results of our chiral fits to the meson mass and decay constant, first at fixed \(N_c\) and then combined with the large \(N_c\) expansion. We also present results for theories with different values of \(N_f\), compare with previous literature and discuss systematic uncertainties. We conclude in Sect. 6.
2 Chiral perturbation Theory predictions
2.1 \(SU(N_f)\) effective theory
2.2 \(U(N_f)\) effective theory
2.3 \(N_f\) versus \(N_c\) dependence
A diagrammatic analysis of fermion bilinear two point functions shows that within the large \(N_c\) expansion, the leading order \(N_c \rightarrow \infty \) limit is \(N_f\) independent and the NLO is \({{\mathcal {O}}}(N_f/N_c)\). We should confirm this expectation also in ChPT formulae above, in particular given the explicit dependence on \(N_f\). It turns out that within the \(U(N_f)\) expansion, the large \(N_c\) expansion yields the expected behaviour: the terms in \(1/N_f\) exactly cancel when the large \(N_c\) expansion is taken at fixed \(M_\pi \). We expect therefore that the LECs should also satisfy this same scaling.
On the other hand within the \(SU(N_f)\) expansion or in the \(U(N_f)\) when \(M_\pi \ll M_{\eta '}\), that is when the chiral limit is taken first, anomalous \(1/N_f\) terms appear coming from an expansion in \(M_\pi /M_{\eta '}\). In the \(U(N_f)\) expansion such dependence is explicit, but in the \(SU(N_f)\) it permeates to the LECs which can no longer be assumed to have the expected \({{\mathcal {O}}}(N_f/N_c)\) dependence, as can be explicitly seen in the matching of \(L_M^{(1)}\) in Eq. (14).
This way, at the order we are working, we can assume the expected scaling in \(N_f\) of the \(U(N_f)\) and \(SU(N_f)\) couplings except in the case of \([L_M^{(1)}]_{SU(N_f)}\).
3 Lattice setup
We have generated ensembles for \(SU(N_c)\) gauge theory with \(N_f=4\) degenerate dynamical fermions, varying \(N_c=3\)-6, using the HiRep code [61]. Some of them have been already presented in Ref. [62]. We have chosen the Iwasaki gauge action (following previous experience with 2+1+1 simulations [63, 64]) and O(a)-improved^{1} Wilson fermions for the sea quarks. Our simulations use the standard Hybrid Montecarlo (HMC) algorithm with Hasenbusch acceleration. We include five layers in each of the fermionic monomials. Interestingly, we observe that the tuning of the integrator at \(N_c=3\) yields similar results at other values of \(N_c\) (at similar pion mass) for the acceptance rate, which we keep at 80–90%. The computational cost of each step in Montecarlo time scales as \(\sim N_c^2\), with the advantage of a more efficient parallelization at large \(N_c\).
Summary of our ensembles: \(\beta \), sea quark bare mass parameter, \(m^s\), and sea pion mass \(M^s_\pi \) . We keep \(c_{sw}=1.69\) throughout
Ensemble | \(L^3 \times T\) | \(\beta \) | \(am^s\) | \(aM^s_\pi \) | \(t_0^{\text {imp}}/a^2\) |
---|---|---|---|---|---|
3A10 | \(20^3 \times 36\) | 1.778 | \(-\) 0.4040 | 0.2204 (21) | 3.263 (50) |
3A20 | \(24^3 \times 48\) | \(-\) 0.4060 | 0.1845 (14) | 3.491 (32) | |
3A30 | \(24^3 \times 48\) | \(-\) 0.4070 | 0.1613 (16) | 3.740 (39) | |
3A40 | \(32^3 \times 60\) | \(-\) 0.4080 | 0.1429 (12) | 3.855 (27) | |
4A10 | \(20^3 \times 36\) | 3.570 | \(-\) 0.3725 | 0.2035 (14) | 3.494 (45) |
4A20 | \(24^3 \times 48\) | \(-\) 0.3752 | 0.1805 (7) | 3.565 (26) | |
4A30 | \(24^3 \times 48\) | \(-\) 0.3760 | 0.1714 (8) | 3.593 (29) | |
4A40 | \(32^3 \times 60\) | \(-\) 0.3780 | 0.1397 (8) | 3.723 (23) | |
5A10 | \(20^3 \times 36\) | 5.969 | \(-\) 0.3458 | 0.2128 (9) | 3.532 (17) |
5A20 | \(24^3 \times 48\) | \(-\) 0.3490 | 0.1802 (6) | 3.614 (18) | |
5A30 | \(24^3 \times 48\) | \(-\) 0.3500 | 0.1712 (6) | 3.664 (24) | |
5A40 | \(32^3 \times 60\) | \(-\) 0.3530 | 0.1331 (7) | 3.776 (19) | |
6A10 | \(20^3 \times 36\) | 8.974 | \(-\) 0.3260 | 0.2150 (7) | 3.619 (17) |
6A20 | \(24^3 \times 48\) | \(-\) 0.3300 | 0.1801 (5) | 3.696 (17) | |
6A30 | \(24^3 \times 48\) | \(-\) 0.3311 | 0.1689 (7) | 3.721 (15) | |
6A40 | \(32^3 \times 60\) | \(-\) 0.3340 | 0.1351 (6) | 3.820 (17) |
4 Scale setting at large \(N_c\)
Results for the \(\left. t_0/a^2\right| _{M_{\text {ref}}}\) and the lattice spacing as a function of \(N_c\). The first error is statistical, the second comes from the uncertainty in \(t_0\) in physical units, the third stems from the difference in the definitions of E(t) after improvement, and the fourth are finite volume effects estimated from Ref. [76]
\(N_c\) | \(\left. t_0/a^2\right| _{M_{\text {ref}}}\) | a (\(\times 10^{-2}\) fm) |
---|---|---|
3 | \(3.71(4)(7)_{t_0}(12)_a(3)_L\) | \(7.53(4)(19)_{t_0}(12)_{a}(3)_L\) |
4 | \(3.64(1)(3)_{t_0}(12)_a(3)_L\) | \(7.60(1)(20)_{t_0}(12)_{a}(3)_L\) |
5 | \(3.69(2)(3)_{t_0}(12)_a(3)_L\) | \(7.54(2)(20)_{t_0}(12)_{a}(3)_L\) |
6 | \(3.76(1)(2)_{t_0}(12)_a(3)_L\) | \(7.48(1)(20)_{t_0}(12)_{a}(3)_L\) |
5 Chiral perturbation theory fits
The results for \(M_\pi \) and \(F_\pi \) in the mixed-action setup are presented in Table 3. We want to compare these results to the expectations in ChPT described in Sec. 2 in order to the extract the LECs and study their \(N_c\) scaling.
Before addressing the fits, we need to explain some technical issues regarding the finite volume effects, the renormalization scale and the fitting strategy. We then perform fits at a fixed value of \(N_c\) to test the ansätze for the \(N_c\) scaling of the LECs in Eqs. 5 and 7. After that, we perform simultaneous chiral and \(N_c\) fits. We present a selection of relevant results for the latter, and conclude the section with a discussion on systematic errors.
5.1 Finite volume effects
Results obtained in the mixed action setup, with Wilson fermions on the sea and twisted mass in the valence sector. We use \(c_{sw}=1.69\), as in the sea sector
Ensemble | \( am_{\mathrm{cr}}\) | \(a\mu _0\) | \(aM^{\mathrm{v}}_\pi \) | \(|am^{\mathrm{v}}_{pcac}|\) | \(aF_\pi \) |
---|---|---|---|---|---|
3A10 | \(-\) 0.4214 | 0.01107 | 0.2216 (20) | 0.0000 (3) | 0.04405 (41) |
3A20 | \(-\) 0.4196 | 0.00781 | 0.1834 (6) | 0.0001 (2) | 0.04023 (24) |
3A30 | \(-\) 0.4187 | 0.00632 | 0.1613 (11) | 0.0008 (2) | 0.03678 (33) |
3A40 | \(-\) 0.4163 | 0.00513 | 0.1423 (7) | 0.0006 (3) | 0.03554 (15) |
4A10 | \(-\) 0.3875 | 0.01030 | 0.2037 (11) | 0.0001 (2) | 0.05131 (37) |
4A20 | \(-\) 0.3865 | 0.00844 | 0.1803 (9) | 0.0000 (4) | 0.05037 (26) |
4A30 | \(-\) 0.3865 | 0.00778 | 0.1717 (9) | 0.0001 (4) | 0.04913 (31) |
4A40 | \(-\) 0.3851 | 0.00546 | 0.1416 (5) | 0.0001 (2) | 0.04608 (15) |
5A10 | \(-\) 0.3611 | 0.01225 | 0.2114 (13) | 0.0003 (4) | 0.06125 (32) |
5A20 | \(-\) 0.3611 | 0.00906 | 0.1799 (10) | 0.0001 (4) | 0.05767 (30) |
5A30 | \(-\) 0.3607 | 0.00824 | 0.1706 (13) | 0.0000 (4) | 0.05647 (40) |
5A40 | \(-\) 0.3596 | 0.00509 | 0.1328 (5) | 0.0002 (2) | 0.05278 (18) |
6A10 | \(-\) 0.3415 | 0.01298 | 0.2142 (6) | 0.0003 (2) | 0.06813 (21) |
6A20 | \(-\) 0.3414 | 0.00956 | 0.1801 (4) | 0.0002 (2) | 0.06435 (25) |
6A30 | \(-\) 0.3414 | 0.00803 | 0.1668 (5) | 0.0002 (2) | 0.06278 (24) |
6A40 | \(-\) 0.3409 | 0.00542 | 0.1342 (4) | 0.0000 (1) | 0.05929 (14) |
5.2 Renormalization scale
5.3 Fitting strategy
5.4 Fit results at fixed \(N_c\)
NLO Fits for \(F_\pi \) for separate values of \(N_c\)
\(N_c\) | \(aF/\sqrt{N_c}\) | \(L_F/N_c\) | \(\chi ^2/dof\) |
---|---|---|---|
3 | 0.0088 (9) | 0.0046 (14) | 0.7/2 |
4 | 0.0155 (6) | 0.0013 (3) | 3.9/2 |
5 | 0.0175 (4) | 0.0011 (2) | 2.2/2 |
6 | 0.0188 (2) | 0.0011 (1) | 0.4/2 |
Fits for \(M_\pi \) for separate values of \(N_c\)
\(N_c\) | aB | \(L_M/N_c\) | \(\chi ^2/dof\) |
---|---|---|---|
3 | 1.564 (55) | 0.00086 (10) | 10.2/2 |
4 | 1.560 (37) | 0.00064 (7) | 1.4/2 |
5 | 1.648 (30) | 0.00031 (6) | 0.1/2 |
6 | 1.610 (20) | 0.00031 (4) | 9.5/2 |
5.5 Simultaneous chiral and \(N_c\) fits
- (i)
- (ii)
Different fits for the decay constant as described in the text
Fit | \({F}_0\) | \({F}_1\) | \(F_2\) | \((FL_F)^{(0)}\) | \((F L_F)^{(1)}\) | \(K_F^{(0)}\) | \(\chi ^2/dof\) |
---|---|---|---|---|---|---|---|
1 | 0.0255 (12) | \(-\) 0.040 (6) | – | \(4.7\;(9.5)\cdot 10^{-6}\) | \(4.8\;(5.1) \cdot 10^{-5}\) | – | 0.79 |
2 | 0.0266 (9) | \(-\) 0.034 (8) | \(-\) 0.033 (14) | \(-\,8\;(10) \cdot 10^{-6}\) | \(5.6\;(4.4) \cdot 10^{-5}\) | \(7.6\;(6.4)\cdot 10^{-7}\) | 0.9 |
Different fits for the meson mass as described in the text
Fit | \(B_0\) | \(B_1\) | \(B_2\) | \((B L_M)^{(0)}\) | \((B L_M)^{(1)}\) | \(K_M^{(0)}\) | \(\chi ^2/dof\) |
---|---|---|---|---|---|---|---|
1 | 1.70 (11) | \(-\) 0.5 (5) | – | \(-\) 0.00046 (29) | 0.0056 (15) | – | 2.0 |
2 | 1.72 (7) | \(-\) 1.8 (5) | 1.8 (1.5) | \(-\) 0.00017 (25) | 0.0066 (10) | \(1.3 (9) \cdot 10^{-6}\) | 2.4 |
Fit | \(L_F^{(0)}\) | \(L_F^{(1)}\) | \(L_M^{(0)}\) | \(L_M^{(1)}\) |
---|---|---|---|---|
1 | \(1\;(4)\cdot 10^{-4}\) | \(23\;(13)\cdot 10^{-4}\) | \(-20\;(15)\cdot 10^{-5}\) | \(29\;(6) \cdot 10^{-4}\) |
2 | \(-3\;(4)\cdot 10^{-4}\) | \(17\;(18)\cdot 10^{-4}\) | \(-1\;(1)\cdot 10^{-4}\) | \(37\;(7)\cdot 10^{-4}\) |
5.6 Selected results
5.7 Comments on systematics
A different estimate comes from the dependence on \(c_{sw}\) in the valence sector. We have recomputed the decay constant for \(c_{sw} =0\) in the 3A10 ensemble, obtaining \([F_\pi ]_{c_{sw} =0} = 0.04303(40)\), within \(2\%\) of the value at the nominal \(c_{sw}\). The effects of a change in \(c_{sw}\) are in principle \(O(a^2)\), which can be estimated at \(\sim 2\%\) for this observable. This concerns however only the charged meson sector, since the neutral pion is known to have higher discretization effects with twisted mass. That issue is out of the scope of this work, and it will be addressed in future publications. in \(N_c\) We end this section with a last word on the chiral fits. We find that our data is well described by ChPT at the order we worked. Still, we cannot exclude that higher order corrections might be relevant in the range of masses we are considering. A robust study on the convergence of ChPT would require simulations at lighter quark masses and a proper continuum extrapolation.
6 Conclusion and outlook
In this work we presented the first lattice determination using dynamical fermions of the \(N_c\) scaling of the couplings in the chiral Lagrangian that contribute to the meson masses and decay constants (see Eqs. (43), (48) and Table 8). We have been able to disentangle the leading and subleading terms and we found that the subleading contributions are typically non negligible. In fact, we find that the value for \(L_M\) at \(N_c=3\) seems to be dominated by the subleading corrections, and the fit result suggests an accidental cancellation of \(2L_8 -L_5\) in the large \(N_c\) limit.
From our chiral fits and theoretical expectations, we have been able to infer the values of the couplings for theories with different numbers of flavours, \(N_f=2\) and \(N_f=3\) at \(N_c=3\). We find that our results nicely agree with those in the literature regarding \(L_F, L_M\) and F (see for example Ref. [45] for a summary of results). For B we need to improve our determination, including a non-perturbatively determined renormalization factor. On the other hand, as long as this factor has a small \(N_f\) dependence, we can estimate the ratio of B and the chiral condensate for \(N_f=2\) and \(N_f=3\). We find excellent agreement with the prediction of paramagnetic suppressions of Refs. [46, 47].
We would like to stress that the results presented in this paper are complementary to similar studies that can be performed in reduced models [16, 17, 18, 19, 20] or the quenched approximations at large \(N_c\) [13], since both of these approaches must yield the leading order result as \(N_c \rightarrow \infty \). Given the strong correlations presents in our results (see Fig. 5), a precise determination of the dominant \(N_c\) term would significantly improve the determination of the subleading \(N_c\) corrections, and hence the determination of the physical values at \(N_c=3\). We are willing to provide the bootstrap samples if requested.
As for the future, we would like to mention that our ensembles have a big potential to study other physical observables. We plan to use them to analyse the scaling of other quantities, such as the \(K \rightarrow \pi \) matrix elements (see [29, 62] for previous results). We also believe that the study of scattering amplitudes is a relevant quantity of study at large \(N_c\): on one hand quantities such as the \(I=2\) \(\pi \pi \) scattering length give access to LECs of the chiral Lagrangian; on the other hand the study of the behaviour resonances at large \(N_c\) is interesting, as it may shed light about their nature [10, 11, 85, 86].
Footnotes
- 1.
For \(N_c=3\), we take the perturbative value of \(c_{sw} = 1 + c^{(1)}_{sw} g^2\) from Ref. [65], where we use the plaquette-boosted coupling \(g^2 = 2 N_c/(\beta P) = O(1/N_c)\). For other values of \(N_c\), we use the fact that the one loop coefficient is dominated by the tadpole contribution, which is of order \(N_c\) (see Eq. 58 in Ref. [65]). This way, \(c_{sw}\) is constant up to subleading corrections in \(N_c\), which have an effect of \(O(a^2/N_c)\) in physical observables. The full result cannot be easily reconstructed from Ref. [65].
Notes
Acknowledgements
We thank Andrea Donini for very useful discussions and previous collaboration on related work, as well as M. García Pérez, A. González-Arroyo, G. Herdoíza, A. Ramos, A. Rusetsky, S. Sharpe, C. Urbach and A. Walker-Loud for useful comments and suggestions. We are particularly grateful to Claudio Pica and Martin Hansen for providing us with a \(SU(N_c)\) lattice code. This work was partially supported by grant FPA2017-85985-P, MINECO’s “Centro de Excelencia Severo Ochoa” Programme under grant SEV-2014-0398, and the European projects H2020-MSCA-ITN-2015/674896-ELUSIVES and H2020-MSCA-RISE-2015/690575-InvisiblesPlus. The work of FRL has also received funding from the European Union Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement no. 713673 and “La Caixa” Foundation (ID 100010434, LCF/BQ/IN17/11620044). Furthermore, CP thankfully acknowledges support through the Spanish projects FPA2015-68541-P (MINECO/FEDER) and PGC2018-094857-B-I00, the Centro de Excelencia Severo Ochoa Programme SEV-2016-0597, and the EU H2020-MSCA-ITN-2018-813942 (EuroPLEx). We thank Mare Nostrum 4 (BSC), Finis Terrae II (CESGA), Tirant 3 (UV) and Lluis Vives (Servei d’Informàtica UV) for the computing time provided.
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