# Extracting the low-energy constant L_0^r at three flavors from pion-kaon scattering

Chaitra Kalmahalli Guruswamy, Ulf-G. Mei?ner, Chien-Yeah Seng

EExtracting the low-energy constant L r at three ﬂavorsfrom pion-kaon scattering Chaitra Kalmahalli Guruswamy a , Ulf-G. Meißner a,b,c , Chien-Yeah Seng a a Helmholtz-Institut f¨ur Strahlen- und Kernphysik and Bethe Center for Theoretical Physics,Universit¨at Bonn, D-53115 Bonn, Germany b Institute for Advanced Simulation, Institut f¨ur Kernphysik and J¨ulich Center for Hadron Physics,Forschungszentrum J¨ulich, D-52425 J¨ulich, Germany c Tbilisi State University, 0186 Tbilisi, Georgia

Abstract

Based on our analysis of the contributions from the connected and disconnected contraction diagramsto the pion-kaon scattering amplitude, we provide the ﬁrst determination of the low-energy constant L r in SU(4 |

1) Partially-Quenched Chiral Perturbation Theory from data of the Extended Twisted MassCollaboration, L r = 0 . · − at µ = 1 GeV.

1. Introduction

In Ref. [1] we performed an analysis of pion-kaon scattering using Partially-Quenched Chiral Pertur-bation Theory (PQChPT). In this framework, additional quark ﬂavors allow one to separate the diﬀerentquark contraction diagrams in the lattice QCD simulation of a given physical observable, which is veryuseful for getting a better handle on the noisier and more expensive contractions. For an introduction tothis method, see e.g. Refs. [2, 3]. However, since this separation is unphysical, it will unavoidably involvenew parameters that cannot be ﬁxed by experiment, but can be determined from lattice simulations.In particular, in PQChPT the Cayley-Hamilton relation can no longer be used, leading to the followingterm in the chiral Lagrangian at O ( p ): L (4) = L Str (cid:0) ∂ µ U † ∂ ν U ∂ µ U † ∂ ν U (cid:1) , (1)where ‘Str’ denotes the supertrace over the ﬂavor space, U is the standard exponential representationof the pseudo-Nambu-Goldstone bosons, and L is a new low-energy constant (LEC) that does notappear in ordinary two-ﬂavor and three-ﬂavor ChPT. It appears in the next-to-next-to-leading order(NNLO) correction to the pion mass and decay constant when the valence and sea quark masses arekept distinct, which is possible on the lattice. Based on this, Ref. [4] has determined the renormalizedLEC L r in the SU(4 |

2) PQChPT, which is equivalent to a two-ﬂavor ChPT in computations of physicalprocesses, at the renormalization scale µ = 1 GeV (using dimensional regularization). The quoted resultis L r = 1 . . · − .Refs. [5, 6] provided an alternative determination of L r through the fact that it appears in separatecontraction diagrams in ππ scattering at next-to-leading-order (NLO), and therefore can be obtained fromthe scattering lengths extracted from linear combinations of lattice correlation functions correspondingto various types of contractions based on L¨uscher’s formula [7]. Using the lattice data of connected ππ correlation functions by the Extended Twisted Mass (ETM) Collaboration [8], Ref. [6] reported L r = 5 . . · − , again for SU(4 | a r X i v : . [ h e p - l a t ] F e b cattering length with isospin I = 3 /

2, we determine, for the ﬁrst time, the LEC L r in SU(4 |

1) PQChPT,which is equivalent to a three-ﬂavor ChPT in the computations of physical processes.This work is organized as follows. In Sec. 2, we display the necessary formalism to extract the LEC L r form the pertinent scattering length a and display the lattice input from the ETM collaboration whichwe use. The results of this analysis are presented in Sec. 3. We end with a short summary.

2. Formalism

As detailed in Ref. [1], the pertinent two contractions in pion-kaon scattering with total isospin I = 3 / |

1) PQChPT: T a ( s, t, u ) = T ( u ¯ s )( d ¯ j ) → ( u ¯ s )( d ¯ j ) ( s, t, u ) ,T b ( s, t, u ) = T ( u ¯ s )( d ¯ j ) → ( d ¯ s )( u ¯ j ) ( s, t, u ) , (2)with s, t, u the conventional Mandelstam variables subject to the constraint s + t + u = 2( M K + M π ), u, d, s are usual light quark ﬂavors and j denotes the additional valence quark (which comes togetherwith a ghost quark) in PQChPT. For total isospin I = 1 /

2, there is one additional scattering amplitude T c ( s, t, u ) ≡ T b ( u, t, s ). The three eﬀective single-channel scattering amplitudes are: T α ( s, t, u ) = T a ( s, t, u ) + T b ( s, t, u ) , (3) T β ( s, t, u ) = T a ( s, t, u ) − T b ( s, t, u ) , (4) T γ ( s, t, u ) = T / ( s, t, u ) = T a ( s, t, u ) − T b ( s, t, u ) + 32 T c ( s, t, u ) . (5)This allows to deﬁne three S-wave scattering lengths, a α , a β , a γ , from which only a β depends on L r . Itis deﬁned as a β = − π √ s (cid:2) T thr a − T thr b (cid:3) , (6)with √ s = M π + M K the threshold energy. The explicit expression for a β then is: a β = − π √ s (cid:20) M π M K F π (cid:18) − L r + 32 L r + 32 L r − L r − L r + 8 L r + 32 L r − L r − π (cid:19) + 8 M π M K L r F π + µ π F π ( M K − M π ) (cid:18) − M K − M π M K + 134 M π M K + 52 M π M K (cid:19) + µ K F π (cid:18) − M K M K − M π − M π M K M K − M π − M π M K M K − M π − M π M K M K − M π ) + M π M K − M π ) (cid:19) + µ η F π ( M K − M π ) (cid:18) − M π M K M η − M π M K M η + 3 M K M π M K + 114 M π M K − M π M K (cid:19) + M π M K ¯ J πK ( s ) F π + ¯ J πK ( u ) F π (cid:18) M K M π M K − M π M K + 12 M π M K + M π (cid:19) + ¯ J Kη ( u ) F π (cid:18) M K

72 + 118 M π M K − M π M K + M π (cid:19) + ¯¯ J πK ( u ) F π (cid:18) − M K − M π M K − M π M K − M π M K − M π (cid:19) + ¯¯ J Kη ( u ) F π (cid:18) − M K − M π M K − M π M K − M π M K − M π (cid:19) − M K π F π − M π M K π F π − M π M K π F π + M π M K F π − M π π F π (cid:21) , (7)2ith µ P = ( M P / π F π ) ln( M P /µ ), u = 2 M π + 2 M K − s , and the functions ¯ J P Q ( s ) , ¯¯ J P Q ( s ) aregiven in Ref. [1]. Further, the { L ri } are the O ( p ) LECs in SU(4 |

1) PQChPT, among which L r − arenumerically identical with those in ordinary three-ﬂavor ChPT, and only L r is new. For our analysis, werewrite the above equation as: M π M K F π (96 L r ) = 8 π √ s a β + (cid:20) M π M K F π (32 L r + 32 L r − L r − L r + 8 L r + 32 L r − L r − π )+ 8 M π M K L r F π + µ π F π ( M K − M π ) (cid:18) − M K − M π M K + 134 M π M K + 52 M π M K (cid:19) + µ k F π (cid:18) − M K M K − M π − M π M K M K − M π − M π M K M K − M π − M π M K M K − M π ) + M π M K − M π ) (cid:19) + µ η F π ( M K − M π ) (cid:18) − M π M K M η − M π M K M η + 3 M K M π M K + 114 M π M K − M π M K (cid:19) + M π M K ¯ J πK ( s ) F π + ¯ J πK ( u ) F π (cid:18) M K M π M K − M π M K + 12 M π M K + M π (cid:19) + ¯ J Kη ( u ) F π (cid:18) M K

72 + 118 M π M K − M π M K + M π (cid:19) + ¯¯ J πK ( u ) F π (cid:18) − M K − M π M K − M π M K − M π M K − M π (cid:19) + ¯¯ J Kη ( u ) F π (cid:18) − M K − M π M K − M π M K − M π M K − M π (cid:19) − M K π F π − M π M K π F π − M π M K π F π + M π M K F π − M π π F π (cid:21) , (8)or symbolically as y = L x + c , (9)where y ∝ a β . So for a given lattice ensemble with a certain value for the scattering length, this canreadily be solved. Before doing so, we need to discuss the pertinent lattice details.In order to pin down down L r , we shall invoke the scattering length values obtained in Ref. [9] for the πK system in the I = 3 / N f = 2 + 1 + 1 twisted mass lattice QCD. For our analysiswe have considered the ensembles A30.32 and A40.24 for the determination of the discrete ground-stateenergies of the pion-kaon system. More precisely, in Ref. [9] two methods to remove the thermal statepollution were used to determine the ground state energy E πK and the corresponding scattering length a β . These methods were labelled as “E1” and “E2”.The pertinent meson masses are given in Table 1 and the pion decay constant is aF π is 0 . . a = 0 . a β corresponding to the method E1 and E2 are displayed in Table 2, where µ πK is the reduced mass of the pion-kaon system.

3. Extraction of L r To pin down the value of the unphysical LEC 10 L r , we require the value for the other LECs. Theseare taken from Ref. [10] and are given for completeness as (in units of 10 − ) L r = 1 . , L r = 1 . , L r = − . , L r = 1 . ,L r = 1 . , L r = 1 . , L r = − . , L r = 0 . . (10)3nsemble aM π aM K aM η A30.32 0.0292 ± ± ± ± ± ± Table 1: Meson masses used in the evaluation. µ πK a β Ensemble E E A30.32 − . ± . − . ± . − . ± . − . ± . Table 2: Scattering length for the discrete energies E and E , evaluated for the ensembles A30.32 and A40.24.Figure 1: Graphical representation of the solution of Eq. (9) for the two methods to deﬁne the ground state energy. E1(left panel) and E2 (right panel). With this input, we can ﬁnd L r from solving Eq. (9). The results are displayed in Fig. 1 and alsocollected in Table 3. We see that the results for L r are stable (independent of the excited state removalmethod), Method E1 : L r = 0 . · − (11)Method E2 : L r = 0 . · − . (12)Since the two results are almost identical, we simply average the two central values and quote a ﬁnalresult of L r = 0 . · .Finally we comment on the relation between this result and the corresponding LEC in SU(4 | x y E1 A30.32 4 . ± .

29 2 . ± . . ± . . ± . . ± .

29 2 . ± . . ± . . ± . Table 3: Numerical values of the results displayed in Fig. 1.

SU(3 | | | |

2) onwards possess an L -dependence at tree-level as itrequires at least four fermionic quarks. Similarly, possible PQChPT extensions of a three-ﬂavor ChPTare SU(4 | | | | |

1) that does not depend on L at tree-level. Therefore,it is not possible to discuss the matching between L in two- and three-ﬂavors based on the theory setupin Ref. [1]. For that, one would have to repeat the calculations using a larger graded algebra, such asSU(5 |

4. Summary

In this work, we have for the ﬁrst time determined the LEC L r (1 GeV) of three-ﬂavor PQChPTthrough an NLO analysis of contraction diagrams in πK scattering, L r = 0 . · − . Utilizing theprecise data from the ETM collaboration for pion-kaon scattering in the I = 3 / . × − , which is better than the previous determinations of thecorresponding LEC for two ﬂavors in Ref. [4] that made use of an NNLO ﬁtting, and in the NLO ﬁttingto the ππ scattering in Refs. [5, 6] that depends on more unknown LECs. Further work is required topin down this LEC even more precisely. Acknowledgements

We are very grateful to Ferenc Pitler for making the ETM collaboration data available to us and forhis detailed explanations concerning these. We thank Hans Bijnens for a useful communication. Thiswork is supported in part by the DFG (Projektnummer 196253076 - TRR 110) and the NSFC (Grant No.11621131001) through the funds provided to the Sino-German CRC 110 “Symmetries and the Emergenceof Structure in QCD”, by the Alexander von Humboldt Foundation through the Humboldt ResearchFellowship, by the Chinese Academy of Sciences (CAS) through a President’s International FellowshipInitiative (PIFI) (Grant No. 2018DM0034), by the VolkswagenStiftung (Grant No. 93562), and by theEU Horizon 2020 research and innovation programme, STRONG-2020 project under grant agreement No824093.

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