Finite volume corrections to LECs in Wilson and staggered ChPT
aa r X i v : . [ h e p - l a t ] D ec Finite volume corrections to LECs in Wilson andstaggered ChPT
Fabrizio Pucci ∗ Fakultät für Physik, Universität Bielefeld, D-33615 Bielefeld, GermanyE-mail: [email protected]
Gernot Akemann
Fakultät für Physik, Universität Bielefeld, D-33615 Bielefeld, GermanyE-mail: [email protected]
We study the simultaneous effect of finite volume and finite lattice spacing corrections in theframework of chiral perturbation theory (ChPT) in the epsilon regime, for both the Wilson andstaggered formulations. In particular the finite volume corrections to the low energy constants(LECs) in Wilson and staggered ChPT are computed to next-to-leading order (NLO) in the e − expansion. For Wilson with N f =2 flavours and staggered with generic N f the partition functionat NLO can be rewritten as the LO partition function with renormalized effective LECs. The 30th International Symposium on Lattice Field TheoryJune 24 - 29, 2012Cairns, Australia ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ inite volume corrections to LECs in Wilson and staggered ChPT
Fabrizio Pucci
1. Introduction
In the last years the lattice QCD simulations near the chiral limit drive renewed interest to under-stand this limit with analytical approaches, and indeed a lot of efforts have been done in the studyof the low energy effective theories. It is important in the numerical simulation to have the latticespacing effects under control that also break explicitly chiral symmetry. Wilson and staggered chi-ral perturbation theory (WChPT and SChPT) provide the framework in which one can study theseUV cut-off effects.In the Wilson and staggered ChPT lagrangians in addition to the continuum Gasser-Leutwylerterms [1, 2] there are additionals O ( a ) contributions, and hence new low energy effective con-stants (LECs) that need to be introduced. More in detail for leading order (LO) WChPT with twoflavors only one new LEC enters, usually denoted with c [3, 4], while for SChPT six new LECs C i need to be introduced [5, 6].Here we study these theories in a finite volume in the so-called e -regime [7, 8], namely when thepion Compton wave length is bigger then the lattice size Lm p L ≪ . (1.1)This regime is extremely intriguing since systematic analytical calculations are possible. In partic-ular it has been shown that for both formulations [9, 10, 11] at LO in the e -expansion these theoriesare equivalent respectively to Wilson chiral Random Matrix Theory (WChRMT) for the Wilsonformulation and to staggered Chiral RMT for the staggered one.In these chiral theories one has also to understand the relative size between the quark mass m andthe lattice spacing a . For example in WChPT there are three possible power countings [12, 13]that are usually applied depending on the appearance of the cut-off effect at the LO, the so-calledAoki-regime, at Next-to-Leading order (NLO) called GSM ∗ regime, or Next-to-Next-to-Leadingorder (NNLO) called GSM regime in which NLO corrections to the spectral density of the WilsonDirac operator have already been computed [14]. Through these proceedings we analyze WChPTand SChPT using the first power counting scheme, namely when m ∼ a L QCD ∼ O ( e ) . (1.2)It is usually known also as large cut-off effect regime since at LO both the mass and the cut-offterms contribute with the same strength to the chiral symmetry breaking.In particular we address the problem of the extension up to O ( e ) of the partition function forWChPT with N f =2 flavors and for SChPT for generic N f . The main result that we present hereand in a forthcoming publication [15] is the possibility to write the NLO partition function for boththeories as the LO one with renormalized LECs. This is analogous to what happens in continuumchiral perturbation theory [16, 17, 18] with the only difference that here the renormalization factorof the LECs can not be written in terms of the geometric data of the lattice alone. This result opensup the possibility to extend the relations WChPT/WChRMT and SChPT/SChRMT up to NLO.2 inite volume corrections to LECs in Wilson and staggered ChPT Fabrizio Pucci
2. Wilson Chiral Perturbation Theory for N f = 2 The Wilson chiral lagrangian for the two-flavor case with degenerate mass m can be written as L LO = F Tr (cid:2) ¶ m U ¶ m U † (cid:3) − S Tr (cid:2) M † U + U † M (cid:3) + a c (cid:0) Tr (cid:2) U + U † (cid:3)(cid:1) . (2.1)As usual S is the chiral condensate, F is the pion decay constant and c is the new LEC of WChPT.The main idea underlying the construction of the theory in the e -regime is that, since the zero modesdominate the path integral, one has to threat them non-perturbatively, in contrast to the propagatingquantum fluctuations. Thus the usual parameterization for the matrix U is given by U ( x ) = U exp " i √ F x ( x ) , (2.2)where U is the two by two unitary matrix describing the zero-modes and x are the fluctuations.Since also the NLO lagrangian needs to be considered in the present calculation, following [19, 20]we write it as L NLO = a c Tr (cid:2) ¶ m U ¶ m U † (cid:3) Tr (cid:2) U + U † (cid:3) + am c (cid:0) Tr (cid:2) U + U † (cid:3)(cid:1) + a d Tr (cid:2) U + U † (cid:3) + a d (cid:0) Tr (cid:2) U + U † (cid:3)(cid:1) , (2.3)where 4 new LECs are introduced, namely c , c , d and d . The idea is to expand the action S = Z d x ( L LO + L NLO ) (2.4)up to O ( e ) using the Aoki regime power counting V ∼ e − , m ∼ e , ¶ ∼ e , x ( x ) ∼ e , a ∼ e . (2.5)At LO the different contributions can be rearranged as S ( ) = Z d x Tr (cid:2) ¶ m x ( x ) ¶ m x ( x ) (cid:3) − mV S Tr h U + U †0 i + a V c (cid:16) Tr h U + U †0 i(cid:17) (2.6) ≡ S ( ) ¶ + S ( ) U . (2.7)Now the trick is to rewrite the partition function by separating the integration over the zero-modesfrom the integration over the Gaussian fluctuations as Z = Z SU ( ) [ d H U ( x )] e − S = Z SU ( ) d H U e − S ( ) U Z x ( U ) (2.8)with Z x ( U ) = Z [ d x ( x )] J ( x ( x )) e S ( ) U − S . (2.9)The factor J ( x ( x )) is the Jacobian arising from the change of integration variables. At this pointwe can expand the function Z x ( U ) up to O ( e ) and then perform all the Gaussian integrals usingthe expression Z [ d x ( x )] exp h − S ( ) ¶ i x ( x ) i j x ( y ) kl = (cid:18) d il d jk − N f d i j d kl (cid:19) D ( x − y ) (2.10)3 inite volume corrections to LECs in Wilson and staggered ChPT Fabrizio Pucci in terms of the propagator D ( x − y ) . We easily find that Z x ( U ) = N (cid:26) + (cid:18) − mV S F D ( ) − a d V (cid:19) Tr h U + U †0 i + (cid:18) a c VF D ( ) − amc V (cid:19) (cid:16) Tr h U + U †0 i(cid:17) − a d V (cid:16) Tr h U + U †0 i(cid:17) (cid:27) (2.11)where N is an overall normalization. In dimensional regularization the propagator D ( ) is finiteand can be written as D ( ) = − b √ V (2.12)with b a numerical coefficient that encodes the geometrical data of the lattice.Now with some algebraic manipulations, using the properties of the SU ( ) group and the relation (cid:18) +
16 ˆ a c (cid:19) h Tr h U + U †0 i i + ˆ m h (cid:16) Tr h U + U †0 i(cid:17) i − ˆ a c h (cid:16) Tr h U + U †0 i(cid:17) i − m = S and c , the renormalized low energy constants S e f f and c e f f defined as S e f f = S (cid:18) − F D ( ) − ˆ a ˆ m √ V (cid:18) a d +
32 ˆ a d − d c (cid:19)(cid:19) (2.14)and c e f f = c (cid:18) − F D ( ) (cid:19) + ˆ m ˆ a (cid:18) c S + d c (cid:19) √ V . (2.15)Here we have defined ˆ m ≡ m S V and ˆ a ≡ a V (2.16)which are of order 1. Thus the NLO partition function reads as Z NLO = N ′ Z SU ( ) d H U exp (cid:20) m S e f f V Tr h U + U †0 i − a c e f f V (cid:16) Tr h U + U †0 i(cid:17) (cid:21) = N ′ N Z LO ( S e f f , c e f f ) . (2.17)Since effective LECs given above at NLO depend in a non trivial way on the additional LECs andnot only on the geometrical data of the lattice, in principle it is possible to use a finite volumescaling analysis to extract the numerical value of these undetermined NLO LECs. Performing thesimulations at two different lattice volume V and V with geometries b and b , WChPT predictsa scaling of the LECs as S e f f ( V ) S e f f ( V ) = + F ( b √ V − b √ V ) √ V V + (cid:18) ad mc S (cid:19) (cid:18) V − V (cid:19) , (2.18) c e f f ( V ) c e f f ( V ) = + F ( b √ V − b √ V ) √ V V . (2.19)4 inite volume corrections to LECs in Wilson and staggered ChPT Fabrizio Pucci
3. Staggered Chiral Perturbation Theory
The staggered chiral lagrangian has been introduced in [5] and [6] respectively for the one-flavortheory and for the general N f case and reads as L LO = F Tr (cid:0) ¶ m U ¶ m U † (cid:1) − S Tr (cid:0) M † U + U † M (cid:1) − a C Tr (cid:0) U g U † g (cid:1) − a C (cid:229) m < n Tr (cid:0) U g mn U † g mn (cid:1) − a C (cid:229) m (cid:2) Tr (cid:0) U g m U g m (cid:1) + h . c . (cid:3) − a C (cid:229) m (cid:2) Tr (cid:0) U g m U g m (cid:1) + h . c . (cid:3) − a C V (cid:229) m (cid:2) Tr (cid:0) U g m (cid:1) Tr (cid:0) U g m (cid:1) + h . c . (cid:3) − a C A (cid:229) m (cid:2) Tr (cid:0) U g m (cid:1) Tr (cid:0) U g m (cid:1) + h . c . (cid:3) − a C V (cid:229) m (cid:2) Tr (cid:0) U g m (cid:1) Tr (cid:0) U † g m (cid:1)(cid:3) − a C A (cid:229) m (cid:2) Tr (cid:0) U g m (cid:1) Tr (cid:0) U † g m (cid:1)(cid:3) . (3.1)The 4 N f × N f unitary matrix U is parameterized as U = u p + K + ... p − d K ... K − ¯ K s ...... ... ... . . . with u , p + , K + ... being 4 × C , C A , C V , C , C , C A , C V , C are introduced in the chiral lagrangian.In the Aoki regime the lagrangian (3.1) describes the LO unrooted theory. If we want to go beyondwe have to consider also the NLO terms that potentially arise from the discretization. However, incontrast to the Wilson theory here the first correction appear only at NNLO, making the computa-tion easily respect to the Wilson case. Thus if we want to study the finite volume correction to theLECs C i we will have only to expand the LO lagrangian up to the O ( e ) order.In order to begin we rewrite the partition function as Z = Z SU ( N f ) [ d H U ( x )] e − S = Z SU ( N f ) D H U e − S ( ) U Z x ( U ) (3.2)where as in the previous section we have separated the integration over the zero-modes U fromthe integration over the fluctuations x . Now we can expand the function Z x ( U ) up to order O ( e ) ,perform the Gaussian integrals over the fluctuations and finally after re-exponentiating all the termswe can absorbs the O ( e ) corrections in the renormalized LECs. At the end we can write Z NLO = N ′ N Z LO (cid:16) S e f f , C e f fi (cid:17) (3.3)where the value of the S e f f and C e f fi are given in the following table.5 inite volume corrections to LECs in Wilson and staggered ChPT Fabrizio Pucci S eff = S (cid:18) − N f − F N f D ( ) (cid:19) C eff = C (cid:16) − N f F D ( ) (cid:17) C eff V = C V − C V ( N f − )+ C N f N f F D ( ) C eff A = C A − C A ( N f − )+ C N f N f F D ( ) C eff = C − C ( N f − )+ C V N f N f F D ( ) C eff = C − C ( N f − )+ C A N f N f F D ( ) C eff V = C V (cid:16) − N f F D ( ) (cid:17) C eff A = C A (cid:16) − N f F D ( ) (cid:17) C eff = C (cid:16) − N f F D ( ) (cid:17) Table 1.
The renormalized SChPT LECs.
From a tree level expansion of the NLO chiral lagrangian we can reads the NLO masses of thenon-neutral mesons composed of quark b and cm = m ( m b + m c ) + a D NLO x B . (3.4)Here the taste splittings D NLO x B depend obviously by the taste state identified by the taste matrix x B .All the states fall into 5 different classes : the Pseudoscalar (PS), Axial-Vector (AV), Tensor (T),Vector (V) and Singlet (S) sector. In such channels the taste splitting [6] can be written at NLOwhen inserting our results from table 1.: D NLOPS = D NLOA = F ( C + C + C + C ) − F N f [ C + C ] + [ C + C ]( N f − ) + [ C V + C A ] N f N f ! D ( ) (3.6) D NLOT = F ( C + C + C ) − F N f C + [ C + C ]( N f − ) + [ C V + C A ] N f N f ! D ( ) (3.7) D NLOV = F ( C + C + C + C ) − F N f [ C + C ] + [ C + C ]( N f − ) + [ C V + C A ] N f N f ! D ( ) (3.8) D NLOI = F ( C + C ) − F [ C + C ]( N f − ) + [ C V + C A ] N f N f ! D ( ) . (3.9)
4. Summary and Discussion
Throughout this paper we have shown that in the e -regime for two-flavor Wilson chiral perturbationtheory and for general N f staggered ChPT the NLO order partition function can be written as theLO one with renormalized effective LECs.This result leads to several consequences that we will expand upon in a forthcoming publication For flavor neutral mesons the situation is more complicated and other terms have to be introduced in the chirallagrangian. inite volume corrections to LECs in Wilson and staggered ChPT Fabrizio Pucci [15]. The first regards the possibility to extend the relations between LO WChPT and SChPT withChiral Random Matrix Theory in its Wilson and staggered version respectively.The second consequence is that in WChPT, due to the fact that the finite volume corrections changethe mean field potential, this effect changes the phase boundaries of the theory.A further point is the extension of our result to Wilson ChPT with general N f . In that case thesituation is more involved since 3 LO and 9 NLO LECs have to be introduced and naturally thechiral lagrangian becomes more involved, including the question of possible constraints on thesigns of individual LECs and combinations of these. Acknowledgments
Partial support by the SFB | TR12 “Symmetries and Universality in Mesoscopic Systems” of theGerman research council DFG is acknowledged (G.A.). F.P. is supported by the Research Execu-tive Agency (REA) of the European Union under Grant Agreement PITNGA- 2009-238353 (ITNSTRONGnet).
References [1] J. Gasser and H. Leutwyler, Annals Phys. (1984) 142.[2] S. Weinberg, Physica A (1979) 327.[3] S. R. Sharpe and R. L. Singleton, Jr, Phys. Rev. D (1998) 074501 [hep-lat/9804028].[4] S. Aoki, Phys. Rev. D (2003) 054508 [arXiv:hep-lat/0306027].[5] W. J. Lee and S. R. Sharpe, Phys. Rev. D (1999) 114503 [arXiv:hep-lat/9905023].[6] C. Aubin and C. Bernard, Phys. Rev. D (2003) 034014 [arXiv:hep-lat/0304014].[7] J. Gasser and H. Leutwyler, Phys. Lett. B (1987) 83.[8] J. Gasser and H. Leutwyler, Phys. Lett. B (1987) 477.[9] P. H. Damgaard, K. Splittorff and J. J. M. Verbaarschot, Phys. Rev. Lett. (2010) 162002[arXiv:1001.2937 [hep-th]].[10] G. Akemann, P. H. Damgaard, K. Splittorff and J. J. M. Verbaarschot, Phys. Rev. D (2011) 085014[arXiv:1012.0752 [hep-lat]].[11] J. C. Osborn, Phys. Rev. D (2011) 034505 [arXiv:1012.4837 [hep-lat]].[12] A. Shindler, Phys. Lett. B (2009) 82 [arXiv:0812.2251 [hep-lat]].[13] O. Bar, S. Necco and S. Schaefer, JHEP (2009) 006 [arXiv:0812.2403 [hep-lat]].[14] S. Necco and A. Shindler, JHEP (2011) 031 [arXiv:1101.1778 [hep-lat]].[15] G. Akemann and F. Pucci, [arXiv:1211.3980 [hep-lat]].[16] P. H. Damgaard, M. C. Diamantini, P. Hernandez and K. Jansen, Nucl. Phys. B (2002) 445[arXiv:hep-lat/0112016].[17] P. H. Damgaard, T. DeGrand and H. Fukaya, JHEP (2007) 060 [arXiv:0711.0167 [hep-lat]].[18] G. Akemann, F. Basile and L. Lellouch, JHEP , 069 (2008) [arXiv:0804.3809 [hep-lat]].[19] M. T. Hansen and S. R. Sharpe, Phys. Rev. D (2012) 054504 [arXiv:1112.3998 [hep-lat]].[20] O. Bar, G. Rupak and N. Shoresh, Phys. Rev. D (2004) 034508 [arXiv:hep-lat/0306021].(2004) 034508 [arXiv:hep-lat/0306021].