The epsilon expansion at next-to-next-to-leading order with small imaginary chemical potential
PPreprint typeset in JHEP style - HYPER VERSION
April 30, 2010, KEK-CP-234, RBRC-842
The epsilon expansion at next-to-next-to-leadingorder with small imaginary chemical potential
Christoph Lehner a,b , Shoji Hashimoto c , and Tilo Wettig a a Institute for Theoretical Physics, University of Regensburg, 93040 Regensburg, Germany b RIKEN/BNL Research Center, Brookhaven National Laboratory, Upton, NY-11973, USA c High Energy Accelerator Research Organization (KEK), Tsukuba 305-080, JapanEmail: [email protected] , [email protected] , [email protected] A BSTRACT : We discuss chiral perturbation theory for two and three quark flavors in the epsilonexpansion at next-to-next-to-leading order (NNLO) including a small imaginary chemical potential.We calculate finite-volume corrections to the low-energy constants Σ and F and determine the non-universal modifications of the theory, i.e., modifications that cannot be mapped to random matrixtheory (RMT). In the special case of two quark flavors in an asymmetric box we discuss how tominimize the finite-volume corrections and non-universal modifications by an optimal choice of thelattice geometry. Furthermore we provide a detailed calculation of a special version of the masslesssunset diagram at finite volume.K EYWORDS : epsilon expansion, imaginary chemical potential, finite-volume corrections,low-energy constants. a r X i v : . [ h e p - l a t ] A p r ontents
1. Introduction 12. The finite-volume effective theory at NNLO 2
3. Optimal geometries: Two quark flavors in an asymmetric box 84. Conclusions 11A. The massless sunset diagram at finite volume 11
A.1 The term P ( m ) P ( m ) P B ( m ) P C ( m ) P D ( m ) P E ( m )
1. Introduction
At low energies the theory of quantum chromodynamics (QCD) can be described by a chiral effec-tive theory. If the theory is confined to a finite volume and considered for small quark masses, the ε -regime power counting applies and replaces the standard p -regime power counting, see Ref. [1].The corresponding systematic expansion is called ε -expansion.At leading order (LO) in the ε -expansion the theory becomes zero-dimensional and is there-fore described by chiral RMT [2], see [3, 4] for reviews. The dimensionless quantities of RMTare mapped to the dimensionful quantities of the chiral effective theory using the LO low-energyconstants (LECs) Σ and F , see, e.g., Ref. [5]. In this way Σ and F , which are of great phenomeno-logical importance, can be obtained from lattice QCD simulations in the ε -regime by a fit to RMTpredictions. While Σ can be determined rather easily, e.g., from the distribution of the small Diraceigenvalues, the extraction of F is more complicated and can be done, e.g., by the inclusion of asuitable chemical potential [6, 7] or by using twisted boundary conditions [8].Lattice QCD simulations in the ε -regime using an exactly chiral lattice fermion formulationare feasible, and recently such a set of lattice configurations, which also includes the effect ofdynamical quarks, was generated by the JLQCD and TWQCD collaborations [9, 10, 11]. In an– 1 –pcoming publication [12] we further investigate the eigenvalue spectrum of the Dirac operator onthese configurations.For typical volumes of state-of-the-art lattice QCD simulations, finite-volume corrections tothe RMT predictions cannot be neglected. These corrections can be calculated in higher orders ofthe ε -expansion. At next-to-leading order (NLO) the analytical results of RMT still apply, but themapping of RMT to chiral perturbation theory is modified, i.e., the LECs Σ and F are replaced byfinite-volume effective LECs Σ eff and F eff , see Refs. [13, 14, 15]. At NNLO, chiral perturbationtheory can no longer be mapped to RMT, i.e., non-universal modifications appear. These non-universal modifications determine the systematic errors in fits of lattice data to RMT predictions.In this paper we calculate the finite-volume corrections at NNLO in the ε -expansion in Eu-clidean space-time. We allow for nonzero imaginary chemical potential and consider its contribu-tion to leading order. In this way we extend results of Hansen [16], which were obtained withoutchemical potential. The paper is structured as follows. In Sec. 2 we calculate the finite-volume ef-fective action and the corresponding finite-volume effective low-energy and high-energy constants(HECs) at NNLO. In Sec. 3 we discuss how to minimize the systematic deviations from RMT aswell as the finite-volume corrections to Σ and F in the special case of two quark flavors on anasymmetric lattice by an optimal choice of the lattice geometry. We conclude in Sec. 4 and providea detailed calculation of a special version of the massless sunset diagram at finite volume in App. A.
2. The finite-volume effective theory at NNLO
In this section we discuss the finite-volume effective theory at NNLO in the ε -expansion. To thisend we give the bare Lagrangian of the p -expansion at NLO and the LO contribution of the invariantintegral measure in Sec. 2.1. In Sec. 2.2 we define the finite-volume effective action, and in Sec. 2.3we calculate the finite-volume effective LECs and HECs. The LECs L , . . . , L and the HEC H that appear in the bare Lagrangian of the p -expansion at NLO are scale-dependent. We renormalizethe theory in Sec. 2.3 and confirm that the scale dependence of L , . . . , L and H is the same asin the ordinary p -expansion [17]. We parametrize the Nambu-Goldstone manifold of chiral perturbation theory with N f quark flavorsby U ( x ) = U exp (cid:20) i √ F ξ ( x ) (cid:21) , (2.1)where ξ is a complex matrix in flavor space of dimension N f with ξ = ξ † and Tr ξ = 0 . Theconstant mode is separated in U , and thus (cid:82) d x ξ ( x ) = 0 , see Ref. [1]. For nonzero imaginarychemical potential the LO Lagrangian of the effective theory in Euclidean space-time is given by L = F ∇ ρ U ( x ) − ∇ ρ U ( x )] − Σ2 Tr[ M † U ( x ) + U ( x ) − M ] (2.2)with ∇ ρ U ( x ) = ∂ ρ U ( x ) − iδ ρ [ C, U ( x )] , (2.3)– 2 –here M = diag ( m , . . . , m N f ) , m f is the mass of quark flavor f , C = diag ( µ , . . . , µ N f ) ,and iµ f is the imaginary chemical potential of quark flavor f , see, e.g., Ref. [14]. We considervery small quark masses such that the Compton wavelength of the pions, given by the Gell-Mann–Oakes–Renner relation m π = 2 m f Σ F , (2.4)exceeds the size of the space-time box. This defines the ε -regime of QCD that was first discussedin Ref. [1]. The corresponding ε -regime power counting [1] is defined by V ∼ ε − , M ∼ ε , C ∼ ε , ∂ ρ ∼ ε , ξ ( x ) ∼ ε (2.5)which gives a consistent perturbative expansion in ε ∼ /F √ V , see, e.g., Ref. [15]. The ε -expansion is applicable if the quantities m f V Σ and µ f F V (2.6)are not larger than O (1) and the volume is sufficiently large, i.e., πF √ V (cid:29) , (2.7)see also the quantitative discussion in Sec. 3.In order to include all NNLO contributions in the ε -expansion we need to include the NLOLagrangian of the p -expansion which for N f = 3 is given by L = − L [Tr( ∇ µ U − ∇ µ U )] − L Tr( ∇ µ U − ∇ ν U ) Tr( ∇ µ U − ∇ ν U ) − L Tr( ∇ µ U − ∇ µ U ∇ ν U − ∇ ν U )+ (cid:18) F (cid:19) L Tr( ∇ µ U − ∇ µ U ) Tr( M U − + M † U )+ (cid:18) F (cid:19) L Tr[ ∇ µ U − ∇ µ U ( M U − + M † U )] − (cid:18) F (cid:19) L [Tr( M U − + M † U )] − (cid:18) F (cid:19) L [Tr( M U − − M † U )] − (cid:18) F (cid:19) L Tr(
M U − M U − + M † U M † U ) − (cid:18) F (cid:19) H Tr( M † M ) , (2.8)where H is a high-energy constant (HEC) corresponding to a contact term that is needed in therenormalization of one-loop graphs, see Refs. [16] and [17]. The field-strength tensors are notincluded since they vanish in the case of a constant vector source [17], of which an imaginarychemical potential is a special case, and therefore the LECs L and L and the HEC H do notappear. Note that in the case of N f = 2 not all of the terms in L are independent and the LECs L , . . . , L can be mapped to the LECs l , . . . , l as described in Ref. [18]. In the case of N f > one needs to include additional terms in Eq. (2.8), see, e.g., Ref. [18].The invariant measure relevant to NNLO in the ε -expansion is given by d [ U ] = d [ U ] d [ ξ ] (cid:18) − N f F V (cid:90) d x Tr[ ξ ( x ) ] (cid:19) , (2.9)see Ref. [16], where d [ U ] is the invariant measure of the constant mode and d [ ξ ] is the flat measureof the fields ξ . – 3 – .2 The finite-volume effective action The volume dependence of the theory is contained entirely in the propagators of the fields ξ [19],and therefore we can obtain a finite-volume effective action in terms of the constant mode U by averaging over the fluctuations in ξ . At NLO this leads to a finite-volume effective actionthat differs from the LO (RMT) result by finite-volume corrections to the LECs Σ and F , seeRefs. [13, 14, 15]. We now perform the expansion of the partition function Z = (cid:90) d [ U ] e − (cid:82) d x ( L + L ) (2.10)in terms of fields ξ to NNLO and average over the fields using computer algebra, resulting in Z = (cid:90) d [ U ] e − S eff . (2.11)The finite-volume effective action S eff contains invariant terms including the constant mode U , themass matrix M , and the chemical potential matrix C . At NNLO in the ε -expansion and to leadingorder C in the imaginary chemical potential, S eff can be written as S eff = − V Σ eff M † U + U − M ) − V F eff CU − CU )+ Υ Σ( V F ) Tr( C )[Tr( U { M † , C } ) + Tr( U − { C, M } )]+ Υ Σ( V F ) Tr( { M † , C } U C + { C, M } CU − + { U , C } U − CU M † + CU { C, U − } M U − )+ Υ Σ( V F ) Tr( U − CU C + C ) Tr( M U − + M † U )+ Υ Σ( V F ) Tr( U − CU C − C ) Tr( M U − + M † U )+ Υ Σ( V F ) Tr([ M † , C ] U C + [ C, M ] CU − + [ U , C ] U − CU M † + CU [ C, U − ] M U − )+ Υ ( V Σ) [Tr( M U − + M † U )] + Υ ( V Σ) [Tr( M U − − M † U )] + Υ ( V Σ) [Tr( M U − M U − ) + Tr( M † U M † U )]+ H V F Tr( C ) + H ( V Σ) Tr( M † M ) + H V F (Tr C ) , (2.12)where { A, B } = AB + BA and Σ eff , F eff , and Υ , . . . , Υ are finite-volume effective LECs thatwill be given in Sec. 2.3. Note that the terms corresponding to H , H , and H do not couple to U , and therefore the H i can be viewed as finite-volume effective HECs. The terms proportional tothe H i only contain external sources and are therefore not relevant for low-energy phenomenology,but they are needed for the computation of operator expectation values, see Ref. [20]. They are alsorelevant to the renormalization of the coupling constants discussed in Sec. 2.3. The finite-volumeeffective HECs H i are also given in Sec. 2.3.Unlike the LECs L , . . . , L and the HEC H , the finite-volume effective LECs Υ , . . . , Υ and HECs H , H , and H are finite and depend on the volume. Specifically, we show in Sec. 2.3 We use a C++ library for tensor algebra developed by one of the authors. – 4 –hat Σ eff , F eff , H = O ( ε ) , H = O ( ε ) , H , Υ , . . . , Υ = O ( ε ) , (2.13)i.e., H , H , and Υ , . . . , Υ vanish in the infinite-volume limit.The terms corresponding to the Υ i cannot be mapped to RMT, see, e.g., Ref. [5]. Thereforetheir magnitude quantifies the systematic deviations from RMT at finite volume. We will return tothis point at the end of Sec. 3. In the following we express the finite-volume effective LECs Σ eff , F eff , and Υ , . . . , Υ and thefinite-volume effective HECs H , H , and H in terms of shape coefficients P , . . . , P defined be-low. The resulting expressions for the effective LECs and HECs are given in terms of the masslessfinite-volume propagator in dimensional regularization, ¯ G ( x ) = 1 V (cid:88) k (cid:54) =0 e ikx k , (2.14)where the sum is over all nonzero momenta, see Ref. [19]. We use the identity ∂ ρ ¯ G ( x ) (cid:12)(cid:12) x =0 = 1 V (2.15)and finally express the result in terms of the shape coefficients P = V ∂ ¯ G (0) , P = √ V ¯ G (0) ,P = √ V (cid:90) d d x [ ∂ ¯ G ( x )] ¯ G ( x ) , P = (cid:90) d d x ¯ G ( x ) ,P = (cid:90) d d x d d y [ ∂ ¯ G ( x + y )] ¯ G ( x ) ¯ G ( y ) , P = V (cid:90) d d x [ ∂ ¯ G ( x )] ¯ G ( x ) , (2.16)where d is the number of space-time dimensions. We use conservation of momentum, by which allone-loop propagators can be related to [19] ¯ G r = Γ( r ) V (cid:88) k (cid:54) =0 k ) r , (2.17)where r ∈ R and Γ( r ) is the Gamma function [21]. The partial derivatives can be evaluated using ∂ ˜ L ν ( ˜ L ν ¯ G r ) = 2Γ( r + 1) V (cid:88) k (cid:54) =0 k ν ( k ) r +1 , (2.18)where ˜ L ν is the length of the space-time box in dimension ν = 0 , , , and ∂ ˜ L ν denotes thepartial derivative w.r.t. ˜ L ν . Note that in Eq. (2.18) no sum over ν is implied. The shape coefficients P , . . . , P contain only one-loop propagators and are given by P = − V ∂ ˜ L ( ˜ L ¯ G ) , P = √ V ¯ G ,P = − √ V ∂ ˜ L ( ˜ L ¯ G ) , P = ¯ G ,P = − ∂ ˜ L ( ˜ L ¯ G ) . (2.19)– 5 –or convenience we state the result of Ref. [19] explicitly as ¯ G r = lim m → (cid:20) π ) d/ Γ( r − d/ m ) d/ − r + g r − Γ( r ) V m r (cid:21) ,V g = β + β m √ V + 12 β m V − log( m √ V )+ V m π ) (cid:20) log( m √ V ) − (cid:21) + O ( m ) ,g r +1 = − ∂g r ∂ ( m ) (2.20)with shape coefficients β n given in Eq. (B.14) of Ref. [19]. We express ¯ G , ¯ G , and ¯ G in termsof β , β , and β and find P = −
12 ˜ L ∂ ˜ L β + 14 , P = − β ,P = 14 β + 12 ˜ L ∂ ˜ L β , P = − λ + β + log √ V (4 π ) ,P = − P −
14 ˜ L ∂ ˜ L β − π ) , (2.21)where we borrow the definition of λ from Ref. [17], λ = µ d − (4 π ) (cid:26) d − − (cid:2) (cid:48) (1) + log(4 π ) (cid:3)(cid:27) → π ) (cid:26) d − − (cid:2) (cid:48) (1) + log( µ ) + log(4 π ) (cid:3)(cid:27) . (2.22)We explicitly include the dependence on the scale µ , which we define with mass dimension one.Note that the logarithms of dimensionful quantities in Eqs. (2.21) and (2.22) can always be com-bined to logarithms of dimensionless quantities. The shape coefficient P is calculated in App. A,see Eq. (A.41). It contains a special version of the massless sunset diagram at finite volume.In the following we state the resulting expressions for Σ eff , F eff , Υ , . . . , Υ , H , H , and H .The finite-volume effective chiral condensate is given by Σ eff Σ = 1 − P F √ V ( N f − N − f ) −
12 (1 − N − f ) P F V + P F V ( N f −
1) + 8 F V (cid:2) ( N f − L + ( N f − N − f ) L (cid:3) , (2.23)which agrees with Eqs. (22) and (23) of Ref. [16]. The finite-volume effective LEC F eff F = 1 − N f P F √ V − N f P F √ V + 2 N f P P F V + 2 N f P F V + N f P F V + N f (2 P + 4 P + P ) F V + 16 F V (cid:2) ( N f − L + L + ( N f − N − f ) L (cid:3) + 16 P F V (cid:2) L + N f L + ( N f − N − f ) L (cid:3) (2.24)– 6 –nd the finite-volume effective HEC H = 12 + N f P F √ V − N f P P F V − N f P F V + 12 N f ( P − P ) F V + 8 F V (cid:2) ( N f − L + L + ( N f − N − f ) L (cid:3) + 8 P F V (cid:2) L + N f L + ( N f − N − f ) L (cid:3) (2.25)contain the contribution of the two-loop propagator in P . The other finite-volume effective LECsand HECs are given by Υ = 12 P + 4 P F V , Υ = − N f Υ , Υ = −
12 Υ , Υ = − P F V − L F V , Υ = − N f P F V − L F V , Υ = − (cid:0) N − f (cid:1) P F V − L F V , Υ = − L F V , Υ = 12 (cid:0) N − f − N f (cid:1) P F V − L F V (2.26)and H = (cid:0) N − f − N f (cid:1) P F V − H F V , H = − ( P + 2 P ) F √ V + N f P + 4 P ) F V + N f ( P + 4 P ) F V + 2 N f P P F V . (2.27)Note that the shape coefficients P , P , and P as well as H and the L i are divergent and needto be renormalized. We separate their scale dependence as P = P r − λ , P = P r + 12 λ , P = P r + 13 λ − P λ ,L i = L ri + Γ i λ with i = 1 , . . . , , H = H r + ∆ λ , (2.28)where the quantities with superscript r are finite. For N f = 3 the divergences in Eqs. (2.23) -(2.27) can be absorbed if and only if we choose Γ = 18 , Γ = 38 , Γ = 11144 , Γ = 0 , Γ = 548 , ∆ = 524 , (2.29)and + 2Γ + 163 Γ = 2Γ + 9Γ + 73 Γ . (2.30)The coefficients Γ , . . . , Γ and ∆ are equal to the coefficients obtained in the one-loop expansionin the p power counting, see Ref. [17]. The renormalization conditions of Eq. (2.30) for Γ , Γ , Γ are also compatible with the result of Ref. [17], Γ = 332 , Γ = 316 , Γ = 0 . (2.31)– 7 –ote that the divergences in H , Υ , Υ , and Υ cancel independently of the choice of Γ i .To summarize, we obtain finite expressions for the finite-volume effective LECs and HECs inEqs. (2.23) - (2.27) if we replace P , P , P , H and the L i by their corresponding renormalizedparts with superscript r . Note that the dependence on the scale µ drops out of the final resultsfor the finite-volume effective LECs and HECs. The renormalization in the case of N f = 2 isdiscussed in Sec. 3.
3. Optimal geometries: Two quark flavors in an asymmetric box
In the following we discuss the finite-volume corrections to Σ and F and the coefficients of thenon-universal terms at NNLO in the ε -expansion. We explicitly consider the case of N f = 2 andan asymmetric box with lattice geometries ( a x ) ˜ L = xL , ˜ L = ˜ L = ˜ L = L , ( b x ) ˜ L = xL , ˜ L = ˜ L = ˜ L = L , (3.1)where x ∈ { , / , , , } . The three-flavor coupling constants L , . . . , L can be related to thetwo-flavor coupling constants l , l , and l by l = 4 L + 2 L , l = 4 L , l = 8 L + 4 L , (3.2)see, e.g., Eqs. (3.15) and (3.16) of Ref. [18]. Therefore Σ eff Σ = 1 − P F √ V − P F V + 3 P F V + 3 l F V (3.3)and F eff F = 1 − P F √ V − P F √ V + 8 P P + 8 P + 4 P F V + 8 P + 16 P + 4 P F V + 12 l + 4 l F V + P (8 l + 16 l ) F V . (3.4)In Ref. [22] the scale dependence of the coupling constants l i with i = 1 , . . . , is separated as l i = l ri + γ i λ , (3.5)where γ = 13 , γ = 23 , γ = 2 . (3.6)It is straightforward to check that the divergences in Eqs. (3.3) and (3.4) cancel with this set of γ , γ , and γ . We therefore obtain finite results for Σ eff and F eff if we replace P , P , P and l , l , l in Eqs. (3.3) and (3.4) by their corresponding renormalized parts with superscript r .The shape coefficients P , P , P and the renormalized shape coefficients P r , P r , P r at scale µ = V − / are given in Table 1 for geometries ( a x ) and ( b x ) . The details of the calculation of P r are given in App. A. – 8 – P P P r P r P r ( a ) 0 . − . . − . . − . a / ) − . − . − . − . . . a ) − . − . − . − . − . . a ) − . . − . . − . . a ) − . . − . . − . − . b / ) 0 . − . . − . . − . b ) 0 . − . . − . . − . b ) 0 . . . . . − . b ) 1 . . . . . . Table 1:
Shape coefficients P , P , P and renormalized shape coefficients P r , P r , P r at scale µ = V − / for geometries ( a x ) and ( b x ) . The error in the last column is due to the extrapolation described in App. A.3. ( a ) ( a / ) ( a ) ( a ) ( a )Σ NLOeff / Σ 1 . . . . . NNLOeff / Σ 1 . . . . . F NLOeff /F . . . . . F NNLOeff /F . . . . . b / ) ( b ) ( b ) ( b ) F NLOeff /F . . . . F NNLOeff /F . . . . Table 2:
Finite-volume corrections to Σ and F at NLO and at NNLO for geometries ( a x ) and ( b x ) at m π √ V = 1 , F = 90 MeV, and L = 1 . fm. The error in Σ NNLOeff is due to the uncertainty in ¯ l , the error in F NNLOeff is due to the uncertainty in ¯ l , ¯ l , and P r . The renormalized coupling constants l ri can be related to scale-independent constants ¯ l i by l ri = γ i π ) (cid:2) ¯ l i + log( m π /µ ) (cid:3) , (3.7)where m π is the mass of the pion, see Ref. [22], and ¯ l = − . ± . , ¯ l = 4 . ± . , ¯ l = 4 . ± . , (3.8)see Ref. [20]. Therefore l ri = γ i π ) (cid:2) ¯ l i + log( m π √ V ) (cid:3) (3.9)at scale µ = V − / . Note again that the finite-volume corrections to Σ and F are independent ofthe choice of scale µ .In Table 2 we give explicit values for the finite-volume corrections to Σ and F at NLO andNNLO for geometries ( a x ) and ( b x ) with m π √ V = 1 , F = 90 MeV, and L = 1 . fm, whichroughly corresponds to the values of the JLQCD lattice simulations. Note that the ε -expansionconverges well for this set of parameters as long as the asymmetry of the lattice is not too strong– 9 – . . . . . x . . . . . NNLOeff / Σ F NNLOeff /F ( a x ) ( b x ) Figure 1:
Finite-volume corrections to Σ and F at NNLO for geometries ( a x ) and ( b x ) at m π √ V = 1 , F = 90 MeV, and L = 1 . fm. (convergence is worst in geometry ( b ) ). We calculate the NLO result by discarding terms of order /F V in Eqs. (3.3) and (3.4). Note that for the same value of x , Σ eff is independent of the choiceof lattice geometry ( a x ) or ( b x ) . In Figure 1 we visualize the NNLO results of Table 2. We confirmthe picture obtained in Ref. [15] at NLO that the finite-volume corrections to F can be largelyreduced by an asymmetric lattice geometry with one appropriately large spatial dimension insteadof one large temporal dimension.We now turn to the non-universal terms that cannot be mapped to RMT. It follows fromEqs. (2.21) and (2.26) that the coefficients Υ , . . . , Υ are independent of the choice of latticegeometry ( a x ) or ( b x ) for the same value of x . The coefficients Υ , Υ , and Υ , however, areaffected by the choice of lattice geometry ( a x ) or ( b x ) , and we have Υ , Υ , Υ ∝ P r + 4 P r . (3.10)We give values for P r + 4 P r in Figure 2 at scale µ = V − / for lattice geometries ( a x ) and ( b x ) .Note that the non-universal contribution of Υ , Υ , and Υ is reduced significantly in lattice geom-etry ( b x ) compared to lattice geometry ( a x ) for the same value of x . In an upcoming publication[12] we perform the corresponding lattice simulation for x = 2 and show numerically that thesystematic deviations from RMT are indeed smaller for lattice geometry ( b ) compared to latticegeometry ( a ) . x ( P r + 4 P r )(4 π ) /x − − − Geometries ( a x ) Geometries ( b x ) Figure 2:
Linear combination of shape coefficients P r + 4 P r for different geometries at scale µ = V − / .We divide by x since V = xL in Eq. (2.26). – 10 – . Conclusions We discussed the ε -expansion at NNLO and determined the finite-volume effective action and thefinite-volume effective LECs and HECs to this order. In the special case of two dynamical quarksconfined to an asymmetric box we have confirmed the picture obtained at NLO that finite-volumecorrections to the LECs Σ and F can be significantly reduced by choosing one appropriately largespatial dimension instead of a large temporal dimension, see Ref. [15]. Furthermore, we haveshown that the systematic deviations from random matrix theory can also be reduced in the setupwith one large spatial dimension. This implies that in order to determine the LECs Σ and F fromeigenvalue correlation functions, as suggested in Refs. [6, 7] and performed in a pilot study inRef. [23], one should choose an asymmetric lattice with one large spatial dimension. This will bedemonstrated numerically in an upcoming publication [12].We would like to add that even though we did not explicitly perform our calculations in apartially quenched setup, it is straightforward to extend our results to the partially quenched case,see, e.g., Ref. [15] for a discussion at NLO, and we expect our findings to be unmodified by thepresence of valence quarks. Acknowledgments
We thank Hidenori Fukaya for stimulating discussions. Two of us (CL and TW) are grateful to theTheory Group of the IPNS, KEK for their hospitality. This work was supported in part by BayEFG(CL), the Grant-in-Aid (No. 21674002) of the Japanese Ministry of Education (SH), and DFG andKEK (TW).
A. The massless sunset diagram at finite volume
In this section we calculate the two-loop contribution defined by P = V (cid:90) d d x [ ∂ ¯ G ( x )] ¯ G ( x ) = − V (cid:88) k (cid:54) =0 k k (cid:88) p (cid:54) =0 , − k p ( p + k ) , (A.1)where the sum is over all nonzero momenta k and p with p + k (cid:54) = 0 , and k is the temporalcomponent of the momentum vector k . We first express the propagators without constant mode asthe limit of ordinary, massive propagators, P = lim m → P ( m ) = lim m → (cid:2) P ( m ) + P ( m ) (cid:3) (A.2)with P ( m ) = 2 m V (cid:88) k k ( k + m ) ,P ( m ) = − V (cid:88) k,p k ( p + m )[( p + k ) + m ]( k + m ) . (A.3)The terms P ( m ) and P ( m ) are calculated separately in the following.– 11 – .1 The term P ( m ) We partition the term P ( m ) into its infinite-volume part and the finite-volume propagator g defined in Eq. (2.20). We find P ( m ) = 2 m V (cid:88) k k ( k + m ) = 1 m ∂ ˜ L (cid:18) ˜ L V (cid:88) k k + m (cid:19) = 1 m π ) d/ Γ(1 − d/ m ) d/ − + 1 m g ( m ) + 1 m ˜ L ∂ ˜ L g ( m ) , (A.4)where g ( m ) = 1 V m − β √ V − m log( m √ V )(4 π ) − m β + O ( m ) . (A.5)Therefore we can express P ( m ) in terms of β , β , and λ as P ( m ) = 2 λ − m √ V ˜ L ∂ ˜ L β − β m √ V − log √ V (4 π ) − ∂ ˜ L ( ˜ L β ) − π ) + O ( m ) . (A.6) A.2 The term P ( m ) The second term P ( m ) = − V (cid:88) k k k + m (cid:88) p p + m )[( p + k ) + m ] (A.7)is more involved. We use Poisson’s summation formula ∞ (cid:88) n = −∞ e πinϕ = ∞ (cid:88) n = −∞ δ ( ϕ − n ) (A.8)and write P ( m ) = − V (cid:88) r,s (cid:90) d d k (2 π ) d d d p (2 π ) d exp (cid:20) i (cid:88) j ˜ L j ( r j k j + s j p j ) (cid:21) × k ( p + m )[( p + k ) + m ]( k + m ) , (A.9)where the sum is over r, s ∈ Z . We partition the sum over r and s into ( A ) r = 0 ∧ s = 0 , ( B ) r (cid:54) = 0 ∧ s = 0 , ( C ) r = 0 ∧ s (cid:54) = 0 , ( D ) r (cid:54) = 0 ∧ s (cid:54) = 0 ∧ s = r , ( E ) r (cid:54) = 0 ∧ s (cid:54) = 0 ∧ s (cid:54) = r . (A.10)Part (A) is given by the infinite-volume sunset diagram, see Ref. [24], which scales with V m d andtherefore vanishes in the massless limit. The parts P B ( m ) , . . . , P E ( m ) are calculated in thefollowing. – 12 – .2.1 The term P B ( m ) Along the lines of Eqs. (A.10) and (A.11) of Ref. [25] we separate P B ( m ) = P B ( m ) + P B ( m ) (A.11)with P B ( m ) = − V (cid:88) r (cid:54) =0 (cid:90) d d k (2 π ) d d d p (2 π ) d k k + m p + m ) exp (cid:18) i (cid:88) j ˜ L j r j k j (cid:19) ,P B ( m ) = − V (cid:88) r (cid:54) =0 (cid:90) d d k (2 π ) d d d p (2 π ) d k k + m exp (cid:18) i (cid:88) j ˜ L j r j k j (cid:19) × (cid:20) p + m )[( p + k ) + m ] − p + m ) (cid:21) . (A.12)The term P B ( m ) contains the ultraviolet divergence and can be calculated explicitly, P B ( m ) = − V (cid:90) d d p (2 π ) d p + m ) (cid:88) r (cid:54) =0 (cid:90) d d k (2 π ) d k k + m exp (cid:18) i (cid:88) j ˜ L j r j k j (cid:19) = − λP − m (4 π ) P + O ( V m d ) , (A.13)where P is the one-loop shape coefficient defined in Eq. (2.16). The term P B ( m ) is finite. Aftera tedious but straightforward calculation along the lines of Ref. [19] we can express P B ( m ) as P B ( m ) = − π ) (cid:88) r (cid:54) =0 (cid:90) ∞ dxdydz K ( x, y, z ) exp (cid:20) − ( x + y + z ) m √ V π (cid:21) (A.14)with K ( x, y, z ) = 1( xy + xz + yz ) (cid:20) x + y ) − ( ˜ L / √ V )[2 r ( x + y )] π ( yz + xy + xz ) (cid:21) × exp (cid:20) − (cid:88) j ( ˜ L j / √ V ) r j ( x + y ) π ( yz + xy + xz ) (cid:21) − x + y ) z (cid:104) − ( ˜ L / √ V )4 πr /z (cid:105) exp (cid:20) − (cid:88) j ( ˜ L j / √ V ) π r j z (cid:21) . (A.15)This expression is suitable for a numerical evaluation of P B ( m ) if we perform the integral over x , y and z in spherical coordinates. In Sec. A.2.4 we discuss how to efficiently calculate infinitesums such as the sums over r , . . . , r with r (cid:54) = 0 in Eq. (A.14). A.2.2 The term P C ( m ) The method used to separate the divergent part of P B ( m ) does not work for the integral over k since it has a power divergence. Nevertheless, we can calculate the divergent sub-diagram I µν ( m, p ) = (cid:90) d d k (2 π ) d k µ k ν ( k + m )[( p + k ) + m ] (A.16)– 13 –xplicitly. The result is given by [26] I µν ( m, p ) = g µν (cid:90) dx [ m + x (1 − x ) p ] log[ m + x (1 − x ) p ]2(4 π ) − p µ p ν (cid:90) dx x (4 π ) (cid:2) m + x (1 − x ) p ] (cid:3) + g µν λ (cid:18) p + m (cid:19) − λp µ p ν . (A.17)We can thus separate the divergent part of P C ( m ) = − V (cid:88) s (cid:54) =0 (cid:90) d d p (2 π ) d I ( m, p ) p + m exp (cid:18) i (cid:88) j ˜ L j s j p j (cid:19) , (A.18)which is given by [ P C ( m )] UV = − λV (cid:88) s (cid:54) =0 (cid:90) d d p (2 π ) d ( p + m ) + 5 m − p p + m exp (cid:18) i (cid:88) j ˜ L j s j p j (cid:19) = − λ − λP + O ( m ) . (A.19)In the calculation of [ P C ( m )] UV we used the identities V (cid:88) s (cid:54) =0 (cid:90) d d p (2 π ) d p + m exp (cid:18) i (cid:88) j ˜ L j s j p j (cid:19) = 1 m + O ( m ) , (cid:88) s (cid:54) =0 (cid:90) d d p (2 π ) d exp (cid:18) i (cid:88) j ˜ L j s j p j (cid:19) = 0 , (cid:88) s (cid:54) =0 (cid:90) d d p (2 π ) d p p + m exp (cid:18) i (cid:88) j ˜ L j s j p j (cid:19) = − P + O ( m d ) . (A.20)Note that the first two identities hold for arbitrary d . Thus there is no finite contribution from theproduct of these integrals with λ .The finite contributions to P C ( m ) are given by [ P C ( m )] finite = − V π ) (cid:88) s (cid:54) =0 J (cid:48) s ( m ) , (A.21)where J (cid:48) s ( m ) = (cid:90) dx (cid:90) d p (2 π ) F ( m , p ) − p F ( m , p ) p + m exp (cid:18) i (cid:88) j ˜ L j s j p j (cid:19) (A.22)with F ( m , p ) = [ m + x (1 − x ) p ] log[ m + x (1 − x ) p ] , F ( m , p ) = 2 x (cid:0) m + x (1 − x ) p ] (cid:1) . (A.23)– 14 – e p Im p i (cid:112) p ⊥ + m i (cid:112) p ⊥ + m /x (1 − x ) Figure 3:
The complex plane of p . We define L si = ˜ L i s i and rotate the coordinate system of p such that J (cid:48) s ( m ) = (cid:90) dx (cid:90) d p (2 π ) F ( m , p ) + (1 / ˜ L ) F ( m , p ) ∂ s p + m exp( iL s p ) (A.24)with L s = [ (cid:80) n =0 ( L sn ) ] / . After differentiating w.r.t. s we find J (cid:48) s ( m ) = (cid:90) dx (cid:90) d p (2 π ) [ F ( m , p ) + F ( m , p ) G s ( p V / )] exp( iL s p ) (cid:16) p − i (cid:113) p ⊥ + m (cid:17)(cid:16) p + i (cid:113) p ⊥ + m (cid:17) (A.25)with G s ( p V / ) = ip L s − ip s ˜ L ( L s ) − p s ˜ L ( L s ) (A.26)and p = p + p ⊥ . (A.27)In Figure 3 we sketch the structure of the integrand of Eq. (A.25) in the complex plane. Thereare two poles at p = ± i (cid:113) p ⊥ + m and a branch cut due to the logarithms in F ( m , p ) and F ( m , p ) . We can close the integration contour in the upper half-plane and find J (cid:48) s ( m ) = [ J (cid:48) s ( m )] p + [ J (cid:48) s ( m )] c , (A.28)where [ J (cid:48) s ( m )] p is the contribution of the pole and [ J (cid:48) s ( m )] c is the contribution of the branch cut.The contribution of the pole is given by [ J (cid:48) s ( m )] p = 1(2 π ) V (cid:90) dx (cid:90) ∞ d ˆ p ⊥ ˆ p ⊥ exp (cid:16) − l s (cid:113) ˆ p ⊥ + m √ V (cid:17) × F ( m , − m ) √ V + F ( m , − m ) G s (cid:16) i (cid:113) ˆ p ⊥ + m √ V (cid:17) √ V (cid:113) ˆ p ⊥ + m √ V (A.29)– 15 –ith l s = L s /V / and ˆ p ⊥ = p ⊥ V / . The contribution of the branch cut is given by [ J (cid:48) s ( m )] c = 1(2 π ) (cid:90) dx (cid:90) d p ⊥ (cid:90) i ∞ i √ p ⊥ + m /x (1 − x ) dp exp( iL s p ) × Disc F ( m , p ) + Disc F ( m , p ) G s ( p V / ) p + m , (A.30)where Disc F ( m , p ) = lim ε → (cid:2) F ( p ⊥ + ( p + ε ) ) − F ( p ⊥ + ( p − ε ) ) (cid:3) = 2 πi [ m + x (1 − x ) p ] , Disc F ( m , p ) = lim ε → (cid:2) F ( p ⊥ + ( p + ε ) ) − F ( p ⊥ + ( p − ε ) ) (cid:3) = 4 πix . (A.31)Therefore [ J (cid:48) s ( m )] c = 2(2 π ) V (cid:90) dx (cid:90) ∞ d ˆ p ⊥ ˆ p ⊥ (cid:90) ∞ (cid:113) ˆ p ⊥ + m √ V /x (1 − x ) dy exp( − l s y ) × m √ V + x (1 − x )(ˆ p ⊥ − y ) + 2 x G s ( iy ) √ Vy − ˆ p ⊥ − m √ V (A.32)with p = iy and thus dp = idy . In Sec. 3 we calculate [ P C ( m )] finite numerically at scale µ = V − / , i.e., we replace F ( m , p ) and F ( m , p ) by F ( m , p ) = [ m + x (1 − x ) p ] log (cid:2) m √ V + x (1 − x ) p √ V (cid:3) , F ( m , p ) = 2 x (cid:0) m √ V + x (1 − x ) p √ V ] (cid:1) . (A.33) A.2.3 The term P D ( m ) The term P D ( m ) is equal to the term P C ( m ) . This can be seen by shifting the integrationvariables p µ → p µ − k µ and using the invariance of the integral under k µ → − k µ . A.2.4 The term P E ( m ) The term P E ( m ) is finite and can be calculated numerically. We rewrite P E ( m ) analogouslyto P B ( m ) as P E ( m ) = − π ) (cid:88) r (cid:54) =0 ,s (cid:54) =0 ,r (cid:54) = s (cid:90) ∞ dxdydz xy + xz + yz ) × (cid:20) x + y ) − ( ˜ L / √ V )[ − s y + 2 r ( x + y )] πyz + xy + xz (cid:21) × exp (cid:20) − (cid:88) j ( ˜ L j / √ V )[ − r j s j y + r j ( x + y ) + s j ( y + z )] πyz + xy + xz (cid:21) × exp (cid:20) − ( x + y + z ) m √ V π (cid:21) . (A.34)– 16 –n the following we discuss how to efficiently calculate the infinite sums over s , . . . , s , r , . . . , r .We first define g = (cid:88) r (cid:54) =0 ,s (cid:54) =0 ,r (cid:54) = s (cid:89) j =0 exp( − a j r j − b j s j + c j r j s j + d j r j + e j s j )= (cid:88) s (cid:54) =0 (cid:20)(cid:88) r (cid:89) j exp( − a j r j − b j s j + c j r j s j + d j r j + e j s j ) − (cid:89) j exp( − b j s j + e j s j ) − (cid:89) j exp[ − ( a j + b j − c j ) s j + ( d j + e j ) s j ] (cid:21) = 2 + (cid:88) r,s (cid:89) j exp( − a j r j − b j s j + c j r j s j + d j r j + e j s j ) − (cid:88) s (cid:89) j exp( − b j s j + e j s j ) − (cid:88) r (cid:89) j exp( − a j r j + d j r j ) − (cid:88) s (cid:89) j exp[ − ( a j + b j − c j ) s j + ( d j + e j ) s j ] , (A.35)where a j , b j , c j , d j , e j ∈ C with Re a j , Re b j > and j = 0 , . . . , . In this way we can write g = (cid:89) j =0 ¯ g ( a j , b j , c j , d j , e j ) − (cid:89) j =0 ν ( b j , e j ) − (cid:89) j =0 ν ( a j , d j ) − (cid:89) j =0 ν ( a j + b j − c j , d j + e j ) + 2 , (A.36)where ν ( a, b ) = (cid:88) n exp( − an + bn ) , ¯ g ( a, b, c, d, e ) = (cid:88) n,m exp( − an − bm + cmn + dn + em )= (cid:88) m ν ( a, cm + d ) exp( − bm + em ) . (A.37)Since ν is a Jacobi theta function it transforms covariantly under inversion of a , ν ( a, b ) = (cid:114) πa exp( b / a ) ν ( π /a, iπ ( b/a )) , (A.38)which is readily shown using Poisson’s summation formula. For a ∈ R with a < π we cantherefore use Eq. (A.38) to achieve a swift convergence of the sum over n in ν ( a, b ) in Eq. (A.37).In fact in the least favorable case of a = π we need only to sum all n with | n − n | ≤ , where n = n ∈ Z maximizes − an + bn , to achieve a precision of − log [exp( − π )] ≈ digits. Inthis way we can express P E ( m ) as P E ( m ) = − π ) (cid:90) ∞ dxdydz R (cid:104) x + y ) − R ˜ l [ − y∂ e + 2( x + y ) ∂ d ] π (cid:105) × g ( e , d ) (cid:12)(cid:12)(cid:12) e = d =0 exp (cid:20) − ( x + y + z ) m √ V π (cid:21) (A.39)– 17 –i) (ii) (iii) extrapolation ( a ) − . − . − . − . a / ) 0 . . . . a ) 0 . . . . a ) 0 . . . . a ) − . − . − . − . b / ) − . − . − . − . b ) − . − . − . − . b ) − . − . − . − . b ) 0 . . . . Table 3:
Best fit of P r for regions (i), (ii), and (iii) and extrapolated values of P r for different geometries. with R = 1 yz + xy + xz , ˜ l j = ˜ L j /V / ,a j = R ˜ l j ( x + y ) π , b j = R ˜ l j ( y + z ) π ,c j = R ˜ l j πy , d = d = d = e = e = e = 0 . (A.40)The integral over x , y and z can be performed conveniently in spherical coordinates, as in the caseof P B ( m ) . A.3 The complete diagram
We combine all contributions to P ( m ) and find that the complete diagram at scale µ = V − / isgiven by ( P ) UV = 13 λ − λP ,P r ( m ) = − m √ V ˜ L ∂ ˜ L β − β m √ V − log( m √ V )(4 π ) P − ∂ ˜ L ( ˜ L β ) − π ) + 2[ P C ( m )] finite − π ) P + P B ( m ) + P E ( m ) ,P r = lim m → P r ( m ) , (A.41)where P r is the renormalized shape coefficient at scale µ = V − / . In Figs. 4 and 5 we show theresult of a numerical calculation of P r ( m ) for different values of m √ V .Note that the linear divergences as well as the logarithmic divergences in /m √ V cancel.We perform a fit to a polynomial of order four for different ranges of m √ V : (i) m √ V < . ,(ii) m √ V < . , and (iii) m √ V < . The result of the fits and the corresponding extrapolatedvalues for m = 0 are given in Table 3. – 18 – i) (ii) (iii) m √ VP r ( m )0 . . . − . − . − . (i) (ii) (iii) m √ VP r ( m )0 . . . . . . ( a ) ( a / ) (i) (ii) (iii) m √ VP r ( m )0 . . . . . . (i) (ii) (iii) m √ VP r ( m )0 . . . − . . . ( a ) ( a ) (i) (ii) (iii) m √ VP r ( m )0 . . . − . − . . ( a ) Figure 4:
Extrapolation of P r = lim m → P r ( m ) for lattice geometries ( a x ) , x ∈ { , / , , , } at scale µ = V − / . We fit a polynomial of order four including numerical data from three different ranges (i), (ii),and (iii). – 19 – i) (ii) (iii) m √ VP r ( m )0 . . . − . − . − . (i) (ii) (iii) m √ VP r ( m )0 . . . − . − . − . ( b / ) ( b ) (i) (ii) (iii) m √ VP r ( m )0 . . . − . − . − . (i) (ii) (iii) m √ VP r ( m )0 . . . − . . . ( b ) ( b ) Figure 5:
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