Implications of Poincaré symmetry for thermal field theories

L. Giusti STRONGnet 23 Graz - September 23 p. /27 Implications of Poincaré symmetry for thermal field theories Leonardo Giusti University of Milano-Bicocca Based on: L. G. and H. B. Meyer JHEP 3 (23) 4, JHEP (2) 87 and PRL 6 (2) 36 L. G. and M. Pepe Lattice 23 and in preparation

L. Giusti STRONGnet 23 Graz - September 23 p. 2/27 Outline Introduction Free-energy density with shifted boundary conditions f ql 2 + z2 = lim V L V ln Z(L, z) Z φ(l, x) = φ(,x z) X Ward Identities in infinite and finite volume Applications on the lattice: step-scaling function for the entropy density Conclusions and outlook

L. Giusti STRONGnet 23 Graz - September 23 p. 2/27 Outline Introduction Free-energy density with shifted boundary conditions q f L + ξ 2 = lim V L V ln Z(L, ξ) Z φ(l, x) = φ(, x L ξ) X Ward Identities in infinite and finite volume Applications on the lattice: step-scaling function for the entropy density Conclusions and outlook

Thermal field theories in the Euclidean path integral formalism From textbooks Z(L ) = Tr ne L H b o where the temperature is T = /L φ(x) = φ(x + V pbc m) m Z 4 V pbc = B @ L L L 2 L 3 C A The basic thermodynamic quantities are defined as f = L V ln Z(L ), e = V ln Z(L ), s = L2 L V n o ln Z(L ) L L which in the thermodynamic limit lead to p = f, s = L (e + p), c v = L L s They are defined in terms of f and of the cumulants of the energy distribution c n = V b H n c L. Giusti STRONGnet 23 Graz - September 23 p. 3/27

L. Giusti STRONGnet 23 Graz - September 23 p. 4/27 Path integrals with shifted boundary conditions: infinite-volume limit (I) We are interested in the partition function Z(L, ξ) = Tr ne o L ( H iξ b P b ) where we have chosen ξ = {ξ,, } φ(x) = φ(x + V sbc m) m Z 4 V sbc = B @ L L ξ L L 2 L 3 C A By making an Euclidean boost rotation γ = q + ξ 2 Λ = B @ γ γ ξ γ ξ γ C A

L. Giusti STRONGnet 23 Graz - September 23 p. 5/27 Path integrals with shifted boundary conditions: infinite-volume limit (II) Assuming that eh has a translationally-invariant vacuum and a mass gap [ξ = {ξ,,}] n Z(L, ξ) = Tr o exp L γ ( eh + iξ ep ), V sbc = Λ V sbc = B @ L /γ L γ ξ L γ L 2 L 3 C A the right hand side becomes insensitive to the phase in the limit L at fixed ξ q f L + ξ 2 = lim V L V ln Z(L, ξ) Thanks to cubic symmetry (infinite volume) V sbc = B @ L /γ L γ L 2 L 3 C A q f L + ξ 2 = lim V L V ln Z(L, ξ), φ(l, x) = φ(, x L ξ) for a generic shift ξ

L. Giusti STRONGnet 23 Graz - September 23 p. 6/27 Thermal field theory in a moving frame If b H and b P are the Hamiltonian and the total momentum operator expressed in a moving frame, the standard partition function is Z(L, v) Tr ne o L ( H v b bp ) If we continue Z to imaginary velocities v = iξ Z(L, ξ) = Tr {e L ( b H iξ bp ) } The functional dependence f(l p + ξ 2 ) is consistent with modern thermodynamic arguments on the Lorentz transformation of the temperature and the free-energy [Ott 63; Arzelies 65; see Przanowski for a recent discussion] In the zero-temperature limit the invariance of the theory (and of its vacuum) under the Poincaré group forces its free energy to be independent of the shift ξ At non-zero temperature the finite length L breaks SO(4) softly, and the free energy depends on the shift (velocity) explicitly but only through the combination β = L p + ξ 2

L. Giusti STRONGnet 23 Graz - September 23 p. 7/27 Cumulants of the energy and the momentum distributions The cumulants of the momentum distribution are k {2n,,} = V bp 2n c = ( )n+ L 2n 2n ξ 2n q f L + ξ 2 ξ=

L. Giusti STRONGnet 23 Graz - September 23 p. 7/27 Cumulants of the energy and the momentum distributions The cumulants of the momentum distribution are k n {2n,,} = ( ) n+ (2n )!! L L o nf q L + ξ 2 ξ= L The cumulants of the energy distribution are c n = V b H n c = ( ) n+ " n n L n + L n L n # q f L + ξ 2 ξ= n = 2,3... which imply that k {2n,,} = (2n )!! (2L 2 )n nx l= (2n l)! l!(n l)! (2L ) l c l the total energy and momentum distributions of a relativistic thermal theory are related

L. Giusti STRONGnet 23 Graz - September 23 p. 7/27 Cumulants of the energy and the momentum distributions The cumulants of the momentum distribution are k n {2n,,} = ( ) n+ (2n )!! L L o nf q L + ξ 2 ξ= L The cumulants of the energy distribution are c n = V b H n c = ( ) n+ " n n L n + L n L n # q f L + ξ 2 ξ= n = 2,3... Up to n = 4 it reads L k {2,,} = c L 3 k {4,,} = 9 c + 3 L c 2, L 5 k {6,,} = 225 c + 9 L c 2 + 5 L 2 c 3, L 7 k {8,,} = 25 c + 4725 L c 2 + 5 L 2 c 3 + 5 L 3 c 4

L. Giusti STRONGnet 23 Graz - September 23 p. 7/27 Cumulants of the energy and the momentum distributions The cumulants of the momentum distribution are k n {2n,,} = ( ) n+ (2n )!! L L o nf q L + ξ 2 ξ= L The cumulants of the energy distribution are c n = V b H n c = ( ) n+ " n n L n + L n L n # q f L + ξ 2 ξ= n = 2,3... Thermodynamic potentials can be extracted from the momentum cumulants k {2,,} = T (e + p) = T 2 s k {4,,} = 3T 4 (c v + 3s) by remembering that p T = s

L. Giusti STRONGnet 23 Graz - September 23 p. 8/27 Thermodynamics from the response to the shift From the previous formulas s = ( + ξ2 ) 3/2 ξ k lim V V ξ k ln Z(L, ξ) or at zero shift s = lim V V 2 ξ 2 k ln Z(L, ξ) ξ= Note that: No reference to field operators Ultraviolet renormalization not needed on the lattice Finite volume effects exponentially small as in correlators

L. Giusti STRONGnet 23 Graz - September 23 p. 9/27 Euclidean Ward identities for correlators of T µν In the path integral formalism L T T c = T T L 3 T T T T c = 9 T 9 T + 3 L T T c... where T = e, T = p, b P it k and T µν (x ) = Z d 3 x T µν (x) Note that: All operators at non-zero distance Number of EMT on the two sides different On the lattice they can be imposed to fix the renormalization of T µν

L. Giusti STRONGnet 23 Graz - September 23 p. /27 Ward identities at non-zero shift When ξ odd derivatives in the ξ k do not vanish anymore, and new interesting WIs hold. The first non-trivial one is T k ξ = ξ k ξ 2 k T ξ T kk ξ which implies s = L ( + ξ 2 ) 3/2 ξ k T k ξ By deriving twice with respect to the ξ k T k ξ = L ξ k 2 X ij Ti T j ξ, c» δ ij ξ i ξ j ξ 2 which implies for instance s = X 2( + ξ 2 ) 3/2 ij Ti T j h ξ, c ξ i ξ j δ ij ξ i iξ j T i ξ T j ξ ξ 2

L. Giusti STRONGnet 23 Graz - September 23 p. /27 Ward identities at non-zero shift When ξ odd derivatives in the ξ k do not vanish anymore, and new interesting WIs hold. The first non-trivial one is T k ξ = ξ k ξ 2 k T ξ T kk ξ which implies s = L ( + ξ 2 ) 3/2 ξ k T k ξ By deriving twice with respect to the ξ k T k ξ = L ξ k 2 X ij Ti T j ξ, c» δ ij ξ i ξ j ξ 2 which implies for instance c v s 2 = X 2( + ξ 2 ) 3/2 ij Ti T j ξ, c ξ i ξ h j T i ξ T j ξ ξ 2 ( 2ξ 2 )δ ij 3 ξ i iξ j ξ 2

L. Giusti STRONGnet 23 Graz - September 23 p. /27 Path integrals with shifted boundary conditions: finite-size effects The leading finite-size contributions to the free energy are q f(v sbc ) f L + ξ 2 = I + I 2 + I 3 + where for L k = L I i = γν 2πL L 3 r d dr h e MLr r i r=ri, r i = γ γ s, γ i = / + X ξk 2 i k i with M and ν being the mass and the multiplicity of the lightest screening state respectively Analogous formula for the entropy by noticing that T k Vsbc T k ξ = ξ k 3X I i +... i=

L. Giusti STRONGnet 23 Graz - September 23 p. 2/27 Ward identities in a finite spatial box (I) In finite volume the SO(4) invariance implies exact relations among partition functions and correlation functions defined with different sets of (generalized) periodic bound. cond. V sbc = B @ L L ξ L L ξ 2 L 2 L ξ 3 L 3 C A V = B @ L γ L γ ξ L /γ L ξ 2 L 2 L ξ 3 L 3 C A Z(V sbc ) = Z(V ) By deriving once with respect to ξ k the first Ward Identity is given by T Vsbc + + ξ2 ξ 2 T V = ξ ξ 2 T Vsbc T Vsbc where T V vanishes also if L kγ 2 k ξ k L = q Z

L. Giusti STRONGnet 23 Graz - September 23 p. 3/27 Ward identities in a finite spatial box (II) By deriving twice with respect to ξ k and then by fixing ξ k = L T k T k Vsbc,c L k e T k T k Vsbc,c = T Vsbc T kk Vsbc where all insertions in the same correlator are at a physical distance from each other and et µν (w k ) = Z h Y ρ k dw ρ i T µν (w) Analogously the fourth derivative leads to L 3 T kt k T k T k Vsbc,c L 3 k e T k e Tk e Tk T k Vsbc,c = o 3 n T Vsbc T kk Vsbc + 3 6 n L 2 T kt k T Vsbc,c L 2 k T e o ktk e T kk Vsbc,c n L k T e o kk T kk Vsbc,c L T T Vsbc,c +...

L. Giusti STRONGnet 23 Graz - September 23 p. 4/27 Ward identities in a finite spatial box (III) R R The commutator of boost with momentum O i [ ˆK k, ˆP k ] = iĥ is expressed in the Euclidean by the WIs Tok X Z R dσ µ (x) K µ;k (x) T k (y ) O... O n c = T (y ) O... O n c when the O i are localized external fields. In a 4D box boost transformations are incompatible with (periodic) boundary conditions. WIs associated with SO(4) rotations must be modified by finite-size contributions The finite-volume theory is translational invariant, and it has a conserved T µν. Modified WIs associated to boosts constructed from those associated to translational invariance L T k (x ) T k (y) Vsbc,c L k e T k (w k ) T k (z) Vsbc,c = T T kk Vsbc,c

L. Giusti STRONGnet 23 Graz - September 23 p. 5/27 Lattice gauge theory [Wilson 74] A Yang-Mills theory can be defined on a discretized space-time so that gauge invariance is preserved Quark fields reside on a four-dimensional lattice, the gauge field U µ SU(3) resides on links L a The Wilson action is S G [U] = β 2 X X x µ,ν» 3 n o ReTr U µν (x) x + ν x + µ + ν where β = 6/g 2 and the plaquette is defined as U ν (x) U µν (x) = U µ (x) U ν (x + ˆµ) U µ (x + ˆν) U ν (x) x x + µ U µ (x) Discrete shifts in the boundary conditions can be implemented straightforwardly

L. Giusti STRONGnet 23 Graz - September 23 p. 6/27 A finer scan in the temperature on the lattice A finer scan in the temperature values become possible at fixed lattice spacing at = β a = r L a 2 + L ξ a 2 + L ξ 2 a 2 + L ξ 3 a 2

L. Giusti STRONGnet 23 Graz - September 23 p. 7/27 Non-perturbative renormalization of T µν (traceless components) On the lattice translational invariance is broken down to a discrete group and standard discretizations of T µν acquire finite ultraviolet renormalizations We focus here on the SU(3) Yang Mills, but the analysis applies to other theories as well [Caracciolo et al. 88, 9] T R = Z T T T R TR = Z T z T (T T ) V T = B @ L L 5 2 2 L L L C A There is a great freedom in choosing the renormalization conditions. A possibility is z T = 3 2 T VT T VT T VT, while Z T can be determined from Z T z T = T VT T 22 VT L T 2 (x ) T 2 (y) VT,c L e T 2 (x 2 ) T 2 (y) VT,c, x y, x 2 y 2

L. Giusti STRONGnet 23 Graz - September 23 p. 8/27 Strategy to compute entropy density and specific heat Once Z T has been computed in a small volume (for instance), the entropy density can be computed as (ξ k ) s = Z T L ( + ξ 2 ) 3/2 ξ k T k Vsbc thanks to the misalignment of the lattice axes with respect to the periodic directions Analogously the specific heat is given by (ξ k and L k chosen to be equal) T c v s = 3 T 2 + T ξ2 T V sbc,c V sbc,c ξ 2 T T T T 2 Vsbc,c V sbc,c Note that: No ultraviolet-divergent power subtraction needed Renormalization constant fixed non-perturbatively by WIs

L. Giusti STRONGnet 23 Graz - September 23 p. 9/27 Step-scaling function of the entropy density A step-scaling function for s(t)/t 3 can be defined as Σ s (T, r) r 3 s(rt) s(t) = ( + ξ2 ) 3 ( + ζ 2 ) 3 ζ k ξ k T k ξ T k ζ where the step is given by r = p + ζ 2 / p + ξ 2 The entropy density at a given T can then be obtained by solving the recursive relation v = s(t ) T 3, v k+ = Σ s (T k, r)v k, T k = r k T Being T the only relevant scale in the problem (no zero-temperature subtraction needed), various orders of magnitude in T can be spanned this way in the spirit of [Lüscher et al. 93-94]

L. Giusti STRONGnet 23 Graz - September 23 p. 2/27 Finite-size effects in the step-scaling function The leading finite-size corrections are given by s = 45 L 4 Tk s SB 32π 2 γ 6 Vsbc T k ξ ξ k The perturbative expression for the lightest screening mass is [Laine, Vepsalainen 9] M 2 = 2 Tg +..., M 3 = 3 T g2 4π +... It is realistic to consider boxes with (LT) > where finite-size effects are negligible. Thanks to the locality of the observable, the cost of the simulation is volume independent at fixed statistical error

L. Giusti STRONGnet 23 Graz - September 23 p. 2/27 Step-scaling function on the lattice We consider the step-scaling function Σ s (T, 2) = 8 T (,,) T (,,) with the energy-momentum tensor defined as T µν = β 6 n FµαF a να a o 4 δ µνfαβ a F αβ a. where F a µν(x) = i nh 4a 2 Tr Q µν (x) Q νµ (x) it ao, Q µν (x) = X In the Yang Mills theory the step-scaling function does not require any ultraviolet renormalization factor, and it has an universal continuum limit

L. Giusti STRONGnet 23 Graz - September 23 p. 22/27 Numerical computation of the step-scaling function (I) id L/a L /a β T L So far we have considered 8 different values of the temperature in the range T c 5Tc T 2, T,..., 4 2 T, 8T where T = /L max.82t c [Capitani et al. 98] For each temperature, 3 (4) values of the lattice spacing have been simulated to extrapolate the step-scaling function to the continuum limit We have chosen aspect ratios (LT) > 2. Finite-size effects checked explicitly with dedicated runs. They are negligible within statistical errors Z 8 3 6.43 3.4 Z a 96 3 6.43 22.6 Z 2 96 4 6.2257 2. Z 3 28 5 6.3875 2.8 A 8 3 6.267 3.4 A a 96 3 6.267 22.6 A 2 96 4 6.4822 2. A 3 28 5 6.6575 2.8 B 8 3 6.5282 3.4 B a 96 3 6.5282 22.6 B 2 96 4 6.7533 2. B 3 28 5 6.983 2.8 C 8 3 6.779 3.4 C a 96 3 6.779 22.6 C 2 96 4 7.2 2. C 3 28 5 7.268 2.8 D 8 3 7.694 3.4 D a 96 3 7.694 22.6 D 2 96 4 7.3 2. D 3 28 5 7.4963 2.8 E 8 3 7.42 3.4 E a 96 3 7.42 22.6 E 2 96 4 7.654 2. E 3 28 5 7.8435 2.8 F 8 3 7.739 3.4 F a 96 3 7.739 22.6 F 2 96 4 7.9489 2. F 3 28 5 8.45 2.8

L. Giusti STRONGnet 23 Graz - September 23 p. 23/27 Numerical computation of the step-scaling function (II) id L/a L /a β T L For the A lattices we fix T = T = /L max In the first three steps the bare coupling constant is fixed by requiring that [Necco, Sommer ] L max /r =.738(6) From the 4 th step, we interpolate quadratically in ln (L/a) each set of data at constant ḡ 2 (L) and we choose [Capitani et al. 98] = L k at k a = L 2 k/2 max a Z 8 3 6.43 3.4 Z a 96 3 6.43 22.6 Z 2 96 4 6.2257 2. Z 3 28 5 6.3875 2.8 A 8 3 6.267 3.4 A a 96 3 6.267 22.6 A 2 96 4 6.4822 2. A 3 28 5 6.6575 2.8 B 8 3 6.5282 3.4 B a 96 3 6.5282 22.6 B 2 96 4 6.7533 2. B 3 28 5 6.983 2.8 C 8 3 6.779 3.4 C a 96 3 6.779 22.6 C 2 96 4 7.2 2. C 3 28 5 7.268 2.8 D 8 3 7.694 3.4 D a 96 3 7.694 22.6 D 2 96 4 7.3 2. D 3 28 5 7.4963 2.8 E 8 3 7.42 3.4 E a 96 3 7.42 22.6 E 2 96 4 7.654 2. E 3 28 5 7.8435 2.8 F 8 3 7.739 3.4 F a 96 3 7.739 22.6 F 2 96 4 7.9489 2. F 3 28 5 8.45 2.8

Extrapolation to the continuum limit of Σ s (PRELIMINARY) Σ s (T, 2 /2 ).3.28.26.24.22.2.8.6.4.2..8.6.4.2.98.96..2.3.4.5.6.7.8.9...2.3.4.5 (a/l ) 2 Discretization effects and statistical errors are at the level of per mille Numerical simulations at the 4 th value of the lattice spacing, L /a = 6, are running. A precision of half a percent in the continuum limit is within reach At fixed statistical error, the cost of the simulation is volume independent thanks to the locality of the observable L. Giusti STRONGnet 23 Graz - September 23 p. 24/27

Extrapolation to the continuum limit of Σ s (PRELIMINARY).22.2.8.6.4 Σ s (T, 2 /2 ).2..8.6.4.2.98 2 3 4 5 6 7 8 9 2 T/T c Discretization effects and statistical errors are at the level of per mille Numerical simulations at the 4 th value of the lattice spacing, L /a = 6, are running. A precision of half a percent in the continuum limit is within reach At fixed statistical error, the cost of the simulation is volume independent thanks to the locality of the observable L. Giusti STRONGnet 23 Graz - September 23 p. 24/27

L. Giusti STRONGnet 23 Graz - September 23 p. 25/27 Results for the entropy density (PRELIMINARY) 8 7 6 5 s/t 3 4 3 2 2 3 4 5 6 7 8 9 2 3 4 5 6 T/T c By fixing (temporarily) the renormalization factor Z T from the literature [L.G., Meyer ], the temperature dependence of s(t)/t 3 is obtained by solving the recursive relation

L. Giusti STRONGnet 23 Graz - September 23 p. 26/27 Conclusions and outlook (I) Lorentz invariance implies a great degree of redundancy in defining a relativistic thermal theory in the Euclidean path-integral formalism In the thermodynamic limit, the orientation of the compact periodic direction with respect to the coordinate axes can be chosen at will and only its length is physically relevant q f L + ξ 2 = lim V L V ln Z(L, ξ) The redundancy in the description implies that the total energy and momentum distributions in the canonical ensemble are related For a finite-size system, the lengths of the box dimensions break this invariance. Being a soft breaking, however, interesting exact Ward Identities survive If the lightest screening mass M, leading finite-size corrections exponentially small in (ML) as in the standard case

L. Giusti STRONGnet 23 Graz - September 23 p. 27/27 Conclusions and outlook (II) When the theory is regularized on a lattice, the overall orientation of the periodic directions with respect to the lattice coordinate system affects renormalized observables at the level of lattice artifacts As the cutoff is removed, the artifacts are suppressed by a power of the lattice spacing The flexibility in the lattice formulation added by the introduction of a triplet ξ of (renormalized) parameters has interesting consequences: A finer scan of the temperature value WIs to renormalize non-perturbatively T µν Simpler ways to compute thermodynamic potentials s = Z T L ( + ξ 2 ) 3/2 ξ k T k Vsbc In the Yang Mills theory the entropy density can be computed over several orders of magnitude in T by introducing a step-scaling function. Discretization errors at the level of per mille. Keeping the statistical errors at the same level is very feasible