Finite-volume effects in $(g-2)^{\text{HVP,LO}}_\mu$

An analytic expression is derived for the leading finite-volume effects arising in lattice QCD calculations of the hadronic-vacuum-polarization contribution to the muon's magnetic moment, $a_\mu^{\text{HVP,LO}} \equiv (g-2)_\mu^{\text{HVP,LO}}/2$. For calculations in a finite spatial volume with periodicity $L$, $a_\mu^{\text{HVP,LO}}(L)$ admits a transseries expansion with exponentially suppressed $L$ scaling. Using a Hamiltonian approach, we show that the leading finite-volume correction scales as $\exp[- M_\pi L]$ with a prefactor given by the (infinite-volume) Compton amplitude of the pion, integrated with the muon-mass-dependent kernel. To give a complete quantitative expression, we decompose the Compton amplitude into the space-like pion form factor, $F_\pi(Q^2)$, and a multi-particle piece. We determine the latter through NLO in chiral perturbation theory and find that it contributes negligibly and through a universal term that depends only on the pion decay constant, with all additional low-energy constants dropping out of the integral.

As the only known, systematically-improvable approach to non-perturbative QCD, numerical lattice QCD is a natural tool in the determination of the LO HVP where a systematic and precise value is of great importance.The most common approach is to estimate a HVP,LO µ ≡ (g − 2) HVP,LO µ /2 via the integral [30] a HVP,LO where α ≈ 1/137 is the fine-structure constant, m µ the * e-mail: maxwell.hansen@cern.ch† e-mail: agostino.patella@physik.hu-berlin.demuon mass and G T,L (x 0 ) ≡ − 1 3 Here j µ (x) = f q f ψ f (x)γ µ ψ f (x) is the Euclideansignature vector current and K 1 (z) a Bessel function.We have used notation to emphasize that the calculation is performed in a finite-volume T × L 3 Euclidean spacetime with periodic geometry.
In Eq. (1) the finite temporal extent is accommodated by cutting the integral at T /2.We leave a detailed analysis of finite-T effects, arising both from the boundary conditions and the treatment of large x 0 in the integral, to a future work.In this work we consider only the finite-L effects, defining a HVP,LO µ (L) ≡ lim T →∞ a HVP,LO µ (T, L).We will show that this quantity has only exponentiallysuppressed finite-volume effects, and the suppression is controlled by the pion mass M π .
Even when T is taken very large, the large-x 0 region of the integral in Eq. ( 1) cannot be calculated from the measured two-point function because of the wellknown exponential degradation of the signal-to-noise ratio.In practice, one can calculate the two-point function G T →∞,L (x 0 ) for x 0 < τ c from numerical simulations (possibly with a mild extrapolation to saturate the T → ∞ limit), and then use additional inputs to reconstruct the x 0 > τ c region.This yields a decomposition where the superscript "recon" stands for reconstructed.The first term is calculated by restricting the integration domain in Eq. (1) to 0 < x 0 < τ c and by using the measured two-point function.The second term is obtained arXiv:1904.10010v2[hep-lat] 8 Apr 2020 from an analogous formula where the integral is taken over τ c < x 0 < ∞ and the reconstructed two-point function is used.
We will see that a µ (L|x 0 < τ c ) approaches the infinitevolume limit exponentially fast.On the other hand, a recon µ (L|x 0 > τ c ) may approach L → ∞ more slowly, depending on the exact prescription used.As an extreme example, if one estimates G ∞,L (x 0 ) for x 0 > τ c by summing over a fixed number of finite-volume states, the resulting contribution to the HVP will have powerlaw L-dependence [31,32].In practice more sophisticated procedures are employed and the resulting scaling must be considered on a case by case basis. 1   Our main result, the formula for the leading exp[−M π L] finite-volume effect to a HVP,LO µ (L), is presented in Sec.II, and is derived in Sec.III by means of a hamiltonian formalism in which quantization along a spatial direction is used to pick out the complete functional form non-perturbatively.In Sec.IV we discuss the implications of our expression for ongoing calculations.We find that the dominant contribution enters through the space-like pion form factor, and, since the latter is readily calculated on the lattice, this provides a viable method for correcting the leading L-dependence.We estimate also the dominant contribution to the finite-volume effects of a µ (L|x 0 < τ c ), which can be useful information when devising a strategy along the lines of Eq. (4).
Our results differ from Refs.[34,35] in that these work to a fixed order in chiral perturbation theory (ChPT) whereas our result is the full non-perturbative expression, to leading-order in the large L expansion.In this regard it is worth emphasizing that the strict chiral expansion is limited by the fact that, at N 3 LO, the momentum-space vector correlator receives a Q 6 contribution that leads to a divergence in the integral defining a HVP,LO µ .

II. RESULT
We define where, as in the introduction, we ignore the effects of the finite temporal extent.These scale as e −MπT and e −Mπ √ T 2 +L 2 .Therefore in the commonly used setup T = 2L, the finite-T corrections are sub-leading and should be dropped.The separation is plausible from the perspective of a generic effective field theory.Volume effects can be encoded via position-space propagators, summed over all periodic images.The propagator's form then leads to exponential decay falling according to the image distance multiplied with the pion mass.The detailed proof of this separation, based on the methods of Ref. [36], is given in a second longer publication.
In Sec.III we show that the leading finite-L corrections are given by where T q is the Compton amplitude in the forward limit.Here |p, q is the relativisticallynormalized state of a single pion with momentum p and charge q, k 2 = k 2 0 − k 2 and k • p = k 0 p 0 − kp are the Minkowski squared norm and scalar product.Following 1 As explained in Ref. [30], one can use the Lellouch-Lüscher formalism [31][32][33], or else some model [20], to extract the time-like pion form factor in infinite volume, and use this as an input in the spectral representation to calculate the contribution of states below the four-pion threshold to aµ(L|x 0 > τc), directly in infinite volume.In this case one trades the finite-volume effects for other systematics that depend on the particular chosen procedure.
the discussion after Eq. ( 5), the subleading exponential, e − √ 2MπL , arisies from an image displaced in two of the spatial directions.
J µ (x) is the Minkowski current.In the Schrödinger picture this is related to its Euclidean counterpart via and the corresponding Heisenberg operators are

III. DERIVATION
Define G Lρ (x 0 ) exactly as G T,L (x 0 ) in Eq. ( 2) but in a volume in which all four directions may differ, i.e. with as the finite-volume residue due to compactification in the 3 direction only.
To determine ∆G 3 (x 0 |L), we study G Lρ (x 0 ) with geometry L ρ = (L ⊥ , L ⊥ , L ⊥ , L) and quantize along the 3 direction.Defining x = (x 1 , x 2 , x 0 ) = (x ⊥ , x 0 ), the Hamiltonian representation of the Euclidean two-point function yields where the Hamiltonian has a discrete finite-volume spectrum of states in L 3 ⊥ and the trace is taken over this Hilbert space.Here we are using L ⊥ to ensure that intermediate expressions are well-defined.This will be sent to infinity at the end of the calculation.For simplicity, in this formula we have assumed periodic boundary conditions for gluons and antiperiodic boundary conditions for fermions in the 3 direction.To account for the commonly-used periodic boundary contitions for fermions one should introduce (−1) F in all traces, where F is the fermion number.This does not change the leading exponential contribution, since this is due to singlepion, hence bosonic, states.
Let |n be a basis of simultaneous eigenstates of the Hamiltonian (eigenvalue E n ), the momentum (eigenvalue p n ), the charge (eigenvalue q n ) operators.Inserting a complete set of such states in both the numerator and the denominator then gives The role of the coordinates x 0 and x 3 in this analysis is potentially confusing.In our final results x 0 plays the role of the time coordinate.This is the coordinate of integration in Eq. ( 1), typically parametrizing the longest Euclidean direction.Here, to identify the leading Ldependence, it is convenient to quantize along the 3 direction.One must only take care that, in any given expression, all energies and all states are consistently defined with respect to the same quantization direction.
Returning to Eq. ( 12), the integral over x 3 can be calculated explicitly.To avoid the need of separating E n = E n terms from the rest, we introduce the following identity, which holds for all values of Substituting into Eq.( 12) and exchanging n ↔ n in certain terms, we obtain This expectation value can be expressed in terms of the (finite-volume) Minkowskian two-point function via which is valid for > 0 and can be easily proven using Eq. ( 10) and integrating over t explicitly.We stress that this is a mathematical identity and the parameter t has no relation to any of the spacetime coordinates in the system.
The expansion about L → ∞ is now straightforward as the exponentials are manifest and one can identify the relevant contribution.Neglecting terms of order e −2MπL we reach where the connected expectation value is defined as n|O|n c ≡ n|O|n − 0|O|0 .At this point we can take the L ⊥ → ∞ limit.This is done by replacing the sum over the states in the one-particle region with the phasespace integral and by replacing the connected expectation value with the forward limit where the definition (7) has been used.In the L ⊥ → ∞ limit, the integrals over t and x ⊥ are readily calculated, yielding delta functions in k 0 = k ⊥ = 0, i.e.
In the final expression note that any contribution to the integrand that is odd in k 3 , p 3 → −k 3 , −p 3 must integrate to zero, which justifies the replacement e ik3x0 → cos(k 3 x 0 ).To complete the derivation we note that the restriction E(p) < 2M π can be dropped, as this amounts to an error of the same order as terms that we are neglecting.Finally, the integral over p ⊥ can be explicitly calculated.We reach Multiplying the result by 3 to account for the three directions with compactification L, we conclude Eq. ( 6). 2  We close by commenting on different choices of boundary conditions.If fermions satisfy e iθ -periodic boundary conditions [37,38], i.e. ψ f (x+L ρ ρ) = e iθ f ρ ψ f (x), Eq. ( 11) should be modified by inserting (−1) F e i f θ f 3 N f in all traces, where N f is the number operator for the flavour f .In this case, Eq. ( 6) is modified by replacing 3 q=0,±1 where we have used T +1 = T −1 ≡ T ±1 as follows from charge-conjugation invariance.

IV. IMPLICATIONS
Having derived the leading-order functional form of ∆a µ (L), we close by considering the implications for ongoing numerical LQCD calculations.Here we mainly focus on periodic boundary conditions but comment again on the role of twisting below.For convenience we define the charge-summed Compton amplitude, T ≡ q T q .
We begin by rewriting Eq. ( 6) as where we have introduced 2 This last step assumes a decomposition similar to that allowing us to neglect finite-T and is demonstrated in detail in a subsequent publication. 3We drop terms of order e − √ 2Mπ L throughout this section.
T 1 E Z 7 x 6 5 q 3 4 B 9 7 9 H y E D q + Y = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " Q m x q X 8 5 W + m w w k d N X n t L A 4 m y F M a 8 = " and T is the second derivative of T with respect to k 2 3 .We next decompose the Compton amplitude into its pole and analytical contributions where F π is the space-like pion form factor and the separation defines T reg .This implies where we have introduced which can be readily reduced to forms well suited to numerical evaluation.The second term in Eq. ( 26) is given by Eq. ( 23) with T → T reg . 4 r 2 j w M d a 6 e M w k b p i w F P 1 9 0 R O I q X G U a A 7 I w J D N e 9 N x P 8 8 J 4 P w 0 s t 5 n G b

µ
< l a t e x i t s h a 1 _ b a s e 6 4 = " Q m x q X 8 5W + m w w k d N X n t L A 4 m y F M a 8 = " > A A A C O 3 i c b V B N S x x B E O 1 R E 3 X z t Z q j l 8 I l Y C D Z z E h A D z l I 4 s G D k I 2 4 q 7 C z D j W 9 t d r Y 8 0 F 3 T X A Z 5 n / l 4 p / w 5 s W L B y V 4 9 Z 7 e d Q 6 J 5 k H D 4 7 1 X d N W L c 6 0 s + / 6 l N z M 7 9 + z 5 / M J i 4 8 X L V 6 / f N J e W e z Y r j K S u z H R m D m O 0 p F V K X V a s 6 T A 3 h E m s 6 S A + / T b x D 3 6 S s S p L 9 3 m c 0 y D B 4 1 S N l E R 2 U t T c + w i B f 7 Q O I a u E L I T b p B k B o z A p Y A 1 2 I X T D D G e R D 1 9 c B o u o D J n O u J R V B e8 / T X N H t b T T 6 3 z Y / V 5 V U b P l t / 0 p 4 C k J a t I S N T p R 8 y I c Z r J I K G W p 0 d p + 4 O c 8 K N

TABLE I .
Contribution to ∆aµ(L) from the Fπ(Q 2 )-term for functional forms as indicated.Here we take mµ/Mπ = 106/137 and M/Mπ = 727/137.As a reference value we take a HVP,LO