Quark mass difference effects in hadronic Fermi matrix elements from first principles

It was recently estimated that the strong isospin-symmetry breaking (ISB) corrections to the Fermi matrix element in free neutron decay could be of the order $10^{-4}$, one order of magnitude larger than the na\"{\i}ve estimate based on the Behrends-Sirlin-Ademollo-Gatto theorem. To investigate this claim, we derive a general expression of the leading ISB correction to hadronic Fermi matrix elements, which takes the form of a four-point correlation function in lattice gauge theory and is straightforward to compute from first principles. Our formalism paves the way for the first determination of such correction in the neutron sector with fully-controlled theory uncertainties.


I. INTRODUCTION
The strength of the Fermi transition in beta decays of hadrons and nuclei at tree level is determined by the Fermi matrix element M F , namely the matrix element of the isospinraising (or lowering) operator between the initial and final states.In the isospin-symmetric limit, M F is fully determined by group theory, but the presence of isospin-symmetry-breaking (ISB) effect induces a small correction that needs to be taken into account for precision tests of the Standard Model (SM).In particular, the Cabibbo-Kobayashi-Maskawa (CKM) matrix element V ud extracted from neutron and nuclear beta decays is approaching a 10 −4 precision [1], which calls for the same precision level for all SM theory inputs to the decay processes.
According to the well-known theorem by Behrends & Sirlin and Ademollo & Gatto (BSAG) [2,3] (see also Ref. [4]), the deviation of M F from its isospin-symmetric limit starts at second order in the ISB interactions.In nuclear beta decays where the primary source of ISB is the Coulomb repulsion between protons that scales as Zα, such correction is at (0.1-1)% level which is substantial.On the other hand, there is no large Z-enhancement in the hadron sector (e.g.pions and neutron), so one expects both strong and electromagnetic ISB corrections to take the natural size.For instance, an explicit calculation using Chiral Perturbation Theory (ChPT) indicates that the ISB effect in pion beta decay is below 10 −5 which can be very safely neglected [5].
The situation is more complicated for neutron decay.While it is widely accepted that the electromagnetically induced ISB effect is small, some recent analyses based on quark models suggested that the strong ISB effect could be on the order of 10 −4 [6,7].This exceeds earlier estimates [8] based on the scaling (m u − m d ) 2 /Λ 2 QCD ∼ 10 −5 , due to the coherent enhancement from the various excited state contributions which avoids the cancellation between electromagnetic and quark mass difference effects present in computing the mass difference.
Given that a 10 −4 correction is relevant to precision experiments, it is highly desirable to check this assertion using first-principles calculations with lattice Quantum Chromodynamics (QCD).A conventional way to study ISB corrections is to set up the calculation in the isospin-symmetric limit and treat the quark mass splitting as a perturbation.Observables such as the hadron mass splitting and the Dashen's theorem breaking parameter that scale linearly to the ISB parameters have been studied with this method [9], but the correction to M F , which is a quadratic effect, remains unexplored.
In this short paper we outline a procedure that allows for straightforward lattice calculations of strong ISB corrections to hadronic Fermi matrix elements to satisfactory precision.It is based on a method similar to the one developed earlier by some of us for nuclear beta decays [10], but without any of the model assumptions present in the latter.
In short, one computes the leading strong ISB correction derived from perturbation theory instead of the full Fermi matrix element; the former takes the form of a hadronic matrix element of two isotriplet scalar operators sandwiching a Green's function 1/(H 0 − ζ) in a fully isospin-symmetric system.One then re-expresses the matrix element in terms of a four-point correlation function which is readily calculable on lattice.Given the smallness of the ISB pre-factor, a ∼20% accuracy for such a matrix element is sufficient.We focus in this paper mainly on the theory formalism, and defer the actual lattice calculation to a future work.

II. FERMI MATRIX ELEMENT
Let us consider a generic allowed β − -decay ϕ i → ϕ f eν e which is triggered by the following charged weak current: The spatial integral of the zeroth component of the vector piece defines the isospin-raising operator: Here, we adopt the particle physics' convention of isospin, i.e.T z (u) = +1/2.
In quantum field theory, plane-wave states are normalized to a delta function: which means zero-momentum states are normalized as: if we restrict ourselves to a finite box of the size L. We may define a quantum-mechanical state |ϕ(s)⟩ from a zero-momentum plane-wave state normalized as ⟨ϕ(s ′ )|ϕ(s)⟩ = δ s,s ′ .With this, the Fermi matrix element in a β − -decay may be defined as: Notice that, according to this definition the Fermi matrix element involves states with vanishing three-momentum and is related to hadronic form factors with four-momentum transfer , rather than q 2 = 0.

III. PERTURBATION THEORY AND ISB CORRECTIONS
To study ISB effects, we split the full Hamiltonian into: where H 0 is the isospin-symmetric part and V is the ISB perturbation term.The states |ϕ i,f (s)⟩ in Eq.( 6) are eigenstates of the full Hamiltonian H while the corresponding eigenstates of H 0 are denoted as |ϕ i,f (s)⟩ 0 with degenerate mass eigenvalue m 0 ϕ .They are also exact isospin eigenstates within the same isomultiplet, connected through the isospin-raising operator as: where M 0 F is the bare Fermi matrix element.Following the notation in nuclear beta decay, we define the ISB correction δ to the Fermi matrix element as: An exact expression of δ can be derived using the Brillouin-Wigner perturbation theory in Quantum Mechanics, which was first adopted in Refs.[11,12] for nuclear beta decays.
Consistently with the BSAG theorem [2,3], one finds that the leading order correction where Λ i,f ≡ 1 − |ϕ i,f (s)⟩ 00 ⟨ϕ i,f (s)| is an operator that projects away the unperturbed state |ϕ i,f (s)⟩ 0 .Eq.( 10) serves as the foundation of our further analysis.
In this work we concentrate on the strong ISB effects, whose only source is the u − d quark mass difference term in the QCD Lagrangian which is purely isovector: where ∆m q = m u − m d is the quark mass splitting.We introduce the following rank-one tensor operators in isospin space for future convenience: With the aid of these operators, we can simplify the right hand side of Eq.( 10) into more elegant expressions.We will do this for pion and neutron decays.
As a first example we consider pion beta decay.As isospin eigenstates, we can label We start from Eq.( 10) and insert a complete set of H 0 eigenstates {|a; T, T z ⟩} with energy E a,T , where a denotes all quantum numbers unrelated to isospin (the single-pion states are automatically excluded by the projection operators).
Due to the isovector nature of V , only T = 0, 1, 2 intermediate states contribute.Applying the Wigner-Eckart theorem where C T,Tz 1,T ′ z ;1,T ′′ z are Clebsch-Gordan coefficients and ⟨a; T || Ô1 ||π⟩ is a reduced matrix element, we obtain: analogous to the result in Ref. [10].
To compute δ π , we define a "generating function" F π (ζ) which involves diagonal matrix elements of two Ô1 i operators and a Green's function of the isospin-symmetric system: where ζ is a free energy parameter, and is a projection operator of the single-particle pion isotriplet states (the subscript π in F π , which can take π ± or π 0 , affects only the external state but not the projection operator), so 1 − Pπ projects away such states.One may check that the combination involves the same relative coefficients between reduced matrix elements of different T as in the correction δ, see Eq. ( 14).Therefore δ is simply obtained by taking the derivative at

B. n → peν e
We work out the same derivation for neutron decay.In the isospin limit we label the Since the expression above involves a sum over all contributing excited states a, a coherent addition could lead to an unexpected enhancement over the naïve power counting result.
This was the main speculation in Refs.[6,7], which can now be tested directly on the lattice.
The generating function is defined similarly as: and one may verify that which implies An obvious advantage of this formalism is that one focuses directly on the small quantity δ N instead of the full M F , which significantly reduces the level of theoretical precision needed.Suppose we consider the estimate of Ref. [7] where δ N ∼ 4 × 10 −4 for neutron decay (after averaging over three models), then a ∼ 20% precision level for the non-perturbative calculation of F ′ N (m 0 N ) would be sufficient to keep the theory uncertainty of M 2 F below 10 −4 .This is in clear contrast to the direct calculation of M F which would require a 10 −4 precision.

IV. LATTICE IMPLEMENTATION
Here we briefly discuss how the aforementioned generating function and its derivative are implemented on the lattice, taking the pion as an example.We define the following four-point correlation function (t ≥ 0) in the isospin-symmetric limit, which can be directly calculated with lattice QCD: where Ô1 = ( Ô1 −1 ) † , Ô2 = Ô1 −1 , and the time dependence of the operator comes from the standard Euclidean space-time Heisenberg picture: The correlation function involves quark contraction diagrams as depicted in Fig. 1.The matrix elements in Eq. ( 23) that involve the zero-momentum pion projection operator Pπ (see Eq.( 16)) can be calculated as the product of two simpler matrix elements which require a separate calculation of the lattice three-point functions with pion initial/final states and a single operator Ô1/2 .
Both the generating function F π (ζ) and its derivative at ζ = m 0 π can now be obtained as: and respectively.The same also works for the nucleon.
One could argue that a 20% precision on a four-point correlation function is not necessarily easier than a 10 −4 precision on the direct calculation of M F ; in the latter, one may compute the derivative in the ISB parameter numerically and exploit the correlation between the calculation performed with different values of the ISB, which has been proven successful in the past studies of meson mass splittings.However, the fact that the ISB correction to M F is a quadratic effect (in contrast to the linear correction to masses) may complicate the procedure.Furthermore, within our theoretical framework, one may also compute the fourpoint function implicitly with numerical derivatives of two-point meson correlation functions with respect to the quark masses.This method is also commonly used in many lattice calculations with comparable precision as calculating the four-point correlation function directly [13][14][15][16].

V. INSIGHTS FROM CHPT
It is useful to gauge first-principles studies of δ with known results in the pion sector.We start by connecting our definition of M F to the relativistic charged weak form factors: where in the isospin limit.Following common practice, we scale out the constant factor f + (0): Also, for the form factor f+ (q 2 ) we need its leading q 2 -dependence, which defines a mean square radius ⟨r 2 W ⟩: Then, by choosing ⃗ p i = ⃗ p f = ⃗ 0 as we advocated above, the full Fermi matrix element can be expressed in terms of the form factors as: For future convenience, we also define the average pion mass and the pion mass splitting as: By parameterizing the deviation of f 2 + (0) from its isospin limit as: where δ f + = O(V 2 ), we can expand Eq.(32) to the leading order in strong ISB parameters, which provides a parameterization of δ π : The four terms at the right hand side represent (1) The deviation of f + (0) from isospin limit, (2) The correction to the external state normalization, (3) The finite-q 2 correction, and (4) The effect of the subdominant form factor f − , respectively.
Predictions of the four terms are available using the three-flavor ChPT [17].To leading chiral order, the pseudoscalar meson octet masses are given in terms of quark masses by ) or higher, and therefore we drop them.This leaves us with δ f + , which we take from [5]: where F 0 is the pion decay constant in the chiral limit.The only contribution quadratic to m u − m d comes from the kaon loop function H K + K 0 , which gives: We take the QCD parameters from the most recent FLAG review in the MS scheme, at the renormalization scale µ = 2 GeV with N f = 2 + 1 [18]: F 0 = 80.3(6.0)MeV [19],  12) [20,[22][23][24][25].This gives δ π ≈ 1 × 10 −5 , a prediction that can be used to benchmark our new method and its implementation in lattice QCD.

VI. CONCLUSION
We summarize our work as follows.In order to investigate the size of strong ISB corrections in the free neutron decay which could affect the precise determination of V ud , we derived an elegant representation of the leading ISB corrections to hadronic Fermi matrix elements, that relies on quantum mechanical perturbation theory and the Wigner-Eckart theorem.Our result is consistent with the BSAG theorem [2,3].The derived correction is particularly suitable for implementation on the lattice, as it requires a four-point correlation function involving readily-computable quark contraction diagrams, and only a 20% theoretical precision is needed to have an impact on phenomenological applications.Furthermore, one can also use the ChPT prediction in the pion sector to benchmark the lattice accuracy.Finally, one could imagine a generalization of the proposed strategy that offers another pathway to study the K 0 → π − transition form factor f Kπ + (0) in addition to the existing methods.This could improve the extraction of V us from semileptonic kaon decays.
where m = (m u + m d )/2, ε = (m d − m u )/(m u + m d ), and the constant B 0 characterizes the strength of the chiral condensate.The π + − π 0 mass splitting induced by the π 3 − η 8 mixing is quadratic to m u − m d .This means the last three terms at the right hand side of Eq.(35) scale as O((m u − m d ) 3