Connected, Disconnected and Strange Quark Contributions to HVP

We calculate all neutral vector two-point functions in Chiral Perturbation Theory (ChPT) to two-loop order and use these to estimate the ratio of disconnected to connected contributions as well as contributions involving the strange quark. We extend the ratio of $-1/10$ derived earlier in two flavour ChPT at one-loop order to a large part of the higher order contributions and discuss corrections to it. Our final estimate of the ratio disconnected to connected is negative and a few % in magnitude.

LECs involved, both old and new. This is done in Sect. 6 and applied there to the light connected and disconnected part. Because of the presence of the LECs we find a total disconnected contribution of opposite sign and size a few % of the connected contribution. The same type of estimates are then used for the strange quark contribution in Sect. 7.
Here we find a very strong cancellation between p 4 and p 6 contributions, leaving the LEC part dominating strongly. A comparison with a number of lattice results is done in Sect. 8. We find a reasonable agreement in some cases. Our conclusions are summarized in Sect. 9.

The vector two-point function
We define the two-point vector function as where the labels a, b specify the involved currents. We label the currents as The divergence of the vector current is given by which means that any current involving equal mass quark and anti-quark is conserved. Assuming isospin for the π + current, Lorentz invariance then implies that we can parametrize the vector two-point functions given above as Π µν ab (q) = (q µ q ν − q 2 g µν )Π ab (q 2 ).
For simplicity we also use Π a = Π aa andΠ a =Π aa In this paper we work in the isospin limit. This immediately leads to a number of relations Π π + = Π π 0 , Π U = Π D , With those one can derive Π EM = 5 9 Π U + 1 9 Π S − 4 9 Π U D − 2 9 Π U S , The two-point functions Π are themselves not directly observable. However, the vector current two-point function in QCD satisfies a once subtracted dispersion relation The imaginary part can be measured in hadron production if there exists an external vector boson like W ± or the photon coupling to the current. ThusΠ(q 2 ) is an observable, but not Π(0). Π(0) depends on the precise definitions used in regularizing the product of two currents in the same space-time point. The two-point functions for the electromagnetic current can be determined in e + e − collisions and Π π + in τ -decays. One main use ofΠ is the determination of the lowest order HVP part of the muon anomalous magnetic moment via the integral over the electromagnetic two-point function 1 3 Chiral perturbation theory and the singlet current ChPT describes low-energy QCD as an expansion in masses and momenta [10,11,12]. The dynamical degrees of freedom are the pseudo-Goldstone bosons (GB) from the spontaneous breaking of the left-and right-handed flavor symmetry to the vector subgroup, or with the 2 × 2 matrix with only the pions in the case of two-flavours. The Lagrangians, as well as the divergences, are known at order p 2 (LO), p 4 (NLO) and p 6 (NNLO) in the ChPT counting [11,12,13,14]. However, the vector currents defined in Sect. 2 contain also a singlet component and the Lagrangians including only this extension are not known. There is work when extending the symmetry to including the singlet GB as well as singlet vector and axial-vector currents at p 4 [15] and p 6 [16]. However this contains very many more terms than we need. If we only add the singlet vector current, in addition to simply extending the external vector field to include the singlet part, there are two extra terms relevant at order p 4 : Since we are only interested in two-point functions of vector currents these will always appear in the combination 2H 3 + H 4 . For the two-flavour case we get H 3 → h 4 and H 4 → h 5 but otherwise similar terms. It should be noted that none of the terms in the extended p 4 Lagrangian contains couplings of the singlet vector-field to the GB. The singlet appearing in commutators vanishes and the terms involving field strengths vanish, except for the combinations above which do not contain GB fields.
At order p 6 there are many more terms, specifically there are terms appearing that contain interactions of the singlet vector field with the GBs. Two examples are The extra terms that contribute to the vector two-point function at order p 6 always contain two field strengths and the extra p 2 needed can come from either two derivatives or quark masses. Setting all GB fields to zero, the only possible extra terms have a structure with F V µν the vector-field field strength andχ the quark mass part of χ. This leads to the possible terms The D i are linear combinations of a number of LECs in the Lagrangian and one can check that they are all independent by writing down a few fully chiral invariant terms. A similar set with D i → d i exists for the two-flavour case.
There is a coupling of the singlet vector current to the GBs already at order p 4 via the Wess-Zumino-Witten (WZW) term. However, due to the presence of ǫ µναβ we need an even number of insertions of the WZW term or higher order terms from the odd-intrinsic-parity sector to get a contribution to the vector two-point functions.

ChPT results up to two-loop order
The vector two-point functions for neutral non-singlet currents were calculated in [17,18]. We have reproduced their results and added the parts coming from the singlet currents.
The expressions for the two-point functions are most simply expressed in terms of the function The one-loop integrals here are defined in many places, see e.g. [18]. The explicit expression is We also need µ is the ChPT subtraction scale. We always work in the isospin limit. The expressions we give are in the three flavour case with physical masses. We will quote the corresponding results with lowest order masses in [7]. The two-point functions only start at p 4 . We therefore write the result as in the chiral expansion. The p 4 results are The obvious relations visible for the G terms will be discussed in Sect. 5. This result agrees with [6] when the appropriate limits are taken. The results at p 6 are somewhat longer but still fairly short.
For the two-flavour case the results can be derived from the above. First, only keep the integral terms with m 2 π , second replace L 9 by −(1/2)l r 6 , L r 10 + 2H r 1 by −4h r 2 and L r 10 by l r 5 . In addition there are also extra counterterms for the singlet current appearing. The results are

Connected versus disconnected contributions
If we look at the flavour content of the two-point functions in the isospin limit, it is clear that Π π + only contains connected contributions while Π U D only contains disconnected contributions. This is derived by thinking of which quark contractions can contribute as shown in Fig. 1. In the same way Π U contains both with Inspection of all the results in Sect. 4 shows that (21) is satisfied. From (7) we thus obtain and Π U S is fully disconnected while Π S has both connected and disconnected parts.

Two-flavour and isospin arguments
In [6], they found, using NLO two-flavour ChPT in the isospin limit, that They also calculated corrections to this ratio due to the inclusion of strange quarks. Their result is in our terms expressed viaΠ which is clearly satisfied for the results shown in (20). Note that Π(0), via the part coming from the LECs, does not satisfy a similar relation due to the extra terms possible for the singlet current. Inspection of (20) shows that the loop part at order p 6 also satisfies (25) but due to the part of the LECs, the relation is no longer satisfied even for the subtracted functionsΠ.
The relation (25) can be derived in more general way. As noted in Sect. 3 the singlet current j µ I2 only couples to GBs at order p 6 or at order p 4 via the WZW term and we need at least two of the latter for the vector two-point function. For the contributions where those couplings are not present, denoted by a tilde, we get The relation (26) if written forΠ has corrections at order p 8 . Eq. (26) together with (21) immediately leads to (25) but for many more contributions. The ratio of disconnected to connected is −1/2 for all loop-diagrams only involving vertices from the lowest-order Lagrangian or from the normal NLO Lagrangian. So the ratio is true for a large part of all higher order loop diagrams and corrections start appearing only in loop diagrams at order p 8 with one insertion from the p 6 -Lagrangian or at p 10 with two insertions of a WZW vertex. The argument includes diagrams with four or more pions.
Using the isospin relations we can derive that Looking at (27), one can see that the ratio (−1/2) is exact for all contributions with isospin I = 1 and only broken due to I = 0 contributions. This can be used as well to estimate the size of the ratio, see below and [19,20]. A corollary is that two-pion intermediate state contributions obey (25) to all orders. The contributions to order p 6 forΠ satisfy the relation (26) up to the LEC contributions. Using resonance saturation, the LECs can be estimated from ρ and ω exchange. In the large N c limit that combination will only contribute to the connected contribution. Since the ρ-ω mass splitting and coupling differences are rather small, we expect that the disconnected contribution from this source will be rather small. This will lower the ratio of disconnected to connected contributions compared to (25).
In [19] it was also noticed that the ratio of −1/2 is valid for all two-pion intermediate states in the isospin limit. They used the slow turn-on of the three-pion channel where the singlet starts contributing to argue for the validity of the one-loop estimate. That slow turn-on follows from the three-pion contribution being p 10 in our way of looking at it. In [20] the difference between ρ and ω measured masses and couplings were used to obtain an estimate of the disconnected contribution of about −1%. We consider that contribution to be within the error of our estimate given in Sect. 6.

Three flavour arguments
It was already noted in [6] that kaon loops violate the relation (25) in NLO three-flavour ChPT and the same is rather visible in the results (18) and (19).
The argument for the singlet current coupling to mesons is just as true in three-as in two-flavour ChPT. However here one needs to use the three-flavour singlet current, j µ I3 , instead. Again denoting with a tilde the contributions from loop diagrams involving only lowest order vertices or NLO vertices not from the WZW term, we have (after using isospin) two relations similar to (26) Note that in this subsection we talk about the three-flavour ChPT expressions. Inspection of the expressions in (18) and (19) show that the relations (28) are satisfied. Note that the relation (28) if written forΠ has corrections at order p 8 . In general we can write using (28) This indicates that corrections to the −1/2 are expected to be small due to the strange quark being much heavier than the up and down quarks. The second relation in (28) allows a relation involving two-point functions with the strange quark current.
Note that a consequence of (28) in the equal mass limit is In this case the disconnected contribution to the electromagnetic two-point function vanishes identically since the charge matrix is traceless.

Estimate of the ratio of disconnected to connected
In order to estimate the ratio of disconnected to connected contributions in ChPT the inputs that appear must be determined. For the plots shown below we use The values for the decay constant and masses are standard ones. The values for the L r i were recently reviewed in [21] and we have taken the values for L r 9 from [22] quoted in [21]. If we only considerΠ, the only other LECs we need are C r 93 and D r 3 . As first suggested in [23] LECs are expected to be saturated by resonances. For C r 93 and D r 3 the main contribution will be from the vector resonance multiplet. Here a nonet approach typically works well and that would suggest that D r 3 ≈ 0. We will set it to zero in our estimates. The value for C r 93 was first determined using resonance saturation in [18] with a value of If we use resonance saturation for the nonet and the constraints from short-distance as used in [24] we obtain for the two-point function Assuming that the pure LEC parts reproduce (33), leads to the value with m V = 770 MeV. Finally fitting the expression for Π π + to a phenomenological form of the two-point function [25] gives The three values are in reasonable agreement. The size can be compared to other vector meson dominated combinations of LECs, e.g. C r 88 − C r 90 = −0.55 10 −4 [22], which is of the same magnitude. In the numerical results we will use the full expression (33) for the contribution from higher order LECs rather than just the terms with C r 93 . In Fig. 2 we have plotted the different contributions toΠ π + . This is what is usually called the connected contribution. As we see, the contribution from higher order LECs, as modeled by (33), is, as expected, dominant. The full result forΠ is the sum of the VMD and the p 4 + p 6 lines. We see that the pure two-loop contribution is small compared to the one-loop contribution but there is a large contribution at order p 6 from the one-loop diagrams involving L r i . In Fig. 3 we have plotted the same contributions but now forΠ U D or the contribution from disconnected diagrams. Note that the scale is exactly half that of Fig. 2. The contributions are very close to −1/2 times those of Fig. 2 except for the pure LEC contribution which is here estimated to be zero.
How well do the estimates of the ratio now hold up. The ratio of disconnected to connected is plotted in Fig. 4. We see that the contribution at order p 4 has a ratio very close to −1/2 and the same goes for all loop contributions at order p 6 . The effects of kaon loops is thus rather small. The deviation from −1/2 is driven by the estimate of the pure LEC contribution. Using the VMD estimate (33) we end up with a ratio of about −0.18 for the range plotted. Taking into account (22) we get an expected ratio for the disconnected to connected contribution to the light quark electromagnetic two-point functionΠ EM 2 of about −3.5%. If we had used the other estimates for C r 93 (and assumed a similar ratio for higher orders) the number would have been about −3%.
An analysis using only the pion contributions, so no contribution from intermediate kaon states, would give essentially the same result.  Figure 2: The subtracted two-point func-tionΠ π + (q 2 ) or the connected part. Plotted are the p 4 contribution of (18) labeled p 4 and the three parts of the higher order contribution: the pure two-loop contribution labeled p 6 R, the p 6 contribution from one-loop graphs labeled p 6 L and the pure LEC contribution as modeled by (33) labeled VMD. Plotted are the p 4 contribution of (18) labeled p 4 and the two non-zero parts of the higher order contribution: the pure two-loop contribution labeled p 6 R and the p 6 contribution from one-loop graphs labeled p 6 L. The the pure LEC contribution is estimated to be zero here.

Estimate of the strange quark contributions
The numerical results in the previous section included the contribution from kaons but only via the electromagnetic couplings to up and down quarks. In this section we provide an estimate for the contribution when including the photon coupling to strange quarks, i.e. we add the terms coming from Π U S and Π S in (23). The loop contributions satisfy the relations shown in (28) with corrections starting earliest at p 8 . Alternatively we can write the first relation as this, together with the ratios shown in Fig. 4 and the second relation in (28), shows that we can expect the extra contributions to be quite small with the possible exception of the pure LEC contribution. The pure LEC contribution is estimated to only apply to the connected part and so contributes only to Π S . Given that the φ mass is significantly larger than the ρ-mass we will for that part need to include this difference. A first estimate is simply by using (33) with m V now the φ-mass of m φ = 1020 MeV. We will call this VMDφ in the remainder.
The estimate we include for Π S includes both connected and disconnected contributions. We would need to go to partially quenched ChPT to obtain that split-up generalizing the  Figure 4: The ratio of the subtracted two-point functionsΠ U D (q 2 )/Π π + (q 2 ) or ratio of the disconnected to the connected part. Plotted are the p 4 contribution of (18) labeled p 4 , the parts of the higher order contribution: the pure two-loop contribution labeled p 6 R and the p 6 contribution from one-loop graphs labeled p 6 L as well as their sum. The ratio of the pure LEC contribution is estimated to be zero. The ratio for all contributions summed is the dash-dotted line.
methods of [6]. Fig. 5 shows the different contributions toΠ S . We did not plotΠ U S since the relations (28) imply that the p 4 , p 6 L and p 6 R are exactly −1/2 the contributions forΠ S and in our estimate the pure LEC part forΠ U S vanishes. The contributions are much smaller than those of the connected light quark contribution shown in Fig. 2. One remarkable effect is the very strong cancellation between the p 4 and p 6 effects give an almost zero loop contribution. This means that vector meson dominance in the coupling is even more clear in this case than for the lighter quarks.

Comparison with lattice and other data
For comparing with lattice and phenomenological data we can use the Taylor expansion around q 2 = 0 from our expressions and the same coefficients evaluated from experimental data or via the time moment analysis on the lattice [26].
We expand the functions asΠ The signs follow from the fact that the lattice expansion is defined in terms of Q 2 = −q 2 and the usual lattice convention for Π has the opposite sign of ours. The coefficients, obtained by fitting an eight-order polynomial to the ranges shown in the plots, are given in Table 1. . Plotted are the p 4 contribution of (18) labeled p 4 , the parts of the higher order contribution: the pure two-loop contribution labeled p 6 R and the p 6 contribution from one-loop graphs labeled p 6 L as well as their sum. The pure LEC contribution is estimated by (33) with the mass of the φ.
[27] is from an analysis of experimental data. [28] are preliminary numbers from the BMW collaboration and we have removed the charm quark contribution from their numbers. These numbers are not corrected for finite volume. For [26,29] we have taken the numbers from their configuration 8, which has physical pion masses and multiplied by 9/5 for the latter to obtain Π π + . Our estimates are in reasonable agreement for the connected contribution. For the disconnected contribution, our results are higher but of a similar order.
There have been many more studies of the muon g − 2 on the lattice and in particular a number of studies of the disconnected part. However, their results are often not presented in a form that we can easily compare to. From our numbers above we expect the disconnected contribution to be a few % and of the opposite sign of the connected contribution. [20] finds −0.15(5)%, much smaller than we expect, [30] finds about −1.5% which is below but of the same order as our estimate.
The same comment applies to studies of the strange contribution, e.g. [31] finds a contribution of about 7% of the light connected contribution which is in reasonable agreement with our estimate.

Summary and conclusions
We have calculated in two-and three-flavour ChPT all the neutral two-point functions in the isospin limit including the singlet vector current. We have extended the ratio of −1/2 (or −1/10 for the electromagnetic current) of [6] to a large part of the higher order   [26,27,28,29] and a comparison with our estimates.
loop corrections. We used the nonet estimates of LECs to set the new constants for the singlet current equal to zero and then provided numerical estimates for the disconnected and strange quark contributions. We find that the disconnected contribution is negative and a few % of the connected contribution, the main uncertainty being the new LECs which we estimated to be zero. A similar estimate for the strange quark contribution has a large cancellation between p 4 and p 6 leaving our rather uncertain estimate of the LECs involved as the main contribution.