Connected, disconnected and strange quark contributions to HVP

Abstract

Abstract

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.

Published for SISSA by Springer

Received: September 8, 2016

Accepted: October 31, 2016 Published: November 14, 2016

JHEP11(2016)086

Connected, disconnected and strange quark contributions to HVP

Johan Bijnens and Johan ReleforsDepartment of Astronomy and Theoretical Physics, Lund University, Solvegatan 14A, SE 223-62 Lund, Sweden

E-mail: mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected]

Abstract: 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 avour ChPT at one-loop order to a large part of

the higher order contributions and discuss corrections to it. Our nal estimate of the ratio disconnected to connected is negative and a few % in magnitude.

Keywords: Chiral Lagrangians, Lattice QCD, Precision QED

ArXiv ePrint: 1609.01573

Open Access, c

[circlecopyrt] The Authors.

Article funded by SCOAP3. doi:http://dx.doi.org/10.1007/JHEP11(2016)086

Web End =10.1007/JHEP11(2016)086

Contents

1 Introduction 1

2 The vector two-point function 2

3 Chiral perturbation theory and the singlet current 3

4 ChPT results up to two-loop order 5

5 Connected versus disconnected contributions 65.1 Two- avour and isospin arguments 75.2 Three avour arguments 8

6 Estimate of the ratio of disconnected to connected 9

7 Estimate of the strange quark contributions 11

8 Comparison with lattice and other data 12

9 Summary and conclusions 14

1 Introduction

The muon anomalous magnetic moment is one of the most precisely measured quantities around. The measurement [1] di ers from the standard model prediction by about 3 to 4 sigma depending on precisely which theory predictions are taken. A review is [2] and talks on the present situation can be found in [3]. The main part of the theoretical error at present is from the lowest-order hadronic vacuum polarization (HVP). This contribution can be determined from experiment or can be computed using lattice QCD [4]. An overview of the present situation in lattice QCD calculations is given by [5].

The underlying object that needs to be calculated is the two-point function of electromagnetic currents as de ned in (2.1). The contribution to a = (g 2)/2 is given by the

integral in (2.9). There are a number of di erent contributions to the two-point function of electromagnetic currents that need to be measured on the lattice. First, if we only consider the light up and down quarks, there are connected and disconnected contributions depicted schematically in gure 1. If we add the strange quark to the electromagnetic currents then there are contributions with the strange electromagnetic current in both points and the mixed up-down and strange case. In this paper we provide estimates of all contributions at low energies using Chiral Perturbation Theory (ChPT).

The disconnected light quark contribution has been studied at one-loop order in ref. [6] using partially quenched (PQChPT). They found that the ratio in the subtracted form

{ 1 {

JHEP11(2016)086

Connected Disconnected

of quarks/gluons

Figure 1. Connected (left) and disconnected (right) diagram for the two-point vector function. The lines are valence quark lines in a sea of quarks and gluons.

factors, as de ned in (2.5), is 1/2 in the case of valence quarks of a single mass and two

degenerate sea quarks. They also found that adding the strange quark did not change the ratio much. Here we give an argument explaining the factor of 1/2 and extend their

analysis to order p6. We also present estimates for the contributions from the strange electromagnetic current.

The nite volume, partially quenched and twisted boundary conditions extensions to two loop order will be presented in [7].

In section 2 we give the de nitions of the two-point functions and currents we use. Section 3 discusses ChPT and the extra terms and low-energy-constants (LECs) needed for a singlet vector current. Our main analytical results, the two-loop order ChPT expressions for all needed vector two-point functions are in section 4. Section 5 uses the observation given in section 3 of the absence of singlet vector couplings to mesons until ChPT order p6 to show for which contributions the ratio 1/2 is valid. Numerical results need an

estimate of the LECs involved, both old and new. This is done in section 6 and applied there to the light connected and disconnected part. Because of the presence of the LECs we nd 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 section 7. Here we nd a very strong cancellation between p4 and p6 contributions, leaving the LEC part dominating strongly. A comparison with a number of lattice results is done in section 8. We nd a reasonable agreement in some cases. Our conclusions are summarized in section 9.

2 The vector two-point function

We de ne the two-point vector function as

ab = i Z

d4xeiqx

JHEP11(2016)086

DT (ja(x)j b(0))

E

(2.1)

where the labels a, b specify the involved currents. We label the currents as

j+ =

d u , jU =

u u , jD =

d d ,

jS =

s s , jEM =

2

3jU

13jD

13jS , jEM2 =

2

3jU

1 3jD ,

j0 =

1p2 jU jD

, jI2 =

1p2 jU + jD

, jI3 =

1p3 jU + jD + jS

. (2.2)

The divergence of the vector current is given by

@

qi qj = i(mi mj)

qiqj , (2.3)

{ 2 {

which means that any current involving equal mass quark and anti-quark is conserved. Assuming isospin for the + current, Lorentz invariance then implies that we can parametrize the vector two-point functions given above as

ab(q) = (qq q2g ) ab(q2). (2.4)

We also de ne the subtracted quantity

^

ab(q2) = ab(q2) ab(0) . (2.5)

For simplicity we also use a = aa and ^

a = ^

aa

In this paper we work in the isospin limit. This immediately leads to a number of relations

+ = 0 , U = D , US = DS . (2.6)

With those one can derive

EM = 59 U +

19 S

4 9 UD

2 9 US ,

EM2 = 59 U

49 UD . (2.7)

The two-point functions are themselves not directly observable. However, the vector current two-point function in QCD satis es a once subtracted dispersion relation

^

(q2) = (q2) (0) = q2 Z

1

threshold ds

JHEP11(2016)086

1 Im (s) . (2.8)

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. Thus ^

(q2) is an observable, but not (0). (0) depends on the precise de nitions 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 + in -decays.

One main use of ^

is the determination of the lowest order HVP part of the muon anomalous magnetic moment via the integral over the electromagnetic two-point function1

aLOHVP = 4 2

Z

1

0 dQ2

1s(s q2)

EM(Q2)g(Q2) ,

g(Q2) = 16m4

Q6

1 +

q1 + 4m2/Q2

^

4 q1 + 4m2/Q2

. (2.9)

3 Chiral perturbation theory and the singlet current

ChPT describes low-energy QCD as an expansion in masses and momenta [10{12]. The dynamical degrees of freedom are the pseudo-Goldstone bosons (GB) from the spontaneous

1The version mentioned here comes from [4] but the result essentially goes back to [8, 9].

{ 3 {

breaking of the left- and right-handed avor symmetry to the vector subgroup, SU(3)L [notdef] SU(3)R ! SU(3)V . The GBs can be parameterized in the SU(3) matrix

U = eip2M/F0 with M = 0

B

@

1p2 0 +

. (3.2)

Since we are only interested in two-point functions of vector currents these will always appear in the combination 2H3 + H4. For the two- avour case we get H3 ! h4 and

H4 ! h5 but otherwise similar terms.

It should be noted that none of the terms in the extended p4 Lagrangian contains couplings of the singlet vector- eld to the GB. The singlet appearing in commutators vanishes and the terms involving eld strengths vanish, except for the combinations above which do not contain GB elds.

At order p6 there are many more terms, speci cally there are terms appearing that contain interactions of the singlet vector eld with the GBs. Two examples are

DFR [notdef]F LU

E+ DFL [notdef]F RU

E, [angbracketleft]FL + FR [angbracketright]

D

[notdef]U + U[notdef]

DUD U

E.(3.3)

The extra terms that contribute to the vector two-point function at order p6 always contain two eld strengths and the extra p2 needed can come from either two derivatives or quark masses. Setting all GB elds to zero, the only possible extra terms have a structure with FV the vector- eld eld strength and

[notdef] the quark mass part of [notdef]. This leads to the

F V

[angbracketleft]

[notdef][angbracketright] + D3 [angbracketleft]@FV [angbracketright]

@F V (3.4)

The Di 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 Di ! di exists for the two- avour case.

There is a coupling of the singlet vector current to the GBs already at order p4 via the Wess-Zumino-Witten (WZW) term. However, due to the presence of [epsilon1] 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.

{ 4 {

1p6 + K+ 1p2 0 +

1p6 K0

K0 2p6

1

C

A

. (3.1)

or with the 2 [notdef] 2 matrix with only the pions in the case of two- avours. The Lagrangians,

as well as the divergences, are known at order p2 (LO), p4 (NLO) and p6(NNLO) in the ChPT counting [11{14]. However, the vector currents de ned in section 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 p4 [15] and p6 [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 eld to include the singlet part, there are two extra terms relevant at order p4:

H3 [angbracketleft]FL [angbracketright]

F L + [angbracketleft]FR [angbracketright]

K

JHEP11(2016)086

F R + H4 [angbracketleft]FR [angbracketright]

F L

possible terms

D1 [angbracketleft]FV [angbracketright]

F V

[notdef] + D2 [angbracketleft]FV [angbracketright]

4 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

G(m2, q2)

1 q2

B22(m2, m2, q2) 1 2A(m2)

(4.1)

JHEP11(2016)086

The one-loop integrals here are de ned in many places, see e.g. [18]. The explicit expression is

G(m2, q2) = 1 162

136 +

1

112 log

m2

[notdef]2 +

q2 4m2 12

Z

0 dx log

1 x(1 x)

q2 m2

= 1

162

112 +

112 log

m2

[notdef]2

q2 12m2

q4 1680m4 + [notdef] [notdef] [notdef]

(4.2)

We also need

m2

[notdef]2 . (4.3)

[notdef] is the ChPT subtraction scale. We always work in the isospin limit. The expressions we give are in the three avour case with physical masses. We will quote the corresponding results with lowest order masses in [7].

The two-point functions only start at p4. We therefore write the result as

= (4) + (6) + [notdef] [notdef] [notdef] (4.4)

in the chiral expansion. The p4 results are

(4)+ = 8G(m2, q2) 4G(m2K, q2) 4(Lr10 + 2Hr1) ,

(4)U = 4G(m2, q2) 4G(m2K, q2) 4(Lr10 + 2Hr1 + 2Hr3 + Hr4) ,

(4)S = 8G(m2K, q2) 4(Lr10 + 2Hr1 + 2Hr3 + Hr4) ,

(4)UD = 4G(m2, q2) 4(2Hr3 + Hr4) ,

(4)US = 4G(m2K, q2) 4(2Hr3 + Hr4) ,

(4)EM = 4G(m2, q2) 4G(m2K, q2)

83(Lr10 + 2Hr1) . (4.5)

The obvious relations visible for the G terms will be discussed in section 5. This result

agrees with [6] when the appropriate limits are taken.

{ 5 {

A(m2) =

m2 162 log

The results at p6 are somewhat longer but still fairly short.

F 2 (6)+ = 4q2 2G(m2, q2) + G(m2K, q2)

2 16q2Lr9

2G(m2, q2) + G(m2K, q2)

32m2Cr61

32(m2 + 2m2K)Cr62 8q2Cr93 ,

F 2 (6)U = 8q2G(m2, q2)2 + 8q2G(m2, q2)G(m2K, q2) + 8q2G(m2K, q2)2

16q2Lr9 G(m2, q2) + G(m2K, q2)

8(Lr9 + Lr10) A(m2) + A(m2K)

8(Lr9 + Lr10) 2A(m2) + A(m2K)

32m2Cr61 32(m2 + 2m2K)Cr62 8q2Cr93 4m2Dr1 4(m2 + 2m2K)Dr2 4q2Dr3 ,

F 2 (6)S = 24q2G(m2K, q2)2 32q2Lr9G(m2K, q2) 16(Lr9 + Lr10)A(m2K)

32(2m2K m2)Cr61 32(m2 + 2m2K)Cr62 8q2Cr93 4(2m2K m2)Dr1 4(m2 + 2m2K)Dr2 4q2Dr3 ,

F 2 (6)UD = 8q2G(m2, q2)2 8q2G(m2, q2)G(m2K, q2) + 4q2G(m2K, q2)2

+ 16q2Lr9G(m2, q2) + 8(Lr9 + Lr10)A(m2) 4m2Dr1 4(m2 + 2m2K)Dr2 4q2Dr3 ,

F 2 (6)US = 12q2G(m2K, q2)2 + 16q2Lr9G(m2K, q2) + 8(Lr9 + Lr10)A(m2K)

4m2KDr1 4(m2 + 2m2K)Dr2 4q2Dr3 . (4.6)

For the two- avour case the results can be derived from the above. First, only keep the integral terms with m2, second replace L9 by (1/2)lr6, Lr10 + 2Hr1 by 4hr2 and Lr10

by lr5. In addition there are also extra counterterms for the singlet current appearing. The results are

(4)+ = 8G(m2, q2) + 16hr2 , (4)U = 4G(m2, q2) + 16hr2 4(2hr4 + hr5) , (4)UD = 4G(m2, q2) 4(2hr4 + hr5) ,

(4)EM = 4G(m2, q2) +

32

3 hr2

JHEP11(2016)086

49(2hr4 + hr5) ,

F 2 (6)+ = 16q2G(m2, q2)2 + 16q2lr6G(m2, q2) 8(2lr5 lr6)A(m2) 32m2cr34 8q2cr56 , F 2 (6)U = 8q2G(m2, q2)2 + 8q2lr6G(m2, q2) 4(2lr5 lr6)A(m2)

32m2cr34 8q2cr56 4m2(dr1 + 2dr2) 4q2dr3 ,

F 2 (6)UD = 8q2G(m2, q2)2 8q2lr6G(m2, q2) + 4(2lr5 lr6)A(m2)

4m2(dr1 + 2dr2) 4q2dr3 . (4.7)

5 Connected versus disconnected contributions

If we look at the avour content of the two-point functions in the isospin limit, it is clear that + only contains connected contributions while UD only contains disconnected

{ 6 {

contributions. This is derived by thinking of which quark contractions can contribute as shown in gure 1. In the same way U contains both with

U = + + UD . (5.1)

Inspection of all the results in section 4 shows that (5.1) is satis ed. From (2.7) we thus obtain

EM2 = 59

+ + 19 UD , (5.2)

and

110 . (5.4)

They also calculated corrections to this ratio due to the inclusion of strange quarks. Their result is in our terms expressed via

^

(4)UD

^

(4)+

e U +

The relation (5.6) if written for ^ has corrections at order p8. Eq. (5.6) together with (5.1)

immediately leads to (5.5) 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 p8 with one insertion from the p6-Lagrangian or at p10 with two insertions of a WZW vertex. The argument includes diagrams with four or more pions.

{ 7 {

19 S . (5.3)

US is fully disconnected while S has both connected and disconnected parts.

5.1 Two- avour and isospin arguments

In [6], they found, using NLO two- avour ChPT in the isospin limit, that

^

DiscEM2 ^

ConnEM2

=

1

2 (5.5)

which is clearly satis ed for the results shown in (4.7). Note that (q2), 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 (4.7) shows that the loop part at order p6 also satis es (5.5) but due to the part of the LECs, the relation is no longer satis ed even for the subtracted functions ^

.

The relation (5.5) can be derived in a more general way. As noted in section 3 the singlet current jI2 only couples to GBs at order p6 or at order p4 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

e U(U+D) =

JHEP11(2016)086

EM = 59

+ + 19 UD

29 US +

=

e UD = 0 , (5.6)

Using the isospin relations we can derive that

UD = 12 ( I2

0 ) (5.7)

Looking at (5.7), 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 (5.5) to all orders.

The contributions to order p6 for ^

satisfy the relation (5.6) up to the LEC contributions. Using resonance saturation, the LECs can be estimated from and ! exchange. In the large Nc limit that combination will only contribute to the connected contribution.

Since the -! mass splitting and coupling di erences 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 (5.5).

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 p10 in our way of looking at it. In [20] the di erence between and ! 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 section 6.

5.2 Three avour arguments

It was already noted in [6] that kaon loops violate the relation (5.5) in NLO three- avour ChPT and the same is rather visible in the results (4.5) and (4.6).

The argument for the singlet current coupling to mesons is just as true in three- as in two- avour ChPT. However here one needs to use the three- avour singlet current, jI3, 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 (5.6)

e U(U+D+S) =

e U +

JHEP11(2016)086

e UD +

e US = 0 ,

e US + e S = 0 . (5.8)

Note that in this subsection we talk about the three- avour ChPT expressions. Inspection of the expressions in (4.5) and (4.6) show that the relations (5.8) are satis ed. Note that the relation (5.8) if written for ^

has corrections at order p8.

In general we can write using (5.8)

e UD

e +

=

1

2

e S(U+D+S) = 2

e US 2

e +

. (5.9)

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.

{ 8 {

The second relation in (5.8) allows a relation involving two-point functions with the strange quark current.

Note that a consequence of (5.8) in the equal mass limit is

mu = md = ms =)

e UD

e +

=

13 . (5.10)

In this case the disconnected contribution to the electromagnetic two-point function vanishes identically since the charge matrix is traceless.

6 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

F =92.2 MeV m =135 MeV mK =495 MeVLr9 =0.00593 [notdef] =770 MeV (6.1)

The values for the decay constant and masses are standard ones. The values for the Lri were recently reviewed in [21] and we have taken the values for Lr9 from [22] quoted in [21].

If we only consider ^

, the only other LECs we need are Cr93 and Dr3. As rst suggested in [23] LECs are expected to be saturated by resonances. For Cr93 and Dr3 the main contribution will be from the vector resonance multiplet. Here a nonet approach typically works well and that would suggest that Dr3 0. We will set it to zero in our estimates.

The value for Cr93 was rst determined using resonance saturation in [18] with a value of

Cr93 = 1.4 104 (6.2)

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

VMD+(q2) =

4F 2 m2V q2

JHEP11(2016)086

. (6.3)

Assuming that the pure LEC parts reproduce (6.3), leads to the value

Cr93 = 1.02 104 (6.4)

with mV = 770 MeV. Finally tting the expression for + to a phenomenological form of the two-point function [25] gives

Cr93 = 1.33 104 (6.5)

The three values are in reasonable agreement. The size can be compared to other vector meson dominated combinations of LECs, e.g. Cr88 Cr90 = 0.55 104 [22], which is of the same magnitude. In the numerical results we will use the full expression (6.3) for the contribution from higher order LECs rather than just the terms with Cr93.

{ 9 {

0

-0.1 -0.08 -0.06 -0.04 -0.02 0

-0.001

-0.002

^

P p +

-0.003

VMD

p4+p6

p4

p6 R

p6 L

-0.004

-0.005

JHEP11(2016)086

-0.006

q2 [GeV2]

Figure 2. The subtracted two-point function ^

+ (q2) or the connected part. Plotted are the p4 contribution of (4.5) labeled p4 and the three parts of the higher order contribution: the pure two-loop contribution labeled p6 R, the p6 contribution from one-loop graphs labeled p6 L and the pure LEC contribution as modeled by (6.3) labeled VMD.

0.003

0 -0.1 -0.08 -0.06 -0.04 -0.02 0

0.0025

p4+p6

p4 p6 R

p6 L

0.002

^

P UD

0.0015

0.001

0.0005

q2 [GeV2]

Figure 3. The subtracted two-point function ^

UD(q2) or the disconnected part. Plotted are the p4 contribution of (4.5) labeled p4 and the two non-zero parts of the higher order contribution: the pure two-loop contribution labeled p6 R and the p6 contribution from one-loop graphs labeled p6 L. The the pure LEC contribution is estimated to be zero here.

In gure 2 we have plotted the di erent contributions to ^

+ . This is what is usually called the connected contribution. As we see, the contribution from higher order LECs, as modeled by (6.3), is, as expected, dominant. The full result for ^

is the sum of the VMD and the p4 + p6 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 p6 from the one-loop diagrams involving Lri.

In gure 3 we have plotted the same contributions but now for ^

UD or the contribution from disconnected diagrams. Note that the scale is exactly half that of gure 2. The contributions are very close to 1/2 times those of gure 2 except for the pure LEC

contribution which is here estimated to be zero.

{ 10 {

0

-0.1

-0.2

^ -0.3

-0.4

-0.5

-0.6

Figure 4. The ratio of the subtracted two-point functions ^

UD(q2)/^

+ (q2) or ratio of the disconnected to the connected part. Plotted are the p4 contribution of (4.5) labeled p4, the parts of the higher order contribution: the pure two-loop contribution labeled p6 R and the p6 contribution from one-loop graphs labeled p6 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.

How well do the estimates of the ratio now hold up. The ratio of disconnected to connected is plotted in gure 4. We see that the contribution at order p4 has a ratio very close to 1/2 and the same goes for all loop contributions at order p6. The e ects 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 (6.3) we end up with a ratio of about 0.18 for

the range plotted. Taking into account (5.2) we get an expected ratio for the disconnected to connected contribution to the light quark electromagnetic two-point function ^

EM2 of

about 3.5%. If we had used the other estimates for Cr93 (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.

7 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 US and S in (5.3).

The loop contributions satisfy the relations shown in (5.8) with corrections starting earliest at p8. Alternatively we can write the rst relation as

e + + 2

{ 11 {

p4+p6+VMD

p4+p6

p4

p6 R

p6 L

^

P UD/

JHEP11(2016)086

-0.1 -0.08 -0.06 -0.04 -0.02 0

q2 [GeV2]

e UD + e US = 0 , (7.1) this, together with the ratios shown in gure 4 and the second relation in (5.8), shows that we can expect the extra contributions to be quite small with the possible exception of the pure LEC contribution.

0.0005

0

-0.0005

-0.001

-0.0015

Figure 5. The subtracted two-point function ^

S(q2). Plotted are the p4 contribution of (4.5) labeled p4, the parts of the higher order contribution: the pure two-loop contribution labeled p6 R and the p6 contribution from one-loop graphs labeled p6 L as well as their sum. The pure LEC contribution is estimated by (6.3) with the mass of the .

The pure LEC contribution is estimated to only apply to the connected part and so contributes only to S. Given that the mass is signi cantly larger than the -mass we will for that part need to include this di erence. A rst estimate is simply by using (6.3) with mV now the -mass of m = 1020 MeV. We will call this VMD in the remainder.

The estimate we include for S includes both connected and disconnected contributions. We would need to go to partially quenched ChPT to obtain that split-up generalizing the methods of [6].

Figure 5 shows the di erent contributions to ^

S. We did not plot ^

US since the

relations (5.8) imply that the p4, p6L and p6R are exactly 1/2 the contributions for

^

S and in our estimate the pure LEC part for ^

US vanishes. The contributions are much smaller than those of the connected light quark contribution shown in gure 2. One remarkable e ect is that the very strong cancellations between the p4 and p6 e ects 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.

8 Comparison with lattice and other data

For comparing with lattice and phenomenological data we can use the Taylor expansion around q2 = 0 from our expressions and the same coe cients evaluated from experimental data or via the time moment analysis on the lattice [26].

We expand the functions as

^

(q2) = 1q2 2q4 + [notdef] [notdef] [notdef] (8.1)

The signs follow from the fact that the lattice expansion is de ned in terms of Q2 = q2 and the usual lattice convention for has the opposite sign of ours. The coe cients,

{ 12 {

^

P S

p4+p6

p4

p6 R

p6 L

VMDf

JHEP11(2016)086

-0.1 -0.08 -0.06 -0.04 -0.02 0

q2 [GeV2]

Reference A 1 (GeV2) 2 (GeV4)

V MD ^

+ 0.0967 0.163 p4 ^

+ 0.0240 0.091 p6 R ^

+ 0.0031 0.014 p6 L ^

+ 0.0286 0.067 sum ^

+ 0.152 0.336 [28] ^

+ 0.1657(16)(18) 0.297(10)(05) [29] ^

+ 0.1460(22) 0.2228(65) p4 ^

UD 0.0116 0.045 p6 R ^

UD 0.0015 0.007 p6 L ^

UD 0.0146 0.032 sum ^

UD 0.0278 0.085 [28] ^

UD 0.015(2)(1) 0.046(10)(04) V MD ^

S 0.0314 0.030 p4 ^

S 0.0017 0.001 p6 R ^

S 0.0000 0.000

p6 L ^

S 0.0013 0.005 sum ^

S 0.0318 0.035 [28] ^

S 0.0657(1)(2) 0.0532(1)(3) [26] ^

S 0.06625(74) 0.0526(11) our result ^

EM 0.0852 0.182 [27] ^

EM 0.0990(7) 0.206(2) [28] ^

EM 0.0972(2)(1) 0.166(6)(3)

Table 1. The Taylor expansion coe cients of ^

of [26{29] and a comparison with our estimates.

obtained by tting an eight-order polynomial to the ranges shown in the plots, are given in table 1. Ref. [27] is from an analysis of experimental data. Ref. [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 nite volume. For [26, 29] we have taken the numbers from their con guration 8, which has physical pion masses and multiplied by 9/5 for the latter to obtain + . 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 g2 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. Ref. [20] nds 0.15(5)%, much smaller than we expect, [30] nds about 1.5% which is below but

of the same order as our estimate.

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JHEP11(2016)086

The same comment applies to studies of the strange contribution, e.g. [31] nds a contribution of about 7% of the light connected contribution which is in reasonable agreement with our estimate.

9 Summary and conclusions

We have calculated in two- and three- avour 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

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 nd 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 p4 and p6 leaving our rather uncertain estimate of the LECs involved as the main contribution.

Acknowledgments

This work is supported in part by the Swedish Research Council grants contract numbers 621-2013-4287 and 2015-04089 and by the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme (grant agreement No 668679).

Open Access. This article is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/

Web End =CC-BY 4.0 ), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

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