Radiative-recoil corrections to hyperfine splitting: Polarization insertions in the muon factor

Michael I. Eides*

Department of Physics and Astronomy, University of Kentucky, Lexington, Kentucky 40506, USA

Valery A. Shelyuto†

D. I. Mendeleev Institute of Metrology, St. Petersburg 190005, Russia(Received 8 July 2009; published 18 September 2009)

We consider three-loop radiative-recoil corrections to hyperfine splitting in muonium due to insertions

of a one-loop polarization operator in the muon factor. The contribution produced by electron polarization

insertions is enhanced by the large logarithm of the electron-muon mass ratio. We obtained all single-

logarithmic and nonlogarithmic radiative-recoil corrections of order �3ðm=MÞEF generated by the

diagrams with electron and muon polarization insertions.

DOI: 10.1103/PhysRevD.80.053008 PACS numbers: 12.20.Ds, 13.40.Em, 31.15.aj, 31.30.jf

I. INTRODUCTION: MUON FACTORCONTRIBUTION TO HFS

Calculation of numerous corrections to hyperfine split-ting (HFS) in muonium has a long history (see, e.g., re-views in [1,2]). By now the largest challenge to the theoryis calculation of three-loop radiative-recoil corrections.Below we consider such corrections generated by twosets of diagrams with electron and muon vacuum polariza-tion insertions in the radiative photon attached to the muonline in Figs. 1 and 2. The muon anomalous magnetic mo-ment is subtracted from all vertices in these diagrams.Technically, the gauge invariant anomalous magnetic mo-ment is the hardest entry in the expression for the vertex atsmall transferred momenta, and this prompts its separation.It turns out that the anomalous magnetic moment does notgenerate radiative-recoil corrections (see, e.g., [3,4]).

The scale of integration momenta in the graphs in Figs. 1and 2 is determined by the muon mass, and is much higherthan external virtualities determined by the characteristicatomic momenta. In order to obtain radiative-recoil cor-rections it is sufficient to calculate matrix elements of thediagrams in Figs. 1 and 2 with the mass-shell externalmomenta with vanishing spatial components. Then thecontribution to hyperfine splitting is given by the sum ofmatrix elements of the gauge invariant sets of diagrams inFigs. 1 and 2 calculated between the free electron andmuon spinors and multiplied by the square of theCoulomb-Schrodinger wave function at the origin.Explicitly, contributions to HFS generated by each of thegauge invariant sets of graphs in Figs. 1 and 2 have theform

�E ¼ Z�

�

m

MEF

Z d4k

�2i

1

ðk2 þ i0Þ2 h��k6 ��i

��

1

k2 þ 4�k0 þ i0þ 1

k2 � 4�k0 þ i0

�L��

� ��Z�

�

m

MEF; (1)

where m andM are the electron and muon masses, respec-tively, � ¼ m=ð2MÞ, k is dimensionless exchange photonmomentum measured in terms of the muon mass, � is thefine structure constant, EF ¼ ð8=3ÞðZ�Þ4ðm=MÞm is theFermi hyperfine splitting energy, L�� is the muon line

factor, and the broken brackets denote matrix elementsbetween hyperfine states. Constant Z measures the muoncharge in terms of the electron charge and is equal one inmuonium, but we keep it as an indicator on the origin ofdifferent contributions.Generically dimensionless recoil contributions �� are

enhanced by powers of the large logarithm of the muon-electron mass ratio (see, e.g., [1,2]). The origin of theselarge logarithms is quite transparent. In the case of theskeleton muon factor L�� (Fig. 3), the momentum integral

in Eq. (1) is linearly divergent at low integration momentaof order mZ� (we temporarily return to dimensionfulmomenta in these considerations). This linear divergencecorresponds to the classic nonrecoil Fermi contribution toHFS and should be subtracted in the calculation of therecoil contribution. After subtraction the skeleton integralin Eq. (1) becomes logarithmic in the wide integrationregion m � k � M and generates the leading recoil cor-

FIG. 1. Electron polarization insertions.

*Also at Petersburg Nuclear Physics Institute, Gatchina, St.Petersburg 188300, Russia.eides@pa.uky.edu,

†shelyuto@vniim.ru

PHYSICAL REVIEW D 80, 053008 (2009)

1550-7998=2009=80(5)=053008(6) 053008-1 � 2009 The American Physical Society

rection to HFS. Considering one-loop radiative insertion inthe muon line in Fig. 4, we need to separate the gaugeinvariant one-loop anomalous magnetic moment and theremaining muon factor. The one-loop anomalous magneticmoment has the same momentum dependence as the skel-eton contribution, and generates only a nonrecoil contri-bution. Because of the generalized low-energy theorem forvirtual Compton scattering [4,5], the remaining one-loopmuon factor is suppressed by an additional factor k2=M2 incomparison with the skeleton muon factor. This suppres-sion not only makes the integral in Eq. (1) convergent inthe infrared region, but also makes the integral nonlogar-ithmic. As a result the one-loop muon factor generates onlynonlogarithmic radiative-recoil corrections obtained in [3–6]. Insertion of vacuum polarization in the radiative pho-tons does not change the low momenta behavior of themuon factor L�� and the integral over exchanged momenta

in Eq. (1) corresponding to the diagrams in Figs. 1 and 2remains convergent and nonlogarithmic. Then the leadingrecoil correction to HFS generated by the diagrams inFig. 2 with the muon polarization insertions does notcontain large logarithms and is a pure number. The caseof electron polarization insertions in Fig. 1 is a bit morecomplicated. While it is still true that integration over theexchanged momentum is nonlogarithmic, another sourceof large logarithms arises in these diagrams. As we alreadydiscussed all characteristic momenta in these diagrams areof the order of the muon mass M. Then the electronpolarization insertions in the diagrams in Fig. 1 enter inthe asymptotic regime, and the leading contribution to HFSgenerated by the diagrams with the electron polarizationinsertions in Fig. 1 is just the product of the leadingasymptotic term in the high momentum expansion of theelectron polarization operator and the radiative-recoil cor-rection to HFS generated by the one-loop muon factor. In

other words, we can say that the leading logarithmic con-tribution to HFS generated by the diagrams in Fig. 1 isobtained from the respective nonlogarithmic contributionof the diagrams in Fig. 4 by substitution of the runningcoupling constant �ðMÞ for radiative photons.

II. ANALYTIC CALCULATIONS

Actual calculations start with consideration of separatecontributions generated by the diagrams with self-energy,vertex, and spanning photon insertions in Figs. 1 and 2.Redefining the muon factor by extracting from it all factorsof �, we write the general expression in Eq. (1) in the form

�E ¼ ���ðZ2�ÞðZ�Þ

�3

m

MEF; (2)

where

�� ¼ ��� þ 2��� þ���: (3)

Explicit expressions for different terms on the right-handside in Eq. (3) are derived in the Feynman gauge. We startwith the respective diagrams in Fig. 4 without polarizationinsertions but with massive radiative (but not exchanged)photons. We account for polarization insertions by consid-ering photon mass squared to be �2 ¼ 16�2=ð1� v2Þ or�2 ¼ 4=ð1� v2Þ, for the electron and muon polarizations,respectively. Then insertion of the polarization operatormeans an additional integration over velocity v with theweight

Z 1

0

dv

1� v2v2

�1� v2

3

�: (4)

Below we collect integral representations for the con-tributions ��i obtained directly from the Feynman dia-grams. For the contribution of the self-energy diagram weobtained

��� ¼ i

2�2

Z 1

0dx

Z x

0dy

Z 1

0

dv

1� v2v2

�1� v2

3

�

�Z d4k

k42k2

k4 � 16�2k20

�3k0

1� x2

�1

� ð3k20 � 2k2Þ

� ð1� xÞ2�1� 2ð1þ xÞy

�0

�1

�1

�

� ���1 þ ���2; (5)

where

FIG. 2. Muon polarization insertions.

FIG. 4. One-loop radiative insertions in the muon line.

FIG. 3. Two-photon exchange.

MICHAEL I. EIDES AND VALERYA. SHELYUTO PHYSICAL REVIEW D 80, 053008 (2009)

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�0ðxÞ ¼ x2 þ �2ð1� xÞ � yð1� xÞa21ðx; yÞ; �1ðx; yÞ ¼ yð1� xÞð�k2 þ 2k0 þ a21 � i0Þ:The spanning photon contribution has the form

��� ¼ � i

4�2

Z 1

0dx

Z x

0dyðx� yÞ

Z 1

0

dv

1� v2v2

�1� v2

3

�Z d4k

k42k2

k4 � 16�2k20

��2ð3k20 � 2k2Þ

�1� 2y

�þ�2þ 2xð1� yÞ � x2

�2þ��2ð1� xÞy

�2

�

� 6bk0

�1� 2y

�þ 2xð1� yÞ þ x2

�2þ �2ð1� xÞð2� yÞ

�2þ 2bk0yð1� yÞ

�2

��

� ���1 þ ���2; (6)

where

�ðx; yÞ ¼ yð1� yÞð�k2 þ 2bk0 þ a2 � i0Þ; a2ðx; yÞ ¼ x2 þ �2ð1� xÞyð1� yÞ ; bðx; yÞ ¼ 1� x

1� y:

The vertex contribution is as follows:

��� ¼ � i

2�2

Z 1

0dx

Z x

0dy

Z 1

0

dv

1� v2v2

�1� v2

3

�Z d4k

k42k2

k4 � 16�2k20

�ð3k20 � 2k2Þ

���½ðx� yÞð1� 2yÞ þ yð1� yÞ� 1

�

þ 2

�1� x� x2

2

� ð1� xÞy��0

� xð1� xÞ ð1� xÞy��0

þ 1

�½1� xþ ðx� yÞ2�

��� 3k0

�k0

xð1� xÞ2y��0

� 1� x

�

�

�Z 1

0du

�ð3k20 � 2k2Þ

�2

�1� x� x2

2

�k2y2ð1� yÞðx� yÞ

�2u�0

� xð1� xÞ k2y2ð1� yÞðx� yÞ

�2u�0

� k2yð1� yÞ�2

u

½1� xþ ðx� yÞ2��� 3k20xð1� xÞ k

2y2ð1� yÞðx� yÞ�2

u�0

��

� ���1 þ ���2 þ���3; (7)

where

�uðx; y; uÞ ¼ yð1� yÞð�k2 þ 2buk0 þ a2uÞ;a2uðx; y; uÞ ¼ ua2ðx; yÞ;

buðx; y; uÞ ¼ 1� y� xuþ yu

1� y:

Our next goal is to calculate the momenta integrals inEqs. (5)–(7) analytically. We explained in Sec. I that due tosoft low-energy behavior of the muon factor L�� the

momentum integral in Eq. (1) is convergent and nonlogar-ithmic. Hence, �� does not depend on the electron-muonratio explicitly (it depends only on the effective photonmass �), and one can safely omit factor � in the electrondenominators in the square brackets in Eq. (1), whichsignificantly simplifies all calculations. However, each ofthe exchanged momentum integrals in Eqs. (5)–(7) corre-sponding to the self-energy, the spanning photon, and thevertex insertion contributions to the muon factor is loga-rithmic in the wide integration momenta region m=M �k � 1. At small integration momenta of order m=M thesewould-be divergences are cut off by the �-dependentelectron denominators. This is the reason why we pre-served parameter � in sums of the electron propagators

2k2=ðk4 � 16�2k20Þ in Eqs. (5)–(7). Separate contributions

in Eqs. (5)–(7) are gauge noninvariant, and one could hopeto find a gauge where individual contributions are infraredfinite even if one neglects the electron mass in the electrondenominators. This is not the case, such a gauge does notexist and infrared divergences survive in individual con-tributions even in the infrared soft Yennie gauge [5].We would like to get rid of the would-be logarithmic

divergencies at the level of integrands, so that divergent for� ¼ 0 momentum integrals would not arise at all. To thisend we wrote all integrals in Eqs. (5)–(7) in terms ofsimilar Feynman parameters and organized them in threegroups, combining contributions from different integrals.We obtained the representation for the energy splitting inthe form

�� ¼ ��� þ 2��� þ ��� ¼ Xi¼3

i¼1

��i; (8)

where

��1 ¼ ���2 þ ���1 þ 2���1; (9)

��2 ¼ ���1 þ ���2 þ 2���2; (10)

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��3 ¼ 2���3: (11)

Momentum integrals in Eqs. (9) and (10) are apparentlylogarithmically divergent at � ¼ 0, and require additionaltransformations.

Consider first ��1 in Eq. (9). After some integrations byparts over the Feynman parameters it acquires the form

��1 ¼ i

�2

Z 1

0dx

Z x

0dy

Z 1

0

dv

1� v2v2

�1� v2

3

�

�Z d4k

k2ðk4 � 16�2k20Þð3k20 � 2k2Þ

�A1

�þ k2B1

�2

�;

(12)

where

A1ðx;y;�Þ ¼ 3

�1� 2xy

�0

�� 5xþ 3x2 þ ð8� 4xÞy

þ�2� 4x

�0

�y2 þ�2ð1� xÞy

�0

ð�9þ 3x� 2yÞ;

and

B1 ¼�x

�1� x

2

�� 2

�1þ x

2

�xð1� xÞy

�0

�

��yð1� yÞ þ ð�2þ xÞ xy

2

�0

�

þ ½�2þ 2xð1� yÞ � x2 � �2ð1� xÞy� y2ðx� yÞ�0

:

After the Wick rotation the integral for ��1 in four-dimensional Euclidean spherical coordinates (k0 ¼ k cos�,jkj ¼ k sin�) has the form

��1 ¼Z 1

0dx

Z x

0dy

Z 1

0

dv

1� v2v2

�1� v2

3

�

�Z 1

0

k2dk2

k4 þ 16�2k2 cos�22

�

Z �

0d�sin2�ð2

þ cos2�Þ�

A1

yð1� yÞ þk2B1

y2ð1� yÞ2@

@a2

�

� k2 þ a2

ðk2 þ a2Þ2 þ 4b2k2cos2�: (13)

Now we see that omitting the term with �2 in the denomi-nator would result in an apparently logarithmically diver-gent integral. However, this is not the case since thecoefficient before the would-be divergence turns intozero due to the identity

Z 1

0dx

Z x

0dy

A1ðx; y; �Þ�0

¼ 0; (14)

that is valid for any �. This means that the momentumintegral in Eq. (13) is convergent even at� ¼ 0. We obtainan explicitly convergent form for this integral by subtrac-tion from the integrand taken at � ¼ 0 its leading low

momentum asymptotic term. This subtraction does notchange the integral since the coefficient before the sub-tracted term is identically zero due to the identity inEq. (14). Of course, we have chosen this particular combi-nation of terms on the right-hand side in Eq. (9) preciselybecause we wanted to organize an infrared finite contribu-tion. After subtraction and momentum integration we ob-tain a finite integral representation for ��1:

��1 ¼Z 1

0dx

Z x

0dy

Z 1

0

dv

1� v2v2

�1� v2

3

��� 9

4

A1

�0

ln1

a2

� 3

4A1

�3

�0

L0 þ 1� y

yð1� xÞ2 ð2L0 þ L1Þ�

� 3

2

B1

y2ð1� xÞ2 ðL0 þ L1Þ�; (15)

where Ln ¼ b2R10 dss

n=ða2 þ b2sÞ, or explicitly

L0 ¼ lna2 þ b2

a2; L1 ¼ 1� a2

b2L0: (16)

Consider now the contribution in Eq. (10) that is definedby the integral

��2 ¼ 3i

2�2

Z 1

0dx

Z x

0dy

Z 1

0

dv

1� v2v2

�1� v2

3

�

�Z d4k

k42k2

k4 � 16�2k20k0

�xð1� xÞ

�0

þ A2

�þ k2xð1� xÞyð1� yÞ

��0

þ k2B2

�2

�; (17)

where

A2ðx; y; �Þ ¼ ð1� xÞð2x� yÞ þ ð�2þ xÞ x2y

�0

;

B2 ¼ y

�3x� 2y� x2

2� 3x2y

2þ xy2

�

þ ð�2þ xÞ�1þ 3x

2� y

�x2y2

�0

:

After the Wick rotation we obtain in four-dimensionalEuclidean spherical coordinates an integral,

��2 ¼ �3Z 1

0dx

Z x

0dy

Z 1

0

dv

1� v2v2

�1� v2

3

�2b

�Z 1

0

k4dk2

k4 þ 16�2k2 cos�22

�

Z �

0d�sin2�cos2�

��� A2

yð1� yÞk2 þxð1� xÞ

�0

� B2

y2ð1� yÞ2@

@a2

�

� 1

ðk2 þ a2Þ2 þ 4b2k2cos2�; (18)

that is apparently logarithmically divergent for � ¼ 0.Like in the case of the contribution ��1, the coefficientbefore the would-be divergence turns into zero due to the

MICHAEL I. EIDES AND VALERYA. SHELYUTO PHYSICAL REVIEW D 80, 053008 (2009)

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identity

Z 1

0dx

Z x

0dy

ð1� xÞy�2

0

A2ðx; y; �Þ ¼ 0; (19)

that is valid for any �. Using this identity we subtract thewould-be divergent at � ¼ 0 integral of the leading lowmomentum asymptotic term in the integrand in Eq. (18),and obtain a finite integral representation for ��2:

��2 ¼ �3Z 1

0dx

Z x

0dy

Z 1

0

dv

1� v2v2

�1� v2

3

�

��ð1� xÞyA2

2�20

�ln

1

a2þ L0

�þ A2

2b�0

L1

þ xð1� yÞ�0

ðL0 � L1Þ þ B2

yð1� xÞ�0

L1

�: (20)

The momentum integral in Eq. (11) is convergent even at� ¼ 0, and since we are interested in the leading recoilcontribution we calculate it in this limit:

��3 ¼ 3Z 1

0dx

Z x

0

dy

yð1� yÞZ 1

0

dv

1� v2v2

�1� v2

3

�

�Z 1

0

du

b2u

�ðL0u þ L1uÞ

�ð2� 3xÞ yðx� yÞ

�0

� ½1� xþ ðx� yÞ2��þ ðL0u � 3L1uÞxð1� xÞ

� yðx� yÞ�0

�: (21)

where

L0u ¼ lna2u þ b2u

a2u; L1u ¼ 1� a2u

b2uL0u: (22)

III. NUMERICAL CALCULATIONS AND RESULTS

Let us notice that the expressions for ��i obtained inEqs. (15), (20), and (21) allow us to reproduce theradiative-recoil correction to HFS generated by the dia-grams in Fig. 4 with one-loop radiative insertions in themuon line. To obtain these corrections we need to omitintegration over velocity v and all explicit factors contain-ing v in Eqs. (15), (20), and (21), and also let the radiativephoton mass � be zero. The resulting integrals admitanalytic calculation and the results are collected inTable I. Restoring the overall dimensional factor we obtain

�E ¼ ��ð0ÞðZ2�ÞðZ�Þ

�2

m

MEF

¼�9

2ð3Þ � 3�2 ln2þ 39

8

� ðZ2�ÞðZ�Þ�2

m

MEF; (23)

which precisely reproduces the old result [3,6].Now we are ready to proceed to calculations of the

contributions to HFS generated by the diagrams inFigs. 1 and 2. The leading logarithmic term correspondingto the diagrams in Fig. 1 with the electron polarizationinsertions in the radiative photon is obtained by substitut-ing the running coupling constant �ðMÞ for the radiativephoton in the result in Eq. (23), and is equal to (we do notinclude � in the definition of ��; it will be restored in thefinal results)

��log ¼ 2

3lnM

m��ð0Þ: (24)

Similar equations are also valid for each of the contri-

butions ��ð0Þi . For practical calculations we defined non-logarithmic contributions to HFS as

�Ci ¼ ��i ��2

3lnM

m� 5

9

���ð0Þi : (25)

We calculated contributions in Eq. (25) numerically usingthe integral representations in Eqs. (15), (20), and (21),with the photon mass squared �2 ¼ 16�2=ð1� v2Þ, andobtained

�C1 ¼ �1:4965ð7Þ; �C2 ¼ 2:2442ð6Þ;�C3 ¼ 5:7905ð5Þ: (26)

Then the total contribution to HFS due to electron polar-ization insertions in Fig. 1 has the form

��e ¼�3ð3Þ � 2�2 ln2þ 13

4

�lnM

mþ 12:227ð2Þ: (27)

The contribution to HFS due to muon polarization in-sertions in Fig. 2 was calculated numerically with the helpof the integral representations in Eqs. (15), (20), and (21).This time �2 ¼ 4=ð1� v2Þ plays the role of the photonmass squared in the integrands, and we obtained

��1 ¼ 0:2987; ��2 ¼ �0:0003;

��3 ¼ �1:2291;(28)

or

��� ¼ �0:9307: (29)

Restoring all dimensional factors we obtain fromEqs. (27) and (29)

TABLE I. One-loop radiative-recoil corrections.

��ð0Þ1 �15ð3Þ þ 6�2 ln2� 5�2 þ 2378

��ð0Þ2158 ð3Þ � 3�2

4 ln2þ �2

4 � 34

��ð0Þ31418 ð3Þ � 33�2

4 ln2þ 19�2

4 � 24��ð0Þ 9

2 ð3Þ � 3�2 ln2þ 398

RADIATIVE-RECOIL CORRECTIONS TO HYPERFINE . . . PHYSICAL REVIEW D 80, 053008 (2009)

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�Ee ¼��

3ð3Þ � 2�2 ln2þ 13

4

�lnM

mþ 12:227ð2Þ

�

� �ðZ2�ÞðZ�Þ�3

m

MEF; (30)

�E� ¼ �0:931ðZ2�Þ2ðZ�Þ

�3

m

MEF: (31)

The total contribution to HFS due to diagrams withelectron and muon polarization insertions in Figs. 1 and2 is

�E ¼��

3ð3Þ � 2�2 ln2þ 13

4

�lnM

mþ 11:297ð2Þ

�

� �3

�3

m

MEF: (32)

This result together with the results [7–10] completescalculation of all radiative-recoil corrections of order�3ðm=MÞEF generated by the diagrams with electron andmuon polarization insertions.

ACKNOWLEDGMENTS

This work was supported by NSF Grant No. PHY-0757928. V.A. S. was also supported in part by RFBRGrants No. 06-02-16156 and No. 08-02-13516, and byDFG Grant No. GZ 436 RUS 113/769/0-3.

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[2] M. I. Eides, H. Grotch, and V.A. Shelyuto, Phys. Rep. 342,63 (2001).

[3] S. G. Karshenboim, V. A. Shelyuto, and M. I. Eides, Zh.Eksp. Teor. Fiz. 94, 42 (1988) [Sov. Phys. JETP 67, 671(1988)].

[4] M. I. Eides, S. G. Karshenboim, and V.A. Shelyuto, Ann.Phys. (N.Y.) 205, 291 (1991).

[5] S. G. Karshenboim, V. A. Shelyuto, and M. I. Eides, Yad.

Fiz. 47, 454 (1988) [Sov. J. Nucl. Phys. 47, 287 (1988)].[6] M. I. Eides, S. G. Karshenboim, and V.A. Shelyuto, Phys.

Lett. B 202, 572 (1988).[7] M. I. Eides, H. Grotch, and V.A. Shelyuto, Phys. Rev. D

65, 013003 (2001).[8] M. I. Eides, H. Grotch, and V.A. Shelyuto, Phys. Rev. D

67, 113003 (2003).[9] M. I. Eides, H. Grotch, and V.A. Shelyuto, Phys. Rev. D

70, 073005 (2004).[10] M. I. Eides and V.A. Shelyuto (unpublished).

MICHAEL I. EIDES AND VALERYA. SHELYUTO PHYSICAL REVIEW D 80, 053008 (2009)

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