DSpace at VNU: Single pion contribution to the hyperfine splitting in muonic hydrogen

Single pion contribution to the hyperfine splitting in muonic hydrogenNguyen Thu Huong,1 Emi Kou,2 and Bachir Moussallam3 1Faculty of Physics, VNU University of Science, Vietnam National

Single pion contribution to the hyperfine splitting in muonic hydrogen

Nguyen Thu Huong,1 Emi Kou,2 and Bachir Moussallam3

1Faculty of Physics, VNU University of Science, Vietnam National University, 334 Nguyen Trai,

Thanh Xuan, Hanoi, Vietnam

2Laboratoire de l’Accélérateur Linéaire, Université Paris-Sud, CNRS/IN2P3, Université Paris-Saclay,

91898 Orsay Cédex, France

3Groupe de physique théorique, IPN, Université Paris-Sud 11, 91406 Orsay, France

(Received 28 December 2015; published 7 June 2016)

A detailed discussion of the long-range one-pion exchange (Yukawa potential) contribution to the

2S hyperfine splitting in muonic hydrogen, which had, until recently, been disregarded, is presented

We evaluate the relevant vertex amplitudes, in particularπ0μþμ−, combining low energy chiral expansions

together with experimental data on π0 and η decays into two leptons A value of ΔEπ

HFS¼ −ð0.09  0.06Þ μeV is obtained for this contribution

DOI: 10.1103/PhysRevD.93.114005

I MOTIVATION The first accurate measurement of the 2SF¼1

1=2 − 2PF¼2

1=2

Lamb shift transition in muonic hydrogen[1]has led, with

the help of the currently accepted theoretical formulas (e.g.,

[2,3]), to a determination of the proton radius rE with a

precision of 0.8 per mil The proton size puzzle arose from

the discrepancy, by 5 standard deviations, between this

result and the CODATA-2010 value[4], which was based

on ordinary hydrogen spectroscopy as well as ep

scatter-ing This has stimulated a number of new theoretical

and experimental investigations (see, e.g., the review in

[5]) In particular, Antognini et al.[6]have measured both

theνt≡ 2SF¼1

1=2 − 2PF¼2

3=2 and theνs≡ 2SF¼0

1=2 − 2PF¼1

3=2

tran-sitions which has confirmed and refined the previous result

on the Lamb shift (increasing the rEdiscrepancy to7σ) and

further provides an experimental value for the2S hyperfine

splitting1

ΔEexp

HFS ¼ 22.8089ð51Þ ðmeVÞ: ð1Þ The hyperfine splitting (HFS) is interesting as it probes

aspects of the proton structure somewhat differently from the

Lamb shift While the influence of the proton radius rE is

suppressed, the main structure dependent contribution

is proportional to the Zemach radius rZ: ΔEZ

HFS¼

−0.1621ð10ÞrZ meV (with rZin fm), as given in the review

[8], and the next main structure dependent contribution is

that associated with the forward proton polarizabilities It has

been estimated in Ref [9] as ΔEpol

HFS¼ ð8.0  2.6Þ μeV

(see also[10]) It is noteworthy that the value of rZthat one determines from the HFS measurement in muonic hydrogen,

rZ ¼ 1.082ð37Þ fm [6], is in agreement with the value computed in terms of the proton form factors GE, GM measured in ep scattering, rZ¼ 1.086ð12Þ fm[11], at the present level of accuracy

A possible role in muonic hydrogen of light, exotic (universality violating) particles, with vector or axial-vector (JPC¼ 1−−;¼ 1þþ) quantum numbers has been

consid-ered[12,13] Similarly, the influence of exchanging a light pseudoscalar particle (JPC¼ 0−þ) was recently studied in

Ref.[14] In that case, the HFS splitting is affected but not the (appropriately defined) Lamb shift

In this article, we point out that a light pseudoscalar particle exists within the standard model, the neutral pion, and we perform the exercise to estimate the influence of the one-pion exchange mechanism onΔEHFS We will show that using chiral symmetry allows one to evaluate the two vertex functions which are needed, represented by blobs

in Fig 1, for small momentum transfer, based on exper-imental data

The coupling of theπ0to a lepton pair proceeds (within

the standard model) via two virtual photons Theμp → μp

FIG 1 Single pion exchange in theμp → μp amplitude

1The2S hyperfine splitting is extracted from the experimental

measurements through equationΔE2S

HFS¼ hνs− hνtþ ΔE2P3=2

HFS −

δ where h is the Planck constant and the 2P hyperfine splitting

ΔE2P3=2

HFS and the 2P F ¼ 1 mixing parameter δ are computed

theoretically[3,7]

PHYSICAL REVIEW D 93, 114005 (2016)

one-pion exchange amplitude can also be viewed as a

two-photon exchange amplitude The pion pole in the

Compton amplitude γp → γp contributes to the so-called

proton backward spin polarizability γπ (e.g [15]) The

corresponding contribution in muonic hydrogen is then

expected to be suppressed by one power ofα as compared

to the forward proton polarizability contribution This

explains why the simple mechanism of Fig 1 does not

seem to have been previously considered until very recently

[16,17] Some enhancement might be expected from the

fact that γπ is numerically large compared to the forward

polarizabilities αp, βp and from the fact that the Yukawa

potential has a relatively long range (on the scale of the

proton size) which increases the overlap with the atomic

wave functions As a final motivation, let us recall that the

π0μþμ− coupling plays a significant role among the

hadronic contributions to the muon g− 2 [18] and it is

thus of interest to probe the level of sensitivity of muonic

hydrogen to this coupling

II PION COUPLING AMPLITUDES TO LEPTONS

AND TO NUCLEONS

A π0-lepton coupling For low momentum transfer, the Plþl− vertex

ampli-tude, where P is a light neutral pseudoscalar meson (π0or

η) and l is a light lepton (e orμ), can be evaluated in

the chiral expansion2[19] At leading order, the amplitude

is given from the two diagrams shown in Fig 2 In the

one-loop diagram, the Pγγ vertex is generated by the

Wess-Zumino-Witten Lagrangian (see[20], Chap 22)

LWZ¼ α

8πFπϵμναβ

π0þ 1ffiffiffi 3

p η



FμνFαβ ð2Þ

with the sign corresponding to the convention ϵ0123¼ 1

(we also useγ5¼ iγ0γ1γ2γ3) This diagram accounts for the

contributions of photons with low energy compared to

1 GeV The higher energy contributions are parametrized

through two chiral coupling constants χ1, χ2 in the

Lagrangian[19],

LSLW ¼ 3iα2

32π2lγμγ5lðχ1hQ2U†D

μU− Q2UD

μU†i

þ χ2hQU†QD

μU− QUQDμU†iÞ; ð3Þ where U is the chiral SUð3Þ matrix,

U¼ expiΦ

Fπ;

Φ ¼

0 B B

π0þ ηffiffi

3

p ffiffiffi

2

p

πþ ffiffiffi

2

p

Kþ ffiffiffi

2

p

π− −π0þ ηffiffi

3

p ffiffiffi 2

p

K0 ffiffiffi

2

p

2

p

K0 −p2ηffiffi3

1 C

and

DμU¼ ∂μU− iðvμþ aμÞU þ iUðvμ− aμÞ; ð5Þ where vμðaμÞ are external vector (axial-vector) sources (see [21]) and Q is the charge matrix, Q¼ diagð2=3; −1=3; −1=3Þ The tree graph shown in Fig 2

is computed from this Lagrangian The coupling constants

χ1, χ2 remove the ultraviolet divergence of the one-loop

graph Assuming the leptons to be on their mass shell, the

Plþl− vertex amplitude can be expressed in terms of a

single Dirac structure,

iTPl þ l − ¼ rP

α2m l

2π2F

πAlððp1− p2Þ2Þulðp2Þγ5u

lðp1Þ; ð6Þ where rP¼ 1; 1=pffiffiffi3if P¼ π, η In practice, dimensional regularization brings in some scheme dependence because

of the presence of the γ5 matrix For instance, the

amplitudes computed in Refs [19] and [22] differ by a constant Some discussion of this point can be found in Ref.[23] For definiteness, we will choose the convention

of[22], which gives AlðsÞ in the form

AlðsÞ ¼ χPðΛÞ þ32logm

2 l

Λ2−52þ ClðsÞ;

with

ClðsÞ ¼ 1

βlðsÞ



Li2βlðsÞ − 1

βlðsÞ þ 1þ

π2

3 þ

1

4log2

βlðsÞ þ 1

βlðsÞ − 1



;

βlðsÞ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 − 4m2

l=s

q

FIG 2 Feynman graphs which generate the π0-lepton vertex

amplitude at leading order in the chiral expansion

2We consider here the coupling mediated by the

electromag-netic interaction The coupling mediated by the weak interaction

is comparatively suppressed by 2 orders of magnitude

Using MS renormalization, the coupling constant

combina-tionχP becomes scale dependent with d=dΛχPðΛÞ ¼ 3=Λ,

which ensures thatAl is scale independent.

The value of χPðΛÞ must be determined from

experi-ment For this purpose, we can use eitherπ0→ eþ −which

was measured recently by the KTeV Collaboration[24]or

η → μþμ− (see[25]) It is convenient to consider the ratio

RP ¼ ΓðP → lþl−Þ=ΓðP → γγÞ which should be less

sensitive to higher order chiral corrections than the

indi-vidual modes It is expressed as follows, in terms of the

amplitude Al:

RP ¼2α2m2l

π2m2 P

βlðm2

PÞjAlðm2

PÞj2: ð9Þ

In the case of theπ0, the quantity measured experimentally

is the branching ratio for the decay mode π0→ eþe−ðγÞ,

including photons in the final state such that seþe ≥

0.95m2

π 0 The ratio which interest us, Rπ0, can be deduced

from this result by removing the bremsstrahlung and the

associated radiative corrections These have been revised

recently in Ref [26] Using the results of that work, one

deduces

Rexpπ0 ¼ ð6.96  0.36Þ × 10−8: ð10Þ

There are two solutions for χP which correspond to this

experimental result,

ðaÞ χPðmρÞ ¼ 4.51  0.97;

ðbÞ χPðmρÞ ¼ −19.41  0.97 ð11Þ

(in which the scale was set toΛ ¼ mρ¼ 0.774 GeV) To

decide which solution to choose, we can compare with the

model proposed in Ref.[22] It is based on a rigorous sum

rule which holds in the large Nc limit of QCD and the

approximation of retaining only the lightest resonance in

the sum This model gives

χLMD

P ðΛÞ ¼11

4 −

3

2log

m2

Λ2−4π2F2

and the uncertainty was estimated in Ref.[22]to be of the

order of 40% Thus, one has

χLMD

P ðmρÞ ≃ 2.2  0.8: ð13Þ This result lies within one sigma of solutionðaÞ and is not

compatible with solutionðbÞ This argument suggests that

solutionðaÞ is more likely to be the physically correct one

Alternatively, we can determine the coupling constantχP

from the decay mode of theη meson, η → μþμ−for which

the experimental branching fraction is (see[25]) BFðη →

μþμ−Þ ¼ ð5.8  0.8Þ × 10−6 leading to

Rexpη ¼ ð1.47  0.20Þ × 10−5: ð14Þ

There are again two solutions forχPcorresponding to this experimental result,

ða0Þ χPðmρÞ ¼ 1.69  0.87;

ðb0Þ χPðmρÞ ¼ 7.96  0.87: ð15Þ None of these solutions is compatible withðbÞ of Eq.(11): one can therefore safely conclude that solution ðbÞ must

be eliminated We can also eliminate ðb0Þ which is not compatible with the model estimate (12) while ða0Þ is

It seems reasonable, for our purposes, to perform an average of theðaÞ and ða0Þ values and thus use

χPðmρÞ ¼ 3.10  1.50; ð16Þ where we have slightly rescaled the error such that the two central values of ðaÞ and ða0Þ lie within the error

B π0-proton coupling

At leading order in the chiral expansion, the pion-nucleon coupling is given, at tree level, from the chiral Lagrangian[27]

LπNN ¼ ψ

iγμΔμ− mNþ igA

2 γμγ5u†DμUu†



where U is the SUð2Þ chiral matrix here, u ¼pffiffiffiffiU

, and

Δμ¼ ∂μþ Γμ;

Γμ¼1

2½u†;∂μu −

1

2iu†ðvμþ aμÞu −

1

2iuðvμ− aμÞu†;

ð18Þ

vμ(aμ) being external vector (axial-vector) sources andψ is

an isospin spinor containing the proton and the neutron,

ψ ¼
ψp

ψn



The coupling constant gA in the Lagrangian(17)is easily identified as the axial charge of the proton and also controls the neutron-proton matrix element of the charged axial current,

lim

q0¼qhpðq0Þjuγμγ5djnðqÞi ¼ gAupðqÞγμγ5u

nðqÞ: ð20Þ

It is determined from neutron beta decay experiments to have the following positive3 value [25]:

3

The absolute value of gAis obtained from the neutron lifetime, and its sign, we remind, is unambiguously determined from the asymmetry parameter of the neutron beta decay which, using

Eq (20), is given by A¼ 2ðgA− g2

AÞ=ð1 þ 3g2

AÞ The experi-mental value is[25]A¼ −0.1184ð10Þ

SINGLE PION CONTRIBUTION TO THE HYPERFINE… PHYSICAL REVIEW D 93, 114005 (2016)

gA¼ 1.2723  0.0023: ð21Þ The pion-proton vertex amplitude is then deduced from the

Lagrangian(17) to be

iTπpp¼ −gπppupðq2Þγ5u

pðq1Þ; gπpp¼gAmp

Fπ : ð22Þ The expression of the coupling constant gπpp at leading

chiral order, in terms of gA, mp, and Fπ, as it appears in the

above expression is, of course, the content of the

Nambu-Goldberger-Treiman relation (e.g., [20], Chap 19) It is

known that the higher order chiral corrections to this

relation do not exceed a few percent

III ENERGY SHIFTS IN MUONIC HYDROGEN

A q2= 0 approximation Having determined the π0μμ vertex [Eq (6)] and the

π0pp vertex [Eq. (22)], it is straightforward to derive

the muon-proton scattering amplitude, μðp1Þpðq1Þ →

μðp2Þpðq2Þ associated with one-pion exchange (Fig 1),

Tμp¼ −4mμmpα2g

AAμððp1− p2Þ2Þ 8π2F2

×uμðp2Þγ5u

μðp1Þupðq2Þγ5u

pðq1Þ

ðp1− p2Þ2− m2 : ð23Þ For our purposes, we can consider that both the muon and

the proton are nonrelativistic; therefore

ðp1− p2Þ2¼ ðq1− q2Þ2≃ −ð~p1− ~p2Þ2≡ −q2: ð24Þ

At first, let us make the approximation to set q2¼ 0 in the

vertex function Aμ We then obtain the nonrelativistic

Yukawa potential in momentum space,

Vμpð~qÞ ¼ − Tμp

4mμmp¼ λAμð0Þ~σμ· ~q~σp· ~q

q2þ m2 ;

λ ¼ α2gA

The contributions to the atomic energy shifts are most

easily performed by Fourier transforming to configuration

space,

Vμpð~rÞ ¼ ~λ½~σμ· ~σpVSSð~rÞ þ S12VTðrÞ;

~λ ¼ −λAμð0Þ m2

where S12¼ 3~σμ·ˆr~σp·ˆr − ~σμ· ~σp is the so-called tensor

operator, and

VSSð~rÞ ¼expð−mπrÞ

m2δ3ð~rÞ;

VTðrÞ ¼

1 þ 3

mπrþ 3

m2r2

 expð−mπrÞ

Making use of the average result(16)for χP, one obtains the following values forAμð0Þ and for the overall coupling

~λ in muonic hydrogen4

Aμð0Þ ¼ −5.37  1.5; ~λ ¼ ð2.61  0.49Þ × 10−7:

ð28Þ

We can now compute the energy shifts of muonic hydrogen caused by the one-pion exchange amplitude

We will consider both the 2S and 2P energy shifts for completeness, and the relevant radial Coulomb wave functions are

ψ2SðrÞ ¼ 1ffiffiffi

2

p exp

−μαr 2



1 −μαr 2



;

ψ2PðrÞ ¼ 1

2p expffiffiffi6

−μαr 2



where μ is the muon-proton reduced mass 1=μ ¼ 1=mμþ 1=mp From these, one computes the expectation values of the components VSS and VT of the Yukawa potential For the S wave, first, one has

h2SjVSSj2Si ≡ YSðmπÞ ¼ −ðμαÞ4

m3

8 þ 11~α þ 8~α2þ 2~α3

4ð1 þ ~αÞ4 ;

~α ¼μα

When computing the expectation value in the2S state, the contribution from the delta function in the potential VSS cancels the leading term inα from the contribution of the first piece As a result of this cancellation, YS scales asα4

and has a negative sign For the2P states, one has

h2PjVSSj2Pi ≡ YP¼ mπ~α5 1

4ð1 þ ~αÞ4;

h2PjVTj2Pi ≡ TP ¼ mπ~α55 þ 4~α þ ~α2

8ð1 þ ~αÞ4 : ð31Þ TableIlists the expressions for the shifts in the2P and the 2S states of muonic hydrogen in terms of the integrals YS,

YP, TP and the overall coupling ~λ [given in Eqs.(26)and (28)] as well as the central numerical values The con-tributions to the 2P3=2 states are particularly suppressed

4We also use Fπ¼ 92.21ð14Þ MeV and mπ¼ mπ 0¼ 134.9766ð6Þ

because the leading terms inα cancel in the combination

YP−2

5TP Finally, in the q2¼ 0 approximation, the

con-tribution from the single pion exchange to the2S hyperfine

splitting in muonic hydrogen is

ΔEπ

HFS¼ 4~λYSðmπÞ ¼ −ð0.19  0.05Þ μeV; ð32Þ

which is small but not irrelevant In contrast, the

contri-butions to the HFS in the2P states, as can be deduced from

TableIare too small to be of physical relevance Our result

(32)disagrees with the one quoted in Ref.[16]which uses

the same approximation We could trace the origin of the

discrepancy, essentially, to an incorrect coefficient for the

delta function in the Yukawa potential

B Influence of the vertex functions momentum

dependence The results quoted above were obtained setting q2¼ 0 in

the vertex functionAμ It was pointed out in Ref.[17]that

this is not a good approximation Plotting Aμð−q2Þ (see

Fig.3) shows indeed that the vertex function has a strong

cusp at q2¼ 0 which induces a rapid variation In the

following we evaluate the corrections induced by the q2

variation ofAμ This is easily done by using the dispersion

relation representation of the functionAμð−q2Þ,

Aμð−q2Þ ¼ Aμð0Þ −q2

π

Z ∞

0 ds

0 ImAμðs0Þ

s0ðs0þ q2Þ: ð33Þ For small values of q2 (compared to1 GeV2) we can use

the leading order chiral approximation which gives, for the imaginary part[19],

ImAlðs0Þ ¼ −πarctan

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4m2

l=s0− 1

q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4m2

l=s0− 1

lÞ;

ImAlðs0Þ ¼ −πarctanh

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 − 4m2

l=s0

q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 − 4m2

l=s0

lÞ ð34Þ

[which is easily verified to be reproduced by the explicit expressions(7)and(8)ofAl] Beyond the low q2region, estimates of the behavior ofAμmay be obtained based on modelings of theπ0γγform factor (e.g.,[28,29]for recent

work; see also[30]where a list of references to earlier work can be found) We will not consider these in detail here and content ourselves with a simple estimate of the role of the

q2≳ 1 GeV2region, taking into account the q2dependence

attached to theπpp vertex In this case, a weak cusp is expected from the three pions threshold at q2¼ −9m2, and

the q2dependence is expected to be smooth in the q2>0 region Models of the nucleon-nucleon interaction suggest

a simple approximation for the behavior in this region[31],

gπppð−q2Þ ≃ Λ2

Λ2þ q2gπppð0Þ ð35Þ withΛπ≃ 1.3 GeV We can now write the μp potential, taking into account a more complete picture of the momentum dependence, as

Vμpðq2Þ ¼ λ Λ2

Λ2þ q2

Aμð−q2Þ~σμ· ~q~σp· ~q

[whereλ is given in Eq.(25)] From this, it is not difficult to compute the Fourier transform, using the representation (33) for Aμð−q2Þ, and then the expectation values using the formulas of the preceding section The result for the2S states can be written in the form

h2SjVμpj2Si ¼ hσμ·σpið−Aμð0Þ þ δA1þ δA2Þ

×λm2

where the two corrective terms δA1 and δA2 have the following expressions:

TABLE I Contributions from the single pion exchange

ampli-tude to the 2S and the 2P energy levels in muonic hydrogen

where YSð≡YSðmπÞÞ, YPand TPare expectation values given in

Eqs.(30)and (31), and ~λ is given by Eq (26)

2PF¼2

5TPÞ −1.3 × 10−7

2PF ¼1

3~λðYP−2

5TPÞ 2.1 × 10−7

2PF ¼1

3~λðYP− 4TPÞ 0.9 × 10−4

2PF¼0

2SF¼1

2SF ¼0

-6

-5

-4

-3

-2

-1

0

1

2

-0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6

FIG 3 Vertex functionAμ as a function of q2.

SINGLE PION CONTRIBUTION TO THE HYPERFINE… PHYSICAL REVIEW D 93, 114005 (2016)

δA1¼ Aμð0Þ mπ

and

δA2¼2

π

Z ∞

0 dx

ImAμðm2x2Þ

xðx2− 1Þ



x4

1 − Rπx2

YSðmπxÞ

YSðmπÞ

1 − Rπ

x2− 1 ð1 − Rπx2ÞRπ

YSðΛπÞ

YSðmπÞþ 1



ð39Þ

with Rπ¼ m2=Λ2 This expression agrees with the result of

Ref.[17]in the limit Λπ→ ∞ and using the leading order

approximation in α of the function YS (which is valid

except when x is very close to zero) Figure4shows that the

integrand in Eq.(39)is peaked at x¼ 0 The effect of Λπis

essentially to cut off the integration region x >Λπ=mπ

which reduces the size of δA2 by 30% approximately

Using the numerical result(28)forAμð0Þ we find, for the

two corrective terms induced by the q2dependence of the

vertices,

δA1≃ −0.52; δA2≃ −2.30; ð40Þ

which reduce the result based onAμð0Þ by roughly 50%

It seems reasonable to affect an uncertainty of ≃30% to

these corrective terms We thus arrive at the following final estimate for the 2S hyperfine splitting induced by the exchange of one pion in muonic hydrogen,

ΔEπ HFS¼ −ð0.09  0.06Þ μeV; ð41Þ which is negative and differs from zero within the error The error is obtained by adding in quadrature the error associated withAμð0Þ [see Eq.(28)] and that arising in the integral givingδA2as discussed above Our result agrees in

magnitude with that of Ref [17] The difference arises mainly from taking into account hadronic form factor effects in the momentum integral forδA2.

IV CONCLUSIONS The recent measurement of the 2S HFS in muonic hydrogen[6] incites one to try to improve the theoretical evaluations of the strong interaction effects, in order to reduce the error in the determination of the Zemach radius

rZ In this context, we have considered here the “simple” one-pion exchange (Yukawa) contribution We have indi-cated how to compute this contribution based on exper-imental results on π0→ eþ −, η → μþμ−, and the

associated low energy chiral expansion as developed, in this sector, in Ref [19] The use of chiral symmetry is important in order to properly fix the signs of the relevant πll and πNN coupling constants and is also necessary in order to perform low-momentum expansions at the vertices The final result for the contribution of one-pion exchange to the HFS is given in Eq.(41) It has a magnitude comparable

to the smallest contributions which are already taken into account in the theoretical evaluation of the HFS (see the list of 28 contributions collected in Table 3 of Ref.[8]) At present, however, the main source of uncertainty affecting the strong interaction effects in the2S HFS is that attached

to the proton forward polarizabilities

ACKNOWLEDGMENTS

We thank Vladimir Pascalutsa, Franziska Hagelstein, and Hai-Qing Zhou for clarifying correspondence

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