Higher order proton structure corrections to the Lamb shift in muonic hydrogen

0.14 MB·English

by  Carl E. Carlson

Higher orderproton structure correctionsto the Lambshiftin muonichydrogen
Carl E. Carlson1,2 and Marc Vanderhaeghen3
1Helmholtz InstitutMainz, JohannesGutenberg-Universita¨t, D-55099 Mainz, Germany
2Department of Physics, College of Williamand Mary, Williamsburg, VA 23187, USA
3Institutfu¨rKernphysik, JohannesGutenberg-Universita¨t, D-55099 Mainz, Germany
(Dated:January31,2011)
Therecentconundrumwiththeprotonchargeradiusinspiresreconsiderationofthecorrectionsthat
enterintodeterminationsoftheprotonsize. Westudythetwo-photonproton-structurecorrections,
withspecialconsiderationofthenon-polesubtractionterminthedispersionrelation,andusingfits
tomoderndatatoevaluate theenergycontributions. Wefindthatindividualcontributions change
morethanthetotal,andpresentresultswitherrorestimates.
1
1
0 The recent measurementof the proton charge radius The (α5)termwasgivenbyFriar[9]as,
2 O
usingtheLambshiftinmuonichydrogen[1]hasgivena

n valuethatisastartling4%,or5ofthepreviousstandard R3 = d3r d3r ~r ~r 3ρ (r )ρ (r ). (5)
a deviations,lowerthanthevaluesobtainedfromenergy (2) Z 1 2| 1− 2| E 1 E 2
J
level shifts in electronic hydrogen [2] or from electron- whereρ isthechargedensityoftheprotonitself. Friar
1 E
3 proton scattering experiments [3, 4]. Specifically, the calledR3 thethirdZemachmoment,becauseitisrem-
newmuonichydrogenmeasurement[1]gives (2)
iniscent of an integral found by Zemach [10] in the re-
]
h R =0.84184(67)fm, (1) latedcontextofhyperfinesplitting.
E
p Inmoderntimes,oneshouldcalculatethe (α5)cor-
O
- comparedtotheCODATAvalue[2] rections field theoretically using the diagram shown in
p
e R =0.8768(69)fm, (2) Fig.1,ashasbeendonebyPachucki[11,12]andbyoth-
h E ers[13]. Wewishtoreexaminethecalculationhere,for
[ orthelatestelectronscatteringvalue[4] the purposeof betterassessingthe connection between
1 theelasticandinelasticcontributions,andtobettereval-
v RE =0.879(8) fm, (3) uatethesubtractiontermneededinadispersionrelation
5 thatispartofthework.
6 where we have addedin quadraturethe severaluncer-
The calculation of the elastic and inelastic contribu-
9 taintiesgivenin[4].
tions should be done together. Perhaps in the future
5 The promise of the muonic hydrogen measurement
. a direct QCD calculation will be possible, and there is
1 was that, because a muon would orbit closer to the
an exploration of the hadronic corrections to the Lamb
0 proton than an electron, the effect of the proton struc-
shift using chiral perturbation theory [14], but for the
1 ture on the energy level splittings would be enhanced
1 present to obtain the required accuracy the calculation
and a more accurate proton radius could be obtained.
: needs to be done dispersively, connecting the off-shell
v Based on the quoted error limit, that promise has been
Compton scattering which is the hadronic side of the
Xi achieved. However, the discrepancyfromthe previous
diagram to information obtained from electron-proton
results requires an explanation, and invites a reconsid-
r scattering. In particular, done thatway the elastic con-
a erationofthetheoreticalcorrectionsthatareinvolvedin
tributions requireno (non-existent) knowledge of form
connecting the experimental energy shift to the proton
factorsforsituations whereaprotonisoff-shell. Italso
chargeradius[5,6]. Inthisnote,wewillfocusononeof
meansthatcertainnon-polecontributionstotheComp-
thecorrections, namelytheorderα5 protonsizecorrec-
tonamplitudesarenotpickedupbythedispersivecal-
tionstotheLambshift.
culationandinfactdonotcontribute.
Theleading (α4)and (α5)protonstructurecontri-
O O Wealsoanalyzemoreconcretelythesubtractionfunc-
butionstothehydrogenLambshiftareoftengivenas
tion that appears because one of the dispersion rela-
2πα 1 tions does not converge if unsubtracted. The subtrac-
∆E = φ (0)2 R2 m αR3 , (4)
3 n E− 2 r (2) tion function depends on the photon four-momentum
(cid:18) (cid:19) squared,Q2,anditsvalueatQ2 =0isgivenintermsof
whereφ2(0)isthesquareofthenS-statewavefunction the proton magnetic polarizability. Its Q2 dependence
n
at the origin (which contains a factor α3) and m is the canbeestimatedbycalculatingatwo-pionloop contri-
r
lepton-proton reduced mass. The quadratic term was bution which couples to the nucleon as a scalar. One
obtained non-relativistically in [7], and one can verify does not need to use a Q2 dependence assumed given
fromarelativisticcalculationthat R isindeedthepro- bythenucleonelectromagneticformfactor,ashasbeen
E
tonchargeradius[8]. donepreviously.
2
γµF (Q2)+(i/2M)σµνq F (Q2) for an incoming pho-
1 ν 2
k k ton,are
1 Q4G2 (Q2)
TB(q ,Q2) = M F2(Q2) ,
q q 1 0 4πM ((Q2−iε)2−4M2q20 − 1 )
p p TB(q ,Q2) = MQ2 GE2(Q2)+τpG2M(Q2), (9)
2 0 π(1+τp) (Q2−iε)2−4M2q20
FIG.1:Theboxdiagramforthe (α5)corrections.
O where τ = Q2/(4M2), and the electric and magnetic
p
formfactorsare
G (Q2) = F (Q2) τ F (Q2),
The Feynman diagram for the two-photon proton- E 1 p 2
−
structurecorrectiontotheLambshiftisshowninFig.1. G (Q2) = F (Q2)+F (Q2). (10)
M 1 2
To the level of accuracy needed here, all external lines
have zero three-momentum. The blob corresponds to TheBorntermsarereliableforobtainingtheimaginary
off-shellforwardComptonscattering,givenintermsof partsofthenucleonpoleterms,butnotreliableingen-
theComptontensor eral, since the given vertex assumes the incoming and
outgoingnucleonsarebothonshell.
Tµν(p,q)= i d4xeiqx p Tjµ(x)jν(0) p CallingthefirstterminT1Bthepoleterm,onecansplit
8πM h | | i thewholeofT intopoletermandnon-poleterms,
Z 1
qµqν
= −gµν+ q2 T1(ν,Q2) T1(q0,Q2) = T1pole+T1. (11)
(cid:18) (cid:19)
1 p q p q The pole term alone evidently allows an unsubtracted
+ pµ · qµ pν · qν T (ν,Q2), (6)
M2 − q2 − q2 2 dispersion relation, and this term calculated from the
(cid:18) (cid:19)(cid:18) (cid:19)
dispersionrelationsimplyreproducesitself.Withaonce
where q2 = Q2, ν = p q/M, and M is the nucleon subtracteddispersionrelationforT ,onehas
− · 1
mass. A spin average is implied and the state normal-
ization is hp|p′i = (2π)32Eδ3(~p−~p′). The functions T1(q0,Q2) = T1pole(q0,Q2)+T1(0,Q2)
T1,2(ν,q2) are each even in ν and their imaginary parts q2 ∞ F (ν,Q2)
are related to the structure functions measured in elec- + 0 dν 1 . (12)
tronormuonscatteringby 2πM Zνth ν(ν2−q20)
pole
1 The nucleon pole is isolated in T and the integral
ImT (ν,Q2)= F (ν,Q2), 1
1 4M 1 begins at the inelastic threshold ν = (2Mm +m2 +
th π π
1 Q2)/(2M).Similarly,asTB containsonlyapoleterm,
ImT (ν,Q2)= F (ν,Q2), (7) 2
2 2
4ν
1 ∞ F (ν,Q2)
wiAthftνer>do0ianngdawWhiecrkerFo1t,a2taioren,swtahnedraerdq [1=5]i.Q andQ~ = T2(q0,Q2)= T2B(q0,Q2)+ 2π Zνth dν ν22−q20 . (13)
0 0
~q,oneobtainsthe (α5)energyshiftas With
O
∆E = 8α2mφ2(0) d4Q ∆E =∆Esubt+∆Einel+∆Eel, (14)
π n
Z weobtain
(Q2+2Q2)T (iQ ,Q2) (Q2 Q2)T (iQ ,Q2)
× 0 1 Q40(Q4+−4m2Q−2) 0 2 0 , ∆Esubt = 4πα2φ2(0) ∞ dQ2γ1(τℓ)T (0,Q2), (15)
0 (8) m n Z0 Q2 √τℓ 1
where m is the lepton mass, and φ2(0) = m3α3/(πn3)
n r k k k k
withm =mM/(M+m).
r
TheT areobtainedusingdispersionrelations. Regge
i q q
arguments[16]suggestthatT satisfiesanunsubtracted q q
2
dispersion relation in ν at fixed Q2, but that T will re- pp p p p
1
quire one subtraction. Before proceeding, we will note
that the Born terms, obtained from the elastic box and FIG.2:Elasticcontributionstotheboxdiagram.
crossed box of Fig. 2 and the vertex function Γµ =
3
2α2 ∞ dQ2
∆Einel = φ2(0)
−mM n Z0 Q2 k+q k−q
∞dν γ1(τ,τℓ)F1(ν,Q2) + γ2(τ,τℓ)F2(ν,Q2) , k k
×Zνth (cid:20) ν Q2/M (cid:21) p p p k k p p k k p
e e (16)
α2m ∞ dQ2 FIG. 3: Diagrams used for estimating the Q2 dependence of
∆Eel =−M(M2−m2)φn2(0)Z0 Q2 thenon-polepartofthesubtractionterm.
γ2(τp) γ2(τℓ) GE2 +τpG2M
×(cid:26) √τp − √τℓ ! τp(1+τp) where,withλ =4m2π/Q2,
− γ√1(ττpp) − γ√1(ττℓℓ)!G2M(cid:27), (17) Floop(Q2)= 32λ(√1+λln√√11++λλ+11 −2)
−
where τ = ν2/Q2 and τℓ = Q2/(4m2). The auxiliary 1 Q2 + (Q4), Q 0
functionsare = − 10m2π O m2π → . (23)
 6m2π ln Q2 2 +O(m4π), Q ∞
γ (τ)=(1 2τ) (1+τ)1/2 τ1/2 +τ1/2,  Q2 m2π − Q4 →
1 − − (cid:16) (cid:17)
γ (τ)=(1+τ)3/(cid:16)2 τ3/2 3τ1/2.(cid:17) (18) WecanidentifyβM = αgS/48π2m2π,andobtain
2
− − 2
β
Both are monotonically falling functions, reducing to 1 T1(0,Q2)= 4πMαQ2Floop(Q2). (24)
atτ =0andfallinglikeτ 1/2atlargeτ. Also
−
TheParticleDataGroupgives[15],
1
γ1(τ,τℓ) = √τℓγ1(τℓ) √τγ1(τ) ,
γe2(τ,τℓ) = ττℓℓ−1 ττ(cid:16) γ√2(ττ) − γ−√2(ττℓℓ) . (cid:17) (19) However,accβorMdi=ng(1to.9s±om0e.5r)e×ce1n0t−a4nfamly3s.es, (25)
− (cid:18) (cid:19)
Theesubtraction function T(0,Q2) has unphysical ar- (4.0 0.7) 10 4fm3 [18]
guments,exceptingthepointQ2 =0. Itcomesfromthe βM = (3.4±1.2)×10−4fm3 [19,20]. (26)
excitationof the proton, and canatlow Q2 (and low ν, (cid:26) ± × −
ingeneral)bedescribedusingtheelectric(αE)andmag- UsingthesubtractionfunctionfromEq.(24),wefind
netic(β )polarizabilitiesandtheeffectiveHamiltonian
M
β
= 14πα ~E2 14πβ ~B2. (20) ∆Esubt =5.3µeV× (3.4 10M4fm3) . (27)
H −2 E − 2 M × −
ForsmallνandQ,thisgives Muchofthesupportfortheintegralisatlow Q2,being
controlled byγ aswellasbythe Q2 dependencefrom
lim T (ν,Q2)= ν2 (α +β )+ Q2β . (21) thepionloop,a1ndhalfthecontributionsto∆Esubtcome
ν2,Q2 0 1 e2 E M e2 M fromQ2 <0.04GeV2,albeitwithalongtail.
→
The ν2 term is shown to connect to known results in Refs.[1∼2]and[21]found∆Esubt tobe1.8and2.3µeV,
another context [17], and the Q2 term was obtained by using βM = 1.5 and 1.9 10−4 fm3, respectively, and
×
Pachucki [11]. With the above result, the integral over usingaQ2falloffrelatedtothenucleonelectromagnetic
T1(0,Q2)convergesatthelowerlimit. formfactor. ForthesameβM,ourresultsareabout30%
For higher Q2, the subtraction function comes from largerduetohavingflatterQ2falloff.
non-nucleon-pole contributions, and the forward am- One can also consider inserting a form factor Fπ for
plitude is dominated by low mass intermediate states. eachincomingphotoncouplingtopions,modifyingthe
WiththeQ2 0limitfixedintermsofβ ,weestimate subtraction function of Eq. (24) by multiplying it with
M
theQ2 depen→dencefrompionloopcontributionswhere Fπ(Q2)2. Obtaining Fπ from the fit of [22], we find
thetwo-pionstatehasascalarcouplingtothenucleon, ∆Esubt =3.8µeV.
asillustratedinFig.3. The inelastic contributions depend on F(1,2)(ν,Q2),
With standardFeynman rulesfor scalarQED and an and good data in the low-Q2 and resonance region is
effective g N¯φ†φN coupling for the lower vertex, one available from Jefferson Lab. Analytic representations
S
obtainsfromtheseterms of this data are given by Christy and Bosted [23], in a
Tµν = gS q2gµν qµqν F (Q2), (22) fitvalidfor0 < Q2 < 8GeV2 andW fromthresholdto
192π3m2 − loop 3.1GeV,whereWisthefinalhadronicmassforinelastic
π
(cid:16) (cid:17)
4
epscattering,W2 = M2+2Mν Q2. FromtheBosted- the individual contributions are larger than the change
−
Christy region, we obtain a 12.2 µeV contribution to inthetotal. Themainchangesoccurredbecausewefeel
∆Einel. WealsousethefitofC−apellaetal.[24],validfor useofalargermagneticpolarizabilityisjustifiedandbe-
data at low and intermediate Q2 above the resonance causeusingadispersivetreatmentthroughoutdoesnot
region, specifically 0 < Q2 < 5 GeV2 and W > 2.5 allowkeepingtheelasticnon-polecontributions.
GeV. This gives a 0.5 µeV contribution using [24] for CEC thanks the National Science Foundation for
W > 3.1 GeV in th−e allowed Q2 region. Contributions support under Grant PHY-0855618 and thanks the
fromhigherQ2arequitesmall(ontheorderof0.002µeV Helmholtz Gemeinschaft in Mainz and the Helsinki
fromQ >5GeV2andW >3.1GeV).Wethushave Institute for Physics for their hospitality. We thank
∆Einel = 12.7µeV. (28) VladimirPascalutsaforusefulcomments.
−
Refs.[12]and[21]quoted 13.9and 13.8µeVforthis
− −
contribution.
Theelasticcontributiondependsonthenucleonform
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