Light Sea Fermions in Electron-Proton and Muon-Proton Interactions

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by  U. D. Jentschura

Light Sea Fermions in Electron–Proton and Muon–Proton Interactions
U. D. Jentschura
Department of Physics, Missouri University of Science and Technology, Rolla, Missouri, MO65409-0640, USA and
MTA–DE Particle Physics Research Group, P.O.Box 51, H–4001 Debrecen, Hungary
The proton radius conundrum [R. Pohl et al., Nature 466, 213 (2010) and A. Antognini et al.
Science 339, 417 (2013)] highlights the need to revisit any conceivable sources of electron-muon
nonuniversalityinlepton-protoninteractionswithintheStandardModel.Superficially,anumberof
perturbative processes could appear to lead to such a nonunversality. One of these is a coupling of
4 the scattered electron into an electronic as opposed to a muonic vacuum polarization loop in the
1 photonexchange of twovalence quarks,which is presentonly for electron projectiles as opposed to
0 muon projectiles. However, we can show that this effect actually is part of the radiative correction
2 to the proton’s polarizability contribution to the Lamb shift, equivalent to a radiative correction
todouble scattering. Weconclude that any conceivable genuinenonuniversality must be connected
n
withanonperturbativefeatureoftheproton’sstructure,e.g.,withthepossiblepresenceoflightsea
a
J fermions as constituent components of the proton. If we assume an average of roughly 0.7×10−7
light sea positrons per valencequark,then we can show that virtualelectron-positron annihiliation
6
processes lead to an extra term in the electron-proton versus muon-proton interaction, which has
1
theright sign and magnitude to explain theproton radius discrepancy.

]
h PACSnumbers:31.30.js,36.10.-k,12.20.Ds,31.15.-p
p
-
m
I. INTRODUCTION charge of the up as compared to the down quarks.
o Still, the investigations [1–4] as well as the proton ra-
t
a Theelectromagneticaspectsoftheprotonandneutron diusconundrum[5,6]highlighttheneedforacloserlook
s. structure are somewhat elusive. It is well known that at the internal structure of the proton if one is inter-
c the mass difference of proton and neutron is responsi- ested in its own “internal” electromagnetic interactions,
si ble for the stability of the universe (the hydrogen atom aswellastheinteractionsoftheprotonwiththe“outside
y would otherwise be unstable against beta decay into an world”. If the interaction of the bound or scattered lep-
h electron-positron pair, a neutrino and a neutron). There tonwiththeprotonisnonuniversal,thenitisconceivable
p havebeenattemptstoexplainthemassdifferenceonthe that the proton radius depends on the projectile parti-
[
basis of the electromagnetic interaction among the con- cle.However,onecanshowthatanumberofperturbative

2 stituent quarks [1–4]. A priori, one would think that the higher-order effects which could appear to lead to such
v electrostatic interaction among the constituent quarks a nonuniversality of electron-proton versus muon-proton
6 leads to an inversion of the mass hierarchy of proton interactionsare in fact absorbedinto correctionterms of
6
versusneutron.Namely,the Coulombinteractionamong known physical origin.
6
3 valence quarks actually lowersthe energy of the neutron Letusconsiderelectromagneticinteractionsamongthe
. as compared to the proton, as a naive counting argu- constituentparticlesoftheproton,forexample,ahigher-
1
ment shows. A hadron with valence quarks uud has in- order effect generated by a coupling of the scattered
0
terquarkelectromagneticinteractionsproportionaltothe projectile (electron or muon) into a vacuum-polarization
4
1 fractionalchargenumbers, 2× −1 +2× −1 +2×2 =0 loop which in turn is inserted into a photon exchanged
3 3 3 3 3 3
: whereasfortheneutronwithv(cid:0)alenc(cid:1)equar(cid:0)ksu(cid:1)dd,wehave betweentwovalencequarks.We hereshowthat,because
v
2 × −1 + 2 × −1 + −1 × −1 =−1. The latter Feynman propagators take care of all possible time or-
i 3 3 3 3 3 3 3
X expr(cid:0)essio(cid:1)n, being(cid:0)neg(cid:1)ativ(cid:0)e, w(cid:1)ould(cid:0)sug(cid:1)gest that the neu- derings of virtual particle creation and annihilation pro-
r tronislighterthantheprotonifthemassdifferencewere cesses, this effect actually constitutes a radiative correc-
a
of electromagnetic origin and due to Coulomb exchange. tion to double scattering,and is absorbed,in Lamb shift
However, the radiative correction is not constrained calculations,intothe radiativecorrectionto the proton’s
to have any particular sign, and warrants further in- polarizability contribution to the Lamb shift. A quanti-
vestigation especially because the electromagnetic wave tative parametric estimate for the order-of-magnitudeof
functions of the valence quarks bound in an MIT bag the effect is provided.
model [1] have a rather peculiar structure [4] and might Thesecondprocessismorespeculativeandconjectures
give rise to significant radiative effects. The conclusion the presence of light sea fermions as a nonperturbative
reached in Refs. [1–4] is that the electromagnetic self- physicalpropertyofthehadron,anadmixturetothegen-
energy of the quarks remains positive for all masses con- uine particle content of the proton. We here show that
sidered. Thus, the quantum electrodynamic (QED) ra- theconceivablepresenceofthesefermionswouldgiverise
diativeenergyshiftcannotexplainthemassdifferenceof to a Dirac-δ potential, in view of a virtual annihilation
proton and neutron, where a negative self-energy would channel, with the right sign to explain the muonic hy-
otherwise be required in view of the larger fractional drogenpuzzle [5, 6]. These two mechanismaredescribed
2
e−,µ− e−,µ− e−,µ− e−,µ−
h h
u u
u u u u
e−
u u u u
d d d d d d
u u
(a) (b)
(a)
FIG.1:Diagram(a)isthestandardscatteringprocessinvolv-
ing an incoming electron or muon, without radiative correc-
e− e−
tions. The hadronic vacuum-polarization contribution from
diagram (b)is taken intoaccount consistently in Lamb shift
calculations and subtracted from scattering data as a radia-
tive correction. The photon is emitted “collectively” by the u u
quarksinsidetheproton.Theprotonasacompountparticle
d d
is encircled bydotted lines.
u u
(b)
in the following two Secs. II and III, respectively. Units
with ~ = c = ǫ are used throughout this paper unless
0
stated otherwise.
e− e−
II. LEPTON–PROTON SCATTERING AND
INTERNAL STRUCTURE OF THE PROTON
u u
u u
Letusfirstrecalltherelevantconventions.Theleading-
orderprocessforthe scatteringofleptons(e.g.,electrons d d
or muons) off of a proton is depicted in Fig. 1(a). A vir-
tualphotonisemittedcollectivelybytheproton,describ- (c)
ingtheelectromagneticinteractionofleptonandhadron.
Anyinsertionsofvirtualparticlesintotheexchangepho-
ton are absorbed into the F and F form factors (or
1 2 FIG. 2: (Color online.) Diagram (a) describes the coupling
SachsGE andGM formfactors)ofthe proton,while the of an incoming electron into the vacuum-polarization loop
protonradiusisdefinedastheslopeoftheSachsGE form of an electromagnetic interquark interaction inside the pro-
factor,withallthoseterms[andradiativecorrections,see ton. Feynman propagators describe all possible time order-
Fig.1(b)]subtractedfromG .Thesewouldotherwisebe ings of the particle creation and annihilation processes and
E
ascribedtoapointprotonwiththe propertiesofastruc- diagram (b)thusdescribesthesameeffect as(a). Thegluon
tureless spin-1/2 Dirac particle. interaction in diagram (b) is representative. The inelastic
contribution to diagram (b), with an “excited” quark in the
Let us briefly reviewthe status (see alsoRef. [7]). The
virtualstate,isidentifiedindiagram(c)asaradiativecorrec-
proton interactionvertex is changed, in view of the non-
tion to the proton’s polarizability contribution to the Lamb
trivial form factors, as
shift.
iσµνq
γµ →γµF (q2)+ ν F (q2), (1)
1 2
2m
p
where F and F are the Dirac and Pauliform factors of
1 2 Onecanonicallyseparatesthe SachsG formfactorinto
E
the proton, respectively. The electric and magnetic GE a QED contribution GQED(q2), which captures all as-
and G Sachs form factors are defined in terms of the E
M pects of the point-particle QED nature of the proton,
F1 and F2 as follows, and a part G (q2) which is due to the proton’s internal
E
structure [7],
q2
G (q2)=F (q2)+ F (q2), (2a)
E 1 4(m c)2 2
p
GM(q2)=F1(q2)+F2(q2), F2(0)=κp. (2b) G (q2)=G (q2)+GQED(q2). (3)
E E E
3
The definition of the proton charge radius then reads as low-energy scattering processes and contributions to the
Lambshift,theproton’sformfactorcanbeapproximated
hr2i =6~2 ∂GE(q2) , (4) very well using a dipole fit [see, e.g., the discussion sur-
p ∂q2 (cid:12) rounding Eq. (74) of Ref. [13]].
(cid:12)q2=0
(cid:12) Let us consider the diagram in Fig. 2(a), which could
(cid:12)
i.e. it measures the internal structure of the proton, af- superficially be assumed to induce a nonuniversality of
ter all QED contributions (“radiative corrections”) have the electron-proton versus muon-proton interaction, on
beensubtracted(andthatincludestheinfrareddivergent the level of higher-order corrections. Namely, the cou-
slope of the QED one-loop contribution to the F1 form plingoftheprojectileelectronintotheelectronicvacuum-
factor). By definition, the subtraction of the QED con- polarizationloop of an electromagnetic interquark inter-
tribution GQED(q2) also includes all virtual loop inser- action is available only for an incoming electron (as op-
E
tionsintothe exchangephotonlinethatwouldotherwise posed to an incoming muon). The Feynman propagators
affect the proton-lepton interaction for a point proton. for the fermions and the leptons in Fig. 2(a) contain all
Oneofthese isthe hadronicvacuum-polarizationloopin possible time orderings, including scattering “backward
Fig. 1(b). in time” which leads to the vacuum-polarization loop.
The proton radius is measured in the low-energy re- The diagram in Fig. 2(b) thus describes the same phys-
gion, where one can use a dipole fit to G (q2) to good ical process as Fig. 2(a). Furthermore, it is necessary to
E
approximation.Let us briefly recallthe fundamental dif- rememberwhatthe“scatteringoffofadefinite quarkin-
ferencesoflow-energyelasticscattering,whichmainlyde- side the proton” [see Fig. 2(a)] physically means in the
terminestheproton’ssize,andhigh-energydeepinelastic characteristic momentum range of an electron or muon
scattering (DIS), which is relevantfor momentum trans- bound to the proton. It implies that the proton’s inter-
fers q2 ≫ m2. For an incoming lepton four-momentum nal state changes in between the two interactions of the
p
ℓ1 and an incoming proton momentum p (outgoing lep- virtual electron with the virtual photons emitted by the
tonmomentumℓ2 andexchangephotonfour-momentum proton [see Fig. 2(c)]. Thus, the process in Fig. 2(a) can
q), the Bjorken scaling law [8] is as follows. After the finally be identified as a radiative correction to proton’s
subtraction of radiative correction, one writes the deep polarizabilitycontributiontotheLambshift,asdepicted
inelastic cross section as in Fig. 2(c).
The contribution of double-scattering processes is
G2(Q2,ν)+τG2 (Q2,ν)
σ =σ E M canonicallysubtractedintheanalysisofscatteringexper-
DIS 0
(cid:20) 1+τ
iments. In the context of bound states, the leading con-
θ tribution from two-photon exchange (without radiative
+2τG2 (Q2,ν) tan2
M (cid:18)2(cid:19)(cid:21) corrections) gives rise to the so-called third Zemach mo-
ment term which is proportional to a convoluted charge
θ
≡σ0 W2(Q2,ν)+2W1(Q2,ν) tan2 , (5) distribution of the proton [7, 14, 15]. The elastic cor-
(cid:20) (cid:18)2(cid:19)(cid:21)
rection to the third Zemach moment due to the proton
where σ is the Mott scattering cross section, ν = structure can be taken into account by inserting proton
0
q · p/m is the energy loss of the lepton, Q2 ≡ −q2, form factors into the two-photonexchange forwardscat-
p
τ =Q2(4m2)−1, and θ is the scattering angle of the lep- tering amplitude [Eqs. (70)–(75) of Ref. [13]], and the
p
ton, i.e., the angle subtended by the spatial components inelastic correction to the third Zemach moment (due
ofℓ andℓ .The formfactorsG (Q2,ν) andG (Q2,ν) to an excited state of the proton in between the photon
1 2 E M
describe inelastic scattering (with energy loss), and the exchanges, also known as the proton polarizability cor-
elastic counterparts are recovered in the limit ν → 0. rection) is numerically too small to explain the proton
Bjorken [8] observed that, if the scattering in the high- radius puzzle [13, 16–19].
energy region were to come from point-like constituents Finally, let us provide a parametric estimate for the
inside the proton, then the structure functions W and contribution of the diagram in Fig. 2(c), based on
1
W should be consistent with scattering from asymptot- the analogy with the two-Coulomb-vertex correction to
2
ically free constituents (“partons” or “quarks”), the self-energy, as given by the calculation reported in
Ref.[20].Theinducedeffectivepotentialforthe diagram
lim νW (Q2,ν)=m F (x), (6a) in Fig. 2(c), by scaling arguments, can be estimated to
2 p 2
Q2→∞ be proportional to
Q2/νconst.
Q2li→m∞ W1(Q2,ν)=F1(x), x≡ 2mQ2pν , (6b) Hvp ∝ [αQED(mm2e2effff/m2e)]3 δ3(r), (7)
Q2/νconst.
Here, α is the running QED coupling which is ap-
QED
where the F andF arenow structurefunctions instead proximatelyequalto1/137.036atzeromomentumtrans-
1 2
of form factors; their argument is the Bjorken x vari- fer, δ3(r) is the three-dimensional Dirac-δ function, and
able. The Bjorken scaling was confirmed by the famous m isaneffectivemassormomentumscaleenteringthe
eff
SLAC–MITexperiments[9–12].However,indealingwith loop in Fig. 2(c). The latter can be estimated as follows.
4
Let λ ∼ r be a characteristic de Broglie wavelength of
p e− e−
the quarks inside the nucleus. Then, the associated mo-
mentum scale is p ∼ h/rp where h is Planck’s unit of e+ e+
action and the corresponding energy scale is obtained as
E ∼ pc ∼ 1.32m , which in turn is commensurate with
p u u
the excitation energy of the proton into its first reso-
nance,the ∆ resonanceat 1232MeV. It is easy to check, d d
based on the running of the QED coupling, that the ef- u u
fective coupling at the scale of the proton’s momentum d
differs from the value of αQED at zero momentum trans- d
fer by less than 5%. The leading finite-size Hamiltonian
is given as follows,
2πα
Hfs = 3mQ2ED m2e rp2 δ3(r). (8) FIG.3:(Coloronline.)TypicalFeynmandiagramillustrating
e (cid:2) (cid:10) (cid:11)(cid:3) thevirtualannihilationofaboundelectronwitha“lightsea
lepton” (positron) inside the proton. The up (u) and down
The ratio is given as
(d)quarks,whichcarrynon-integerchargenumbers,interact
electromagnetically. The dashed lines mark theformation of
hH i m2 1
R∝ vp ∼α2 e ∼2.2×10−6, (9) the asymptotic state of the proton in the distant past or
hHfsi QEDm2eff m2e rp2 future, with its valence and sea quark contents, and with
(cid:10) (cid:11) a light sea lepton that annihilates with the bound electron.
wherewetakeintoaccountthatm2 r2 ∼(1/386)2,and The given Feynman diagram is not included if the proton is
e p
m /m ∼m /m ∼5.4×10−4.Th(cid:10)era(cid:11)tioRistoosmall treated perturbatively as a spin-1/2 particle with charge e.
e eff e p An exemplary quantum chromodynamic (QCD) interaction
tomakeasignificantcontributiontoasolutionofthepro-
via a blue-antigreen gluon also is indicated in thefigure.
tonradiuspuzzle.Itisinterestingtonotethatthesimple-
minded parametric estimate described above, with one
radiativefactorα fromtheself-energyloopexcluded,
QED
givesthe rightorder-of-magnitudeforthe leadingproton excluded by any known experiments. In fact, a signifi-
polarizability contribution evaluated in Ref. [15]. cant photon content of the proton is well confirmed in
the so-called Deep Inelastic Compton Scattering (DICS,
see Refs. [21–26] and Fig. 3).
III. LIGHT SEA FERMIONS If the proton contains these electron-positron pairs,
which are not accounted for in any perturbative higher-
order QED term, then the interaction between the pro-
Let us consider the possible presence of light sea
ton and electron is given by both photon exchange and
fermionsasnonperturbativecontributionstotheproton’s
annihilationdiagrams.Innaturalunits,thephotonanni-
structure, inspired by a possible importance of virtual
hilation diagram in the case of positronium leads to the
electron-positron pairs in the lepton nonuniversality in
effective interaction [27]
interactions with protons. We consider a thought exper-
iqmuaenrkt:sIifnwsideeswthitechperdotooffn,ththeeelreecsturlotwinegak“pinrotetroanc”tiownosulodf δH = πα2mQE2D (3+~σ+·~σ−) δ3(r). (10)
e
of course be neutral but otherwise rather comparable
in its mass and in its nuclear properties to a real pro- ThisHamiltoniangivesanonvanishinginteractionofthe
tonwithsomenonperturbativequantumchromodynamic bound electron and the light sea positron if their spins
(QCD) “wave function.” Now, if we include back the adduptoone.Assumingthattheelectron-positronpairs
electroweak interactions of quarks, virtual photons and within the proton are not polarized, we can replace ~σ+·
electron-positron pairs would backreact on the previous ~σ− →0 after averaging over the polarizations of the sea
“wave function” leading to a reshaping, and the actual leptons.For atomic(electronic)hydrogen,the additional
“proton wave function” which now additionally contains interaction of the electron with the proton due to the
photons and the electron-positron pairs. Due to highly annihilation channel therefore is of the form
nonlinearnonperturbativenatureofQCD,thisreshaping 3πα
can be much larger than the electromagnetic perturba- Hann =ǫp 2mQ2ED δ3(r), (11)
tion itself, and therefore there is a room for the conceiv- e
able presence of electron-positron pairs inside the pro- where ǫ measures the amountof electron-positronpairs
p
ton,whichcannotbeaccountedforbyperturbativeQED withinthe proton.Formuonic hydrogen,the effect is ex-
considerationsalone (see Fig. 3). This density (probabil- pected to vanish because the dominant contribution to
ity) of electron-positron pairs, because of the inherently the sea leptons comes from the lightest leptons, namely,
nonperturbativenature ofQCD,is difficult if notimpos- electron-positronpairsandthustheannihilationchannel
sible to quantitatively estimate, but its presence is not is not available. By comparison, the finite nuclear size
5
effect is givenby Eq.(8). For anS state, the ratioof the estimate would entail the observation that the carbon
corresponding energy shifts is nucleusisthreetimesbiggerthantheproton.So,naively
and classically, three nucleons fit into the diameter of
hHanni 9 ǫp ! 0.882−0.842 the 12C nucleus. If every one of these has its effecetive
= = =0.089. (12)
hHfsi 4 m2e rp2 0.882 diameter reduced by 5%, then the overall radius is re-
(cid:10) (cid:11) ducedbyonly1.7%.Whilethe tablesofRef.[38]suggest
The equality marked with the exclamation mark has to agreement to (slightly) better than 1% for independent
holdifwearetoexplainthediscrepancyoftheelectronic experimental determinations of the 12C charge radius, it
andmuonichydrogenvaluesofthe protonchargeradius, isnoteworthythatthisagreementhasbeenachievedonly
whichareroughly0.88fmand0.84fm,respectively[5,6, after earlier discrepancies had been resolved.
16, 28]. The parameter ǫp thus can be as low as Veryinterestingly,apossibleelectron-muonnonuniver-
salityhas beenseenin ascatteringexperiment[40]some
ǫ =2.1×10−7, (13)
p fourty years ago, comparing the scattering of electrons
versusmuonsoff ofprotons,andwascautiously ascribed
and still explain the effect the different proton radii ob-
by the authors of Ref. [40] to an incorrect normaliza-
tainedfromelectronicandmuonichydrogen.Pervalence
quark, one thus has a fraction of ǫ /3 = 0.7×10−7 sea tion of the scattering data. The observed 8% difference
p in the cross sections observed in Ref. [40] translates into
fermion pairs. The interaction due to the annihilation
a 4% difference in the proton radius, with the same sign
channel has the right sign: it enhances the nuclear size
and magnitude as that observed in muonic spectroscopy
effect for electronic as opposed to muonic hydrogen and
experiments [5, 6]. If one ignores the possiblility of an
thus makes the proton appear larger for electronic sys-
incorrectnormalizationof the data in Ref. [40], then the
tems.
proton,“seen” with the “eyes” of a muon, appears to be
4%÷5% smaller as compared to its “appearance” when
“seen” through the “eyes” of an electron [5, 6, 40]. The
IV. CONCLUSIONS
experiment [40] urgently needs to be confronted with an
independent investigation.
Let us include some historical remarks. In the 1970s,
AccordingtoRefs.[17,41]andothertheoreticalworks
transition frequencies in muonic transitions were found
which came to the same conclusion, it is hard to imag-
to be in disagreement with theory [29]. After a sign er-
ine any perturbative process (direct exchange of a vir-
ror in the calculation of the two-loop vacuum polariza-
tual “subversive” particle, or insertion of a “subversive”
tion correction [30] was eliminated [31, 32] and a stan-
particle into the exchange photon line) which could ex-
dard γ-ray spectrometer used in the experiments was
plainthemuonichydrogendiscrepancywithoutseriously
recalibrated [33], other experiments later found agree-
questioningthevalidityofothermeasurementsandcorre-
ment of theory and experiment in muonic systems (e.g.,
spondingtheory,suchasthemuongfactormeasurement.
Refs. [34, 35]). Nuclear radii of some carbon, nitro-
Furthermore, any other perturbative insertions into the
gen and oxygen isotopes [36] were determined by ana-
photon-protonvertex,conceivablyinvolvinginternalcon-
lyzing muonic transitions, and the resulting radii were
stituents of the proton, are absorbed in the definition of
found to be in agreement with electron scattering radii
theprotonradiusandthuscouldnotexplainthediscrep-
to better than 5%. Later, the radius of 12C was up-
ancy(seeSec.II).Withoutquestioningthevalidityofthe
dated in Refs. [37, 38], finally “converging” to a value
Maxwell equations or quantum electrodynamics (QED),
of r = 2.478(9)fm, in good agreement with the value
C andwithoutintroducinganyadditionalvirtualparticles,
from muonic x-ray studies. Muonic atom and ion spec-
itisperhapspermissibletospeculatethatanonperturba-
troscopy is meanwhile regarded an established tool for
tivemechanismsuchastheoneproposedinSec.IIImight
the determination of nuclear radii [39].
beafeasiblecandidateinthecaseoffurtherexperimental
However, the light sea fermions discussed in Sec. III
confirmations of the proton radius discrepancy [5, 6, 40]
are distributed only inside the nucleons as opposed to
betweenelectronicasopposedtomuonicboundsystems.
the entire nucleus which is held together by meson ex-
change, because the local electromagnetic (EM) field is
strongest inside the protons and neutrons. The size of
theprotoncouldbedeterminedbythelightseafermions, Noteadded.Theexperimentalobservationofslightly
amongotherthings,butthe sizeofacomposedlargenu- smaller cross sections in muon-proton versus electron-
cleusisdeterminedbythenuclearmeson-mediatedforce. proton scattering has been made indepdendently in
Expressed differently, the muonic hydrogen experiments Refs. [40, 42]. One may consult the first row of graphs
probe one and only one nucleon, whereas other exper- in Fig. 15 of Ref. [42] (which pertain to elastic as op-
iments involving, say, a muon bound to a 12C nucleus, posed to inelastic cross sections), the general remarks
probethechargeradiusofanensembleofnucleons,which made in Sec. VII of Ref. [42], and the discussion sur-
is mainly determined by the arrangement of the nucle- rounding Eq. (48) of Ref. [42], which is consistent with
ons inside the 12C nucleus. Thus, the effect proposed in an 8% difference in the electron versus muon cross sec-
Sec. III shouldbecome smaller for largernuclei. A rough tions.
6
Acknowledgments authorwishes toacknowledgesupportfromthe National
Science Fundation (grant PHY–1068547) and from the
The author acknowledges insightful discussions with National Institute of Standards and Technology (preci-
Professor K. Pachucki. Helpful conversations with Pro- sion measurement grant).
fessor K. Meisser are also gratefully acknowledged. The
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