Two Photon Exchange Contributions to Elastic e + p --> e + p Process in a Nonlocal Field Formalism

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by  Pankaj Jain

Two Photon Exchange Contributions to Elastic ~e + p → e + p~ Process in a
Nonlocal Field Formalism
Pankaj Jain, Satish D. Joglekar and Subhadip Mitra
Department of Physics, IIT Kanpur, Kanpur - 208016,India
Abstract: We compute the two photon exchange contributions to elastic scattering of polarized elec-
tronsfromtargetprotons. Weuseanonlocalfieldtheoryformalism forthiscalculation. Theformalism
maintainsgaugeinvarianceandprovidesasystematicprocedureformakingthiscalculation. Theresults
depend on one unknown parameter¯b. We compute the two photon exchange correction to the ratio of
electric to magnetic form factors extracted using thepolarization transfer experiments. The correction
8 is found tobesmall if¯b∼1. Howeverfor larger valuesof¯b>3, thecorrection can bequitesignificant.
0
The correction to thepolarization transfer results goes in the right direction toexplain their difference
0
2 withtheratiomeasuredbyRosenbluthseparationmethod. Wefindthatthedifferencebetweenthetwo
experimental results can be explained for a wide range of values of the parameter ¯b. We also find that
n
a the corrections due to two photon exchange depend on the photon longitudinal polarization ε. Hence
J wepredictanεdependenceoftheform factorratioextractedusingthepolarization transfertechnique.
1 Finally we obtain a limit on ¯b by requiring that the non-linearity in ε dependence of the unpolarized
2 reduced cross section is within experimental errors.

]
h
p
- 1 Introduction given by:
p
e
[h The observed discrepancy [1, 2] in the proton electromag- M0 =−qe22u¯(k′)γµu(k,se)U¯(p′,sp′)ΓµU(p), (1)

1 netic form factors in the JLAB polarization transfer ex-
where
v periments [3, 4, 5, 6] andthe SLAC Rosenbluthseparation
iκ
6 experiments[7,8,9]hasbeenstudiedextensivelyinthelit- Γµ =F1(q)γµ+ p F2(q)σµαqα. (2)
2M
2 p
erature. The two photon exchange contributions are most
1
likely the source of the difference [10, 11, 12, 13, 14, 15]. Here e is the charge of proton, k, k′ are the momenta of
3
. In a recent paper we constructed a nonlocal Lagrangian the initial and final electron, p, p′ are the initial and final
1
0 to model the electromagnetic interaction of extended ob- proton momenta, Mp is the proton mass, κp =1.79 is pro-
jects such as the proton [16]. The model maintains gauge ton anomalous magnetic moment and q = p′−p = k−k′
8
0 invariance in the presence of a form factor at the electro- is the momentum transfer.
: magnetic vertex. We truncate the Lagrangian to include Using (1), one can calculate the tree level cross-section
v
i only operators with dimension five or less. The dimension forthescatteringoflongitudinallypolarizedelectronsfrom
X five operator is necessary if we include the contribution protons:
ar proportional to the Pauli form factor F2. This truncation dσ1γ = |M0|2Ee′2 , (3)
is reliable as long as the off-shellness of the proton propa- dΩ 64M2π2E2
e p e
gator is small compared to the hadronic momentum scale.
TheresultingLagrangiandependsonthetwoon-shellform where |M0|2 represents the amplitude squared, summed
factors, F1 and F2, and contains one unknown parameter over final electron spin and averaged over initial proton
whichwedenoteas¯b. Wefoundthatforsmallvaluesofthis spin and is given by:
parameter¯b∼1,the results ofthe SLAC Rosenbluthmea-
1
2 2
surement, after the two photon exchangecorrection,arein |M0| = |M0|
2
agreement with the JLAB polarization transfer results. In sX′e,sp
the present paper we compute the two photon exchange e4 h
corrections to the polarization transfer experiment. = 8q4 tr k/+me 1+ mγ5k/ γν k/′+me)γµ
Inpolarizationtransfer experiments, longitudinally po- h(cid:0) (cid:1)(cid:0) (cid:1) (cid:0) (cid:1)i
larized electrons are scattered from fixed target protons. ×tr /p+Mp γ0Γ†νγ0 /p′+Mp 1+γ5/sp′ Γµ .
The tree level amplitude for the process (See Fig. 1) is h(cid:0) (cid:1) (cid:0) (cid:1)(cid:0) (cid:1) i(4)
1
k′ p′ polarized recoiling proton. Let us define,
dσ1γ α2E′cos2 θe I1γ
0 = e 2 0 , (7)
q dΩe 8Ee3sin4 θ2e !1+τ
d(∆σ1γ) α2E′cos2 θe I1γP1γ
L = e 2 0 L , (8)
dΩe 8Ee3sin4 θ2e ! 1+τ
d(∆σ1γ) α2E′cos2 θe I1γP1γ
k p dΩeT = 8Ee3esin4 θ2e2 ! 10 +Tτ . (9)
Fthieguerlaest1i:c eTlehcetroonnepprhoototonnsecxatcthearningge.dHiaegrreamk,cko′ntrreifbeurttinogthtoe Using these we can rewrite dσ1γ/dΩe as
′
initialandfinalelectronmomentaandp,p totheinitialandfi-
nalprotonmomentarespectively. Thesymbolq=k−k′ =p′−p dσ1γ = dσ01γ +h d(∆σL1γ)(nˆp′ ·zˆ)
denotes themomentum exchanged. dΩe dΩe dΩe
d(∆σ1γh)
T
Here sµ is the spin four-vector of the final proton. In + dΩ (nˆp′ ·xˆ) . (10)
p′ e
general the spin four-vector s for a particle with mass m i
Hence, the ratio of the form factors is given by:
and momentum p = (E; p~) can be written in terms of the
ufrnaimtethorfeet-hveecptaorrtinˆc,les,pbeycifyingthespindirectionintherest R = GE =−PT1γ Ee+Ee′ tanθe
GM PL1γ 2Mp 2
sµ =(cid:18)nˆm·~p; nˆ+~pm(mnˆ·+p~E)(cid:19). (5) = −dd((∆∆σσLT11γγ))//ddΩΩeeEe2M+pEe′ tanθ2e. (11)
For the incident electron, energy E is much greater than
e
The contribution of the two photon exchange diagrams
its mass m , and we have approximated the spin four vec-
e
tor, sµ ≈hkµ/m , where h=nˆ ·kˆ. to the electron-proton elastic scattering cross section can
e e e be written as
Let the scattering plane be the X−Z plane where the
mi.eo.,mzˆen=tumpˆ′.ofTthheeYrecaoxiilsedispdreofitonnedp~a′sdyeˆfi=neskˆt×hekˆ′Z. aTxhisis, dσ2γ = 2Re(M∗0M2γ)Ee′2 +O(α4), (12)
dΩ 64M2π2E2
also defines the X axis as xˆ = yˆ×zˆ. With this choice of e p e
coordinate axes, the tree level cross section can be written
where M2γ is the total amplitude of the two photon ex-
as:
change diagrams. As in the case of tree level process we
can define d(∆σ2γ)/dΩ and d(∆σ2γ)/dΩ from the terms
ddσΩ1eγ = α82EEe3e′scions42θ2θe2e!(cid:18)1+1 τ(cid:19) irnesdpσe2cγti/vdeΩlye.tThhatLena,repreoportionaltToh(nˆp′e·zˆ)andh(nˆp′·xˆ)
× I01γ +hI01γPL1γ(nˆp′ ·zˆ)+hI01γPT1γ(nˆp′ ·xˆ) ,(6) dσ2γ dσ2γ d(∆σ2γ)
h i = 0 + N (nˆp′ ·yˆ)
dΩ dΩ dΩ
where e e e
d(∆σ2γ) d(∆σ2γ)
L T
I1γ = G2 + τG2 , +h dΩ (nˆp′ ·zˆ)+ dΩ (nˆp′ ·xˆ) . (13)
0 E ε M h e e i
I01γPL1γ = EeM+pEe′ τ(1+τ)G2Mtan2 θ2e, Hspeirneodfσt02hγe/dfiΩnealrperporteosnenatsndthde(∆teσrN2mγs)/idnΩdeepceonrdreesnptonodfsthtoe
I01γPT1γ = −2 τ(1p+τ)GMGEtanθ2e. theEnxoprmeraiml penoltaarlilzyattihoenpoofltahreizfiatniaolnpcrootmopno.nents, PL and
p PT are measured for different q2. From these the ratio of
Here τ = −q2/4M2, ε = 1/[1+2(1+τ)tan2(θ /2)] is the
p e the form factors is extracted by the following relation:
longitudinal polarization of the photon and θ is the elec-
e
turnopnoslacraitzteedricnrgosasn-sgelec.tioIn01γ. Ii0s1γpPrLo1γpo(rIt01iγoPnTa1lγ)toistphreotproeretiloenvaell RE =−PPTL Ee2M+pEe′ tanθ2e. (14)
to the term in the tree level cross-sectionthat corresponds
toscatteringalongitudinallypolarizedelectronfromanun- In the tree level approximation RE is identical to R. Let
polarized proton producing a longitudinally (transversely) GM be the experimentally measured value of GM. Then,
e
2
k′ p′ k′ p′ k′
p′
q−l
q−l q−l
k−l p+l k−l p′−l k−l
l
l l
p
k p k p k
(a) (b) (c)
Figure 2: The two-photon exchange diagrams contributing to theelastic electron proton scattering: (a) box diagram, (b) cross-
box diagram and (c) diagram proportional to¯b2. Anotherdiagram proportional to¯b2 is obtained byinterchanging l↔(q−l) in
(c)
with the knowledge of GM and RE one can define the fol- and (b) show the box and cross-box diagrams respectively
lowing quantities: and Fig. 2(c) shows a diagram proportional to ¯b2. The
e amplitudes for the box and cross-box diagrams are given
d(∆σL) = α2Ee′ cos2 θ2e I0PL, (15) by:
dΩe 8Ee3sin4 θ2e !1+τ
d(d∆ΩσeT) = α82EEe3e′scions42θ2θe2e!1I0+PTτ (16) iMB = e4Z (2dπ4)l4"u¯(k((′k)γ−µ(lk/)2−−/lm)γ2eνu+(kiξ,)sk)#
1
where
×
(l2−µ2+iξ)(q˜2−µ2+iξ)
E +E′ θ " #
I0PL = e e τ(1+τ)G2Mtan2 e, (17)
I0PT = −2Mpτ(1p+τ)G2MREetanθe. 2 (18) ×"U¯(p′,sp′)(F1(q˜)γµ+i2κMppF2(q˜)σµαq˜α)
2
From these one obtainps the ratioeR¯ corrected for the two- × (p+/pl+)2/l−+MM2p+iξ
photon exchange processes by the following relation: ( p )
R¯ = −"dd((∆∆σσTL))//ddΩΩee−−dd((∆∆σσLT22γγ))//ddΩΩee# ×(F1(l)γν +i2κMppF2(l)σνβlβ)U(p)#, (23)
×Ee2M+pEe′ tanθ2e (19) iMCB = e4Z (2dπ4l)4"u¯(k(′k)γ−µ(lk/)2−−/lm)γ2eνu+(kiξ,se)#
= RE11−−∆∆TL (20) × (l2−µ2+iξ)1(q˜2−µ2+iξ)
" #
where
κ
∆T = d(∆σT2γ)/dΩe, (21) ×"U¯(p′,sp′)(F1(l)γν +i2MppF2(l)σνβlβ)
d(∆σT)/dΩe
d(∆σ2γ)/dΩ /p+/q−/l +Mp
L e ×
∆L = . (22) (p+q˜)2−M2+iξ
d(∆σL)/dΩe ( p )
κ
× F1(q˜)γµ+i p F2(q˜)σµαq˜α U(p) .(24)
2 Calculations and Results ( 2Mp ) #
As explained in [16] there are three diagrams that con- Here q˜=q−l andξ is aninfinitesimal positive parameter.
tribute in the two-photon exchange processes. Figs. 2(a) The amplitude for the diagram proportional to ¯b2 can be
3
q2 = -2.47 GeV2 q2 = -2.97 GeV2
0.02 0.02
0.00 0.00
-0.02 -0.02
-0.04 -0.04
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
ε ε
(a) (b)
q2 = -3.50 GeV2 q2 = -3.97 GeV2
0.04 0.04
0.02 0.02
0.00 0.00
-0.02 -0.02
-0.04 -0.04
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
ε ε
(c) (d)
q2 = -4.75 GeV2 q2 = -5.54 GeV2
0.08 0.12
0.08
0.04
0.04
0.00
0.00
-0.04 -0.04
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
ε ε
(e) (f)
Figure 3: Contribution of box and cross-box diagrams to ∆L (unfilled circles) and ∆T (filled circles) for different q2 - (a)
q2 = −2.47GeV2, (b) q2 = −2.97GeV2, (c) q2 = −3.50GeV2, (d) q2 = −3.97GeV2, (e) q2 = −4.75GeV2, (f) q2 = −5.54GeV2.
∆T is defined in (21) and ∆L is defined in (22).
4
q2 = -2.47 GeV2 q2 = -2.97 GeV2
0.002 0.002
0.001
0.000
0.000
-0.002
-0.001
-0.002 -0.004
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
ε ε
(a) (b)
q2 = -3.50 GeV2 q2 = -3.97 GeV2
0.004 0.005
0.002
0.000
0.000
-0.005
-0.002
-0.010
-0.004
-0.006 -0.015
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
ε ε
(c) (d)
q2 = -4.75 GeV2 q2 = -5.54 GeV2
0.010 0.010
0.000
0.000
-0.010
-0.010 -0.020
-0.030
-0.020
-0.040
-0.030 -0.050
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
ε ε
(e) (f)
Figure 4: Contribution of the diagrams proportional to ¯b2 to ∆L (unfilled circles) and ∆T (filled circles) for different q2 - (a)
q2 = −2.47GeV2, (b) q2 = −2.97GeV2, (c) q2 = −3.50GeV2, (d) q2 = −3.97GeV2, (e) q2 = −4.75GeV2, (f) q2 = −5.54GeV2.
∆T is defined in (21) and ∆L is defined in (22). Here¯b=1.
5
written as: 1.2
e4¯b2 d4l u¯(k′)γµ(k/−/l)γνu(k,s )
iM¯ = e 1.0
b 8M2 (2π)4 (k−l)2−m2+iξ
p!Z " e #
1 M 0.8
× G
"(l2−µ2+iξ)(q˜2−µ2+iξ)# /E
G
iκ µp 0.6
× U¯(p′,sp′) p F2(q˜)σµαq˜α (/p+/l −Mp)
2M
" ( p
(cid:16) (cid:17) 0.4
iκ iκ
× p F2(l)σνβlβ + p F2(l)σνβlβ
2M 2M
p p
(cid:16) (cid:17) (cid:16) (cid:17) 0.2
×(/p+/q−/l −Mp) iκp F2(q˜)σµαq˜α U(p) . 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5
2Mp ) # Q2
(cid:16) (cid:17)
(25)
Figure 5: The ratio µpGE/GM from polarization transfer ex-
periments(filledcircles)andaftercorrectionforthetwo-photon
The total two photon exchange amplitude, exchangediagramswith¯b=0(unfilledsquares),¯b=2(unfilled
circles), ¯b = 3 (unfilled triangles), ¯b = 4 (unfilled diamonds),
M2γ =MB+MCB+M¯b. (26) ¯bco=rre5ct(eudnvfiallleudesphenavtaegboene)naonffdse¯bt=by6.∆91Q2(fi=lle0d.1trGiaenVg2lefso)r. cTlahre-
ity. The filled diamonds and the filled squares represent the
In the numerical calculation we have dropped m . The
e
RosenbluthextractionexperimentresultsobtainedatJLABand
model used for the form factors [17, 18] is the same as
SLACrespectively.
Model I described in [16]. The values of the parameters
are repeated here in Appendix A for convenience. The re- 1.2
sults of our calculation may show some dependence on the
precise form factor model. This appears to be the main
1.0
uncertainty in our calculation which we hope to explore in
a future publication.
0.8
Contributions from box and cross-box diagrams are M
G
computedat10differentvaluesofµ2(from0.005to0.0095) /E
for each value of q2 and ε. The cross-box diagram is well Gp 0.6
µ
defined but for the evaluation of the box diagram a small
imaginary term ξ is kept in the propagators. This makes 0.4
the integral in the infrared limit well defined in the case
me = 0. For each value of q2, ε and µ2 we have calcu- 0.2
lated the box diagram amplitude for 4 different values of
ξ (between 0.001 and 0.00175). The final µ2 dependent 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5
box diagram amplitudes are obtained by extrapolation to Q2
ξ = 0. The diagram proportional ¯b2 has no infrared (IR)
divergentterm. Sothecontributioncomingfromitiscom- Figure 6: Theratio µpGE/GM from SLACRosenbluthextrac-
puted keeping µ2 =0. t(iuonnpoelxapriezreimd)entwtso-(pfihlloetdonsqeuxacrheasn)gaenddiaagfrtaemr scowrriethcti¯bon=fo0r(uthne-
Following Mo and Tsai [19, 20, 21, 22] one can show filled squares), ¯b = 2 (unfilled circles), ¯b = 3 (unfilled trian-
thatinthelimitµ2 →0theleadingtermfromtheboxand gles), ¯b= 4 (unfilled diamonds), ¯b= 5 (unfilled pentagon) and
cross-box diagram can be expressed as: ¯b = 6.91 (filled triangles). The filled diamonds represent the
JLAB Rosenbluth extraction experiment results and the filled
M2IRγ = α[K(p′,k)−K(p,k)]M0, (27) circles represent the results of the polarization transfer experi-
π ments.
where
Hence,the leadingcontributionsto the crosssections com-
1 ing from the box and cross-box diagrams are given by:
dx
K(p ,p ) = (p .p )
i j i (jxpZi0+((1xp−i+x)p(1j)−2 x)pj)2 d(∆σL2γ,T) = 2α[K(p′,k)−K(p,k)]d(∆σL1γ,T). (28)
× ln .
µ2 dΩe (cid:12) π dΩe
(cid:20) (cid:21) (cid:12)IR
(cid:12)
(cid:12)
(cid:12)
6
q2 = -2.47 GeV2 q2 = -2.97 GeV2
0.720 0.640
0.635
0.716
0.630
0.712
0.625
0.708
0.620
0.704
0.615
0.700
0.610
0.696 0.605
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
ε ε
(a) (b)
q2 = -3.50 GeV2 q2 = -3.97 GeV2
0.605 0.515
0.595 0.505
0.585 0.495
0.575 0.485
0.565 0.475
0.555 0.465
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
ε ε
(c) (d)
q2 = -4.75 GeV2 q2 = -5.54 GeV2
0.430 0.320
0.420 0.310
0.410 0.300
0.400 0.290
0.390 0.280
0.380 0.270
0.370 0.260
0.360 0.250
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
ε ε
(e) (f)
Figure 7: Dependanceof theform factor ratio µpGE/GM on εfrom differentq2 - (a) q2=−2.47GeV2,(b)q2 =−2.97GeV2,(c)
q2 = −3.50GeV2, (d) q2 = −3.97GeV2, (e) q2 = −4.75GeV2, (f) q2 = −5.54GeV2. For each q2, the ratio, µGE/GM is plotted
for different¯b values -¯b=0 (unfilled squares),¯b=2 (unfilled circles),¯b=3 (unfilled triangles),¯b=4 (unfilled diamonds),¯b=5
(unfilled pentagon) and¯b=6.91 (filled triangles).
7
Let unpolarized electrons from protons, σR with the following
function:
d(∆σ2γ )
L,T ≡air +bir lnµ2. (29)
dΩe (cid:12) L,T L,T σR =P0+P1ε¯+P2ε¯2, (31)
(cid:12)IR
(cid:12)
To remove the IR p(cid:12)(cid:12)art from d(∆σL2γ,T)/dΩe we fit these where ε¯=ε− 21. As the result obtained by JLAB [23, 24]
with the following functions: contains much smaller error-bars than the SLAC data we
2 2
use JLAB data to obtain P2 for different Q =−q . Then
d(∆σ2γ ) we fit the contribution of the box and cross-box diagrams
dΩL,T =aL,T(µ2)+bL,T(µ2)lnµ2 (30) (σRBCB)andthecontributionproportionalto¯b2 (σR¯b)tothe
e
reduced cross-sectionwith similar functions, i.e.,
with
BCB BCB BCB BCB 2
σ = R +R ε¯+R ε¯ , (32)
R 0 1 2
2 0 1 2 4
aL,T(µ ) = aL,T+aL,Tµ +O[µ ] σ¯b = ¯b2(R¯b +R¯bε¯+R¯bε¯2). (33)
R 0 1 2
bL,T(µ2) = biLr,T+b1L,Tµ2+O[µ4].
To put the limit on ¯b we first obtain a limit on the
Hence, a0L,T give the IR removed d(∆σL2γ,T)/dΩe. nonlinearity parameter P2. The 2 sigma limit is obtained
Fig. 3showsthecontributionto∆L,T comingfrombox by finding the largest value of −P2 such that χ2 deviates
and cross-box diagrams for different q2. The contribution fromitsminimumvalueby4units. Weconsiderthelargest
from the diagram proportional to ¯b2 is shown in Fig. 4. value of −P2 since the two photon exchange contributions
Fig. 5showstheratioµpGE/GM frompolarizationtransfer are found to give a negative value of P2. The parameters
experiments(filledcircles)andaftercorrectionforthetwo- P0,P1 arealsoallowedtovarywhiledeterminingthemax-
photon exchange diagrams with ¯b = 0 (unfilled squares), imum value allowed for −P2. The best fit value of P2 and
¯b = 2 (unfilled circles), ¯b = 3 (unfilled triangles), ¯b = 4 the 1 and 2 sigma limits are given in Table 1. The best
(unfilleddiamonds),¯b=5(unfilledpentagon)and¯b=6.91 fit values of RBCB and R¯b are given in Table 2. Assuming
2 2
(filled triangles). The highest value of ¯b used is obtained that the dominant source of nonlinear behaviour are the
from the limit on nonlinearity of JLAB Rosenbluth data box, cross-box and ¯b diagrams, we obtain the limit on ¯b2
(see Section3). Inthe range3.5≥Q2 ≥2.5GeV2, the po- by the following relation,
larization transfer experiment used the kinematic regime
with ε lying roughly between 0.7 and 0.8 [5]. For simplic- ¯b2 = P2limit−RB2CB (34)
ity, here we assume a fixed value ε = 0.75. Fig. 6 shows max R¯b
the effectof¯b onthe unpolarizedSLAC data. The method 2
usedtoobtainthesecorrectionsisdescribedinSection6of
whereP2limitistheoneortwosigmalimitontheparameter
[16]. P2. FromTable1we seethatthe moststringentlimiton¯b
The form factor ratio GE/GM depends only on Q2. isobtainedforQ2 =2.64GeV2 andis givenby: |¯b|.6.91.
However the ratio RE, defined in (14), may show a de- The nonlinearity of the reduced cross section σR has also
pendence on ε due to higher order contributions. We pre-
been computed in [25] using the GPD formalism.
dict this dependence by assuming that the electron-proton
elastic scattering cross section can be well approximated
by including only one and two photon exchange diagrams.
The uncorrectedformfactor ratioRE canbe calculatedby Q2 P2 P2 P2 ¯b2
using (20), where the corrected ratio R¯ is independent of in best 1 sigma 2 sigma 2 sigma
GeV2 fit limit limit limit
ε. Fig. 7 shows the predicted dependance of the uncor-
rected form factor ratio µpRE on ε for different ¯b and q2.
Here again we have assumed that the kinematic regime of
the polarizationtransferexperimentcorrespondsto afixed 2.64 2.02 −3.82 −9.64 47.74
ε=0.75. Thismaybeusedinfuturetofixthevalueofthe
parameter¯b. 3.20 1.10 −3.70 -8.52 70.61
4.10 2.68 −2.92 -8.48 152.93
¯
3 Limit on b
As described in [16]¯b is a free parameter. But the fact Table 1: The best fit and the one and two sigma limit on the
that σR¯b varies nonlinearly with ε allows us to put a limit phaavraembeeteenrsPca2l(eddebfiyne1d04in. T(3h1e))cfoorrredsipffoenrednintgQt2w.oAslilgvmaaluleismoitfPon2
on¯b. We fit the reduced cross-section for the scattering of theparameter¯b2 is also given.
8
Q2 in GeV2 RBCB R¯b a A′ B′ m Γ′
2 2 a a a a
2.64 −4.04×10−4 −11.73×10−6 1 −2.882564 −3.177877 0.8084 0.2226
+i1.944314 +i2.123389
3.20 −3.69×10−4 −6.84×10−6 2 2.882564 3.177877 0.9116 0.1974
−i1.944314 −i2.123389
4.10 −1.95×10−4 −4.27×10−6 3 −1.064011 −0.608148 1.274 0.5712
−i3.216318 −i5.685885
4 1.064011 0.608148 1.326 0.5488
Table 2: Values of the fitting parameters RB2CB (defined in +i3.216318 +i5.685885
(32)) and R¯b2 (definedin (33)) for different Q2. 5 0 3.211388 1.96 1.02
+i0.693412
6 0 −3.211388 2.04 0.98
4 Conclusion −i0.693412
Wehavecomputedthetwophotonexchangecorrections Table 3: Masses, widths and parameter values for GM/µp and
′ ′
to the proton electromagnetic form factor ratio µpGE/GM GE fits. A s and B s are defined in (A.1) and (A.2).
using a gauge invariant nonlocal Lagrangian. The La-
grangian is truncated to dimension five operators and de-
pends ononeunknownparameter¯b. Thehigherdimension References
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10