Constraints on the virtual Compton scattering on the nucleon in a new dispersive formalism

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by  Irinel Caprini

Constraints on the virtual Compton scattering on the nucleon in a new
dispersive formalism
Irinel Caprini1
1Horia Hulubei National Institute for Physics and Nuclear Engineering,
POB MG-6, 077125 Bucharest-Magurele, Romania
The dispersive representation of the virtual Compton forward scattering amplitude has been
recently reexamined in connection with the evaluation of the Cottingham formula for the proton-
neutron electromagnetic mass difference and the proton radius puzzle. The most difficult part of
the analysis is related to one of the invariant amplitudes, denoted as T1(ν,Q2), which requires
a subtraction in the standard dispersion relation with respect to the energy ν at fixed photon
momentumsquaredq2 =−Q2. Weproposeanalternativedispersiveframework, whichimplements
analyticity and unitarity by combining the Cauchy integral relation at low and moderate energies
with the modulus representation of the amplitude at high energies. Using techniques of functional
6
analysis, we derive a necessary and sufficient condition for the consistency with analyticity of the
01 subtractionfunction S1(Q2)=T1(0,Q2),thecrosssections measuredat lowandmoderateenergies
and the Regge model assumed to be valid at high energies. From this condition we obtain model-
2
independentconstraintsonthesubtractionfunction,confrontingthemwiththeavailableinformation

r on nucleon magnetic polarizabilities and results reported recently in the literature. The formalism
p
can beused also for testing theexistence of a fixed pole at J =0 in theangular momentum plane,
A
butmore accurate data are necessary for a definiteanswer.

4
1
I. INTRODUCTION Choosing the subtraction point at ν =0 and denoting

]
h S (Q2) T (0,Q2), (2)
1 1
The virtual Compton forward scattering on the nu- ≡
p
- cleon is of interest for the calculation of the proton- the dispersion relation satisfied by the amplitude T at
1
p
neutron electromagnetic mass difference by the Cotting- fixed Q2 is written in the form [4]
e
hamformula[1,2]. Anearlydispersiveevaluationofthis
h
[ formula, performed in [3], was updated recently in [4] T (ν,Q2)=S (Q2)+2ν2 ∞ dν′ V1(ν′,Q2) , (3)
2 and the problem was examined also in other recent pa- 1 1 Z0 ν′ ν′2−ν2−iǫ
pers[5]-[10]. Inaddition,thevirtualComptonscattering
v taking into account the fact that the amplitude is an
is of interest for the proton radius puzzle (for a review
7 even function of ν at fixed Q2. The dispersive integral
see [11]). It has been considered also by severalauthors,
8
can be evaluated using the structure function V (ν,Q2)
7 in particular in the context of Finite Energy Sum Rules 1
expressed in terms of the electron-proton cross sections
2 (FESR), in connection with the topic of a fixed J = 0
0 pole in the amplitude [12–16]. measuredatlowandmoderateenergiesandparametrized
. by Regge model at high energies.
1 The nucleonmatrix element of the time-orderedprod-
The problem considered in this work is the determi-
0 uct, pTjµ(x)jν(y)p , where jµ is the electromagnetic
6 currehnt|,isdescribed|initermsoftwoinvariantamplitudes nation of the subtraction function S1(Q2). We note
1 T (ν,Q2), i = 1,2, free of kinematical singularities and that at Q2 = 0 the subtraction function is expressed in
: i terms of the nucleon polarizabilities. The view adopted
v zeros, which depend on ν = p q/m and Q2 = q2,
i where p and q denote the nucleon· and photon mom−enta in Refs. [5]-[10] is that the subtraction function S1(Q2)
X for nonzero values of Q2 cannot be calculated in terms
andmthenucleonmass. ThefunctionsT (ν,Q2)atfixed
ar Qan2d≥ν0 are rνea,lwahnearlyetνic inistthheeνlopwlaensteucunititaarloitnygtνhr≥esνh0- opfutp.hyAsciccaolrdoibnsgelyrv,asbpleecsifiacndmordepelrsesfeonrtsS1a(nQ2a)rbhiatrvaerybeienn-
≤ − 0 0 adopted in these references in order to evaluate the dis-
old. Following [4], we denote the structure functions as
persionrelation(3). Combinedwiththepoorexperimen-
V (ν,Q2):
i tal knowledge of the magnetic polarizability of the neu-
tron[17],thearbitraryQ2 dependenceofthesubtraction
ImT (ν+iǫ,Q2)=πV (ν,Q2), ν ν , Q2 0. (1)
i i 0 functionrepresentsthemainsourceofuncertaintyinthe
≥ ≥
estimate of the proton-neutron mass difference obtained
Due to the contributions arising from Regge exchanges in these references.
which exhibit a growth at infinity, V (ν,Q2) να with Of course, the subtraction function in a dispersion re-
1
α > 0, the standard dispersion relation for T∼(ν,Q2) as lation is arbitrary if the available information refers ex-
1
a function of ν at fixed Q2 requires a subtraction, while clusivelyto the discontinuityofthe amplitude acrossthe
for T (ν,Q2) an unsubtracted relation is valid. cut (which for real-analytic functions is related to the
2
2
imaginary part). However, in the present case more in- is possible to derive rigorousupper and lower bounds on
formation is available on the amplitude at large energies thevaluesofthefunctionatallpointsinsidetheholomor-
intheframeofReggemodel. InRefs. [3,4]itwasshown phy domain, in particular at ν =0. Therefore, although
that,iftheamplitudeT atfixedQ2doesnothaveafixed thesubtractionfunctioncannotbecalculatedexactly,its
1
pole at J = 0 in the angular momentum plane, the sub- value is not completely arbitrary.
traction function itself can be calculated in terms of the In the present paper we propose a dispersive formal-
physical electroproduction cross sections. The detailed ismwhichusestheCauchyintegralrelationtoaccountfor
analysisperformedin[4]shows thatthe results obtained theknowledgeoftheimaginarypartonapartofthecut,
fromthisassumptionforthedifferenceoftheprotonand and the modulus representation on the remaining part.
neutron polarizabilities are consistent with experiment As we shallsee, in generalthis input does not determine
and somewhat more precise, which may be viewed as a uniquelytheamplitude,butpredictsawholeclassof“ad-
nontrivial test of the hypothesis that the Compton am- missible” functions to which the physical amplitude be-
plitude is free of fixed poles. longs. By means of the solution of a functional extremal
In the present work we develop an alternative disper- problem on this class, it is possible to formulate a rigor-
sive framework for the investigation of the subtraction ousnecessaryandsufficientconditionfortheconsistency
function in the virtual Compton scattering, in which no oftheinputgivenontheboundarywiththevaluesofthe
explicitassumptionaboutthe presenceorthe absenceof amplitude inside the holomorphy domain. This can be
the fixed poles is made. We start from the remark that exploitedtoconstrainthesubtractionfunctionandtoin-
the Regge model predicts at large energies not only the vestigatethepossiblefixedpolesintheReggeasymptotic
imaginary part of the amplitude, but the amplitude it- behavior. The method can be applied to both invariant
self. The high-energybehaviorof the amplitude is ofthe amplitudes, however we restrict in this paper to the am-
standard Regge asymptotic form [18, 19] plitude T . We note that the method is applicable to a
1
general Regge behavior, including the Pomeron contri-
πβ (t)
T(s,t) α exp[ iπα(t)]+τ sα(t), (4) bution, allowing therefore the separate discussion of the
∼−sinπα(t){ − } proton and neutron amplitudes.
The paper is organized as follows: in the next section
wheresisthec.menergysquared(connectedtothevari-
we review the physical information used as input, con-
ableν bys=2mν+m2 Q2)andtthemomentumtrans-
− sistingofthe structurefunctionatlowandmoderateen-
fer (in the present case t = 0 and the signature τ = 1).
ergies and the Regge model at high energies. In Sec. III
Therefore, from the Regge parametrization of the imag-
we formulate an extremal problem whose solution pro-
inary part of the amplitude it is possible to recover also
vides a necessary and sufficient condition for the consis-
the real part, which is in a particular relation to it, as
tency with analyticity of the cross sections, the Regge
follows from (4).
modelandthe subtractionfunction. The solutionofthis
If one knows both the real and imaginary parts of a
problem is given in the Appendix, while in Sec. IV we
function along a part of the unitarity cut, the function
discuss the numerical evaluation of the quantities enter-
can be predicted in principle everywhere in the com-
ing the solution. In Sec. V we present the results of our
plex plane by the uniqueness of analytic continuation.
analysis: we first discuss the consistency of the input at
However, it is known that in practice this task is im-
Q2 = 0, where the subtraction function is related to the
possible, due to the fact that analytic continuation is
magneticpolarizability. Wederivethenparametrization-
an ill-posed problem in the Hadamard sense [20]: this
free upper andlowerbounds onthe subtractionfunction
meansthatsmalluncertaintiesoftheinputareamplified
S (Q2) and compare them with the results obtained re-
in anuncontrolledway outside the originaldomain. The 1
cently in [4] and with the parametrization proposed in
analytic continuation can be nevertheless “stabilized” in
[10]. Section VI summarizes our main results and con-
some cases, for instance if one restricts the class of ad-
clusions.
missible functions to a compact set. Such a condition is
provided, for instance, by the knowledge of the modulus
of the function along the whole boundary (see [21] and
II. PHYSICAL INPUT
referencestherein). Guidedbytheseideas,inthepresent
workwedevelopaformalismwhichusesasinputathigh
energies the modulus of the amplitude, provided by the Weconsidertheso-called“inelastic”amplitudedefined
Regge model. asin[4]bysubtractingfromthetotalamplitudetheelas-
Itisusefultorecallthatinthemodulusrepresentation, tic contribution
where one constructs an analytic function starting from
Tinel(ν2,Q2)=T (ν2,Q2) Tel(ν2,Q2), (5)
its modulus on the boundary of the analyticity domain 1 1 − 1
ratherthanfromitsdiscontinuity(imaginarypart),much
where Tel has a simple expression [4]
weaker assumptions at infinity are needed (since the in- 1
tegralrepresentationcontains the logarithmof the input
4m2Q2[G2(Q2) G2 (Q2)]
modulus). As a consequence, if one assumes that the Tel(ν2,Q2)= E − M (6)
modulus of a function is known along the boundary, it 1 − (4m2ν2 Q4)(4m2+Q2)
−
3
interms ofthe Sachselectromagneticformfactorsofthe Finally, since we are interestedin the subtractioncon-
nucleon, G (t) and G (t). In (5), the notation empha- stant, we consider the condition
E M
sizes the fact that the amplitudes, being even functions
of ν, depend actually on ν2. At fixed Q2, the ampli- Tinel(0,Q2)=Sinel(Q2), (10)
1 1
tude Tinel(ν2,Q2) is a real-analytic function in the ν2-
1
complex plane, cut along the real axis above νt2h, where where Sinel(Q2) is the inelastic part of the subtraction
ν = M +(M2 Q2)/2m is the threshold due to the 1
th π π − function (2). It satisfies the low-energy theorem [4]
πN intermediate state. The discontinuity across the cut
is related to the imaginary part
κ2 m
Sinel(0)= β , (11)
ImT1inel(ν2+iǫ,Q2)=πV1(ν,Q2), νt2h ≤ν2 ≤νh2, (7) 1 −4m2 − αem M
where the structure function V1 is expressed in terms of whereαem isthe finestructureconstant,κisthe anoma-
thelongitudinalσL andtransversalσT electroproduction lous magnetic moment of the particle and βM its mag-
cross sections as [4] netic polarizability.
As we shall see later, the conditions (5), (7), (9) and
mν
V (ν,Q2)= k(ν,Q2) σ¯ (ν,Q2) σ (ν,Q2) , (10) do not specify uniquely the physical amplitude. If
1 L T
2α −
em (cid:8) (cid:9) the input conditions are compatible among themselves,
ν2 one can find a whole class of functions analytic in the
σ¯ (ν,Q2) σ (ν,Q2),
L ≡ Q2 L ν2 complex plane cut along ν2 ν2 , which satisfy these
1 ν Q2/2m conditions. Obviously,thephys≥icalthamplitudeT1inel must
k(ν,Q2) − . (8) belong to this “admissible” class. On the other hand, if
≡ 2π2 ν(ν2+Q2)
the conditions are not mutually consistent, there is no
function that can satisfy all conditions simultaneously,
In our treatment we shall assume that V (ν,Q2) is ex-
1 i.e. the admissible class is empty. As we shall show in
perimentally accessible fromthe crosssections measured
the next section, it is possible to formulate a necessary
for Q2 0 below a certain high-energy value of ν2, de-
noted a≥s ν2. and sufficient condition for the existence of at least one
h function in this class.
At higher energies, Tinel(ν2,Q2) is approximately de-
1
scribed by Regge model. In the standard dispersion re-
lations, only the structure function V (ν,Q2) is obtained
1
from the Regge parametrization. However, the Regge
model predicts also the real part, which is related to the
III. OPTIMAL CONSISTENCY CONDITION
imaginary part through the standard form (4). There-
fore, we shall assume that the modulus of the amplitude
along the cut above ν2 satisfies the condition Inthis sectionwe express the informationavailableon
h
the amplitude in a suitable formthat allowsthe applica-
Tinel(ν2,Q2) = TRegge(ν2,Q2), ν2 ν2, (9) tionofstandardmathematicalresults[22],andformulate
| 1 | | 1 | ≥ h anextremumproblemwhosesolutiondescribesinanop-
where TRegge(ν2,Q2) denotes the dominant Regge con- timal way the consistency of the input conditions1.
1
tributions, given by the Pomeron P and the degener- We define first a so-calledouter function [22], denoted
ate f and a trajectories. From the arguments given as C(ν2,Q2), which is analytic and without zeros in the
2
in the next section, it will be clear that the bounds on ν2 plane cutalongthe semiaxisν2 ν2 atfixed Q2,and
≥ h
the subtraction function remain the same even if the whose modulus on this cut is given by
equality sign in (9) is replaced by the sign. More-
≤
over, as we shall prove in Sec. IV, if instead of a given C(ν2,Q2) = TRegge(ν2,Q2), ν2 ν2. (12)
TRegge(ν2,Q2) we use an upper estimate of it, the con- | | | 1 | ≥ h
| 1 |
straintsonthesubtractionfunctionbecomeweaker. This
Withthisdefinition,wewritetheboundarycondition(9)
monotonicitypropertywillbeveryusefulforcheckingthe
in the equivalent form
stabilityofthemethodtowardthevariationoftheinput.
AswewilldiscussinSec. IV,alargeerror,of30%,willbe
Tinel(ν2,Q2)
adopted for the modulus of the Regge parametrization. 1 =1, ν2 ν2. (13)
(cid:12) C(ν2,Q2) (cid:12) ≥ h
Thus, if the Compton amplitude contains a contribution (cid:12) (cid:12)
(cid:12) (cid:12)
from a fixed pole, then that contribution is included in (cid:12) (cid:12)
the r.h.s. of (9). We do not impose any constraints on
the Q2-dependence of the fixed pole term, but allow it
1 Similarmethodshavebeenappliedtootherphysicalsituationsin
to be arbitrary, subject only to the constraint that the
[23–26]. Inparticular,inRefs. [24,25]itallowedthederivation
contribution from the fixed pole does not exceed 30% of
of model-independent constraints on the real proton Compton
the leading terms from Pomeronand Reggeons. scattering.
4
The function C(ν2,Q2) is obtained by the explicit for- Recallingthatz˜(0)=0,wenotethatF(z,Q2)satisfies
mula2 also the condition
∞ F(0,Q2)=S(Q2) (21)
ν2 ν2 ln TRegge(ν′2,Q2)
C(ν2,Q2)=exp h− dν′2 | 1 | .
p π Z ν′2 ν2(ν′2 ν2) where
 νh2 p − h −  Sinel(Q2)
(14) S(Q2) 1 . (22)
≡ C(0,Q2)
We consider now the change of variable
The constraints (19) and (21) are fully implemented by
the representation3
1 1 ν2/ν2
z˜(ν2)= − − h, (15)
1+p1−ν2/νh2 F(z,Q2)=S(Q2)+ z 1 dxσ(x,Q2) +zg(z,Q2),
p π Z x(x z)
which maps the ν2 plane cut along the semiaxis ν2 ν2 zth − (23)
≥ h
onto the interior of the unit disk z < 1 cut along the
segment z x 1 of the real a|xi|s, where z z˜(ν2) where g(z,Q2) is, for every fixed Q2 0, an arbitrary
th ≤ ≤ ≡ ≥
and function analytic in z < 1, free of any constraints at
| |
interior points. The presence of this function shows that
z =z˜(ν2 ). (16) the low-energy conditions do not specify uniquely the
th th
amplitude.
Wenotealsothatz˜(0)=0andtheupper(lower)edgeof We exploit now the boundary condition (13), which
the cut along ν2 ν2 becomes the upper (lower) semi- becomes, in the z-variable,
≥ h
circle of the circle z =1.
It is convenient t|o|define the new function F(ζ,Q2) =1, ζ =1. (24)
| | | |
Tinel(ν˜2(z),Q2) By inserting in this condition the general representation
F(z,Q2) 1 , (17) (23) and dividing both terms of the equality (24) by the
≡ C(ν˜2(z),Q2)
factor ζ 1, we obtain with no loss of information the
| |≡
relation
where ν˜2(z), defined as
g(ζ,Q2) h(ζ,Q2) =1, ζ =1, (25)
4z | − | | |
ν˜2(z)= ν2, (18)
(1+z)2 h where
S(Q2) 1 1 σ(x,Q2)
is the inverse of the function z˜(ν2) from (15). Since by h(ζ,Q2)= dx (26)
construction C(ν2,Q2) is analytic and has no zeros in − ζ − π Zzth x(x−ζ)
the ν2-plane cut along the semiaxis ν2 ν2, which in
≥ h is a calculable complex function defined in terms of the
the z-variable corresponds to the unit disk z < 1, the input information, and g(z,Q2) is, at fixed Q2 0, an
function F(z,Q2) is, for each fixed Q2, a r|ea|l analytic ≥
arbitrary function analytic in z <1. In general, we ex-
function in the unit disk z < 1 except for a cut along | |
pect that the boundary condition (25) for a given input
| |
the real segment z x 1. The discontinuity across
th h is satisfiedby manyanalytic functions g. Onthe other
the cut, discF(x,Q2)≤=F≤(x+iǫ,Q2) F(x iǫ,Q2)=
hand, it is possible that the input h is such that no an-
− −
2 i ImF(x+iǫ,Q2), is related to the imaginary part
alytic function satisfying this condition exists. So, the
question is to decide whether the class of the functions
ImF(x+iǫ,Q2) σ(x,Q2), z x 1, (19)
≡ th ≤ ≤ g satisfying the condition (25) contains at least one ele-
ment. In order to answer this question, we consider the
which is obtained as
following functional extremal problem: find
σ(x,Q2)= πV1(qν˜2(x),Q2). (20) µ0(Q2)=gm∈Hin∞kg−hkL∞, (27)
C(ν˜2(x),Q2)
where the L∞ norm is defined as [22]
In writing this expression we took into account the fact
that the function C(ν˜2(z),Q2) is realand has no discon- kFkL∞ ≡ sup |F(eiθ)|. (28)
θ∈(0,2π)
tinuity across the real interval z x 1.
th
≤ ≤
3 The transition point z = 1 requires a special treatment in the
2 Anequivalent representation can be written inthe canonical z- intermediate steps, see. Refs. [24,26]. Weomitithere,sinceit
variabledefinedin(15),see[22]. does notaffectthefinalresultquotedintheAppendix.
5
At each fixed Q2, the minimization in (27) is done with and intermediate energies enter through the function
respecttothefunctionsg(z,Q2)analyticinthedisk z < V (ν,Q2), and the Regge model is included in the outer
1
| |
1andboundedonits boundary(this classoffunctions is function C(ν2,Q2) defined in (14).
denoted as H∞ [22]). For the experimental input we rely on the detailed
Forwhatfollowsitisimportanttonotethat,ifatleast analysis performed in the recent paper [4]. At low en-
onefunctiong thatsatisfiesthecondition(25)exists,the ergies, for W = √s 1.3GeV, we use the data on the
≤
minimal norm µ (Q2) satisfies the condition cross sections measured in [27]-[34]. More precisely, we
0
useascentralvaluestheaverageofMAID[28]andDMT
µ (Q2) 1. (29)
0 [33] data, and ascribe to them an error equal to their
≤
difference. In the range 1.3 < W < 3GeV we use the
Ontheotherhand,iftheminimalnormµ (Q2)isstrictly
0
parametrization of Bosted and Christy [35, 36]. For the
greaterthan1,thedifferenceintheleft-handsideof(25)
protonamplitude,theerrorwasobtainedbyattachingan
will be strictly greater than 1 for every analytic func-
uncertaintyof8%totheprotontransverseandlongitudi-
tion g, which means that there are no analytic functions
nalcrosssectionsandaddingtheerrorsupinquadrature.
which can satisfy the constraint (25). Therefore, the in-
For the proton-neutron difference an uncertainty of 30%
equality (29) is a necessary and sufficient condition for
was assumed (to allow the comparison with the results
the existence of at least one admissible function g that
reportedin[4],forQ2 0.5GeV2 theerrorwasassumed
satisfies the input constraints imposed on the physical ≥
to be 8% from the proton cross section, calculated as
amplitude. Implicitly, it represents a consistency condi-
mentioned above).
tionofthe inputquantities enteringthefunction h,from
SeveralReggeparametrizationsofthecrosssectionsat
which we can derive correlations and bounds on various
high energies have been proposed in the literature [37]-
input quantities.
[40]. We shall use at W W , where W = 3 GeV, the
From the definition of the minimal norm, one can see ≥ h h
parametrizations given by the vector-meson dominance
that the problem is not changed if the equal sign in the
model of Alwall and Ingelman [37]:
condition (9) is replaced by the sign. In the next sec-
≤
tionweshallshowalsothatbyusinginsteadoftheright- σ =βT(Q2)sαP−1+βT(Q2)sαR−1,
T P R
handsideof(9)anupperestimateoftheReggemodulus,
weobtain,forafixedsubtractionconstant,alowervalue σL =βPL(Q2)sαP−1+βRL(Q2)sαR−1, (31)
of µ (Q2), which means that the allowed range for the
0 wheres=W2 isthesquareofthecenter-of-massenergy.
subtractionfunctionderivedfromthe condition(29)will
A “soft” Pomeronwith intercept α =1.091is adopted,
be larger. P
while the Reggeons with the quantum numbers of f and
The solution of the problem (27) and the algorithm
a are represented by a single contribution with α =
for calculating the quantity µ (Q2) are presented in the 2 R
0 0.55. The Pomeron residues of proton and neutron are
Appendix. In the next section we discuss the numerical
usually assumed to be the same:
calculation of the quantities entering the solution, based
on the experimental input on the cross sections and the βT(Q2)n =βT(Q2)p, βL(Q2)n =βL(Q2)p. (32)
P P P P
Regge parametrization.
while for the Reggeons one uses [41]:
βT(Q2)n =ξβT(Q2)p, βL(Q2)n =ξβL(Q2)p, (33)
R R R R
IV. NUMERICAL ANALYSIS OF THE INPUT
with ξ 0.74.
The s≃tructurefunctionV (ν,Q2)atν <ν isobtained
1 h
As shown in (A.14), the parameter µ0(Q2) is given by directlyfromthecrosssectionsusing(8). Athigherener-
thesquarerootofthegreatesteigenvalueofthepositive- gies, at ν ν , we can recoverlocally the full amplitude
h
semidefinitematrix † ,where isaHankelmatrixde- Tinel from≥the parametrizationofthe imaginarypartob-
finedin(A.10)interHmsHoftheFoHuriercoefficientsck(Q2) ta1ined from (31). Using the standard Regge expression
defined in (A.11) or (A.12). (4), it is easy to see that this is achieved by making in
Having in view the expression (26) of the function (31) the replacements
h(ζ,Q2), it is convenient to use for the calculation of
ck(Q2) the definition (A.12) as a contour integral. By β 1+exp[−iπαj]β , j =P,R. (34)
applying the residue theorem, we obtain the convenient j →− sinπα j
j
expression
By this prescription we ensure that the real and imagi-
1 1 nary parts are related as required by the Regge model:
c (Q2)= S(Q2)δ + xk−2σ(x,Q2)dx, k 1,
k − k,1 π Z ≥ for the Pomeron, the real part is much smaller than the
zth
imaginary one, while for the Reggeons, using α =0.55,
(30) R
the real and imaginary parts are almost equal.
where S(Q2), z and σ(x,Q2) are defined in Eqs. (22), In Fig. 1 we show for illustration the modulus of
th
(16) and (20), respectively. The cross sections at low the amplitude TRegge obtained with this prescription for
1
6
quence,theallowedrangeoftheparameterSinel(Q2),de-
1
rived from the consistency condition (29), will be larger.
24 This property has important consequences. First, it
proton
allows us to use at all energies above W = 3GeV the
neutron
20 parametrization of the Pomeron by a “soft” trajectory
W)| with an intercept αP slightly larger than 1. This model
ReggeT(116 ipsrevdailcidtssatrnicatlsyymsppetaoktiicnginocnrleyasaetifinncitoenfleincetrgwieitsh, suinncitearit-
| 12 ity. Indeed, universality implies that the same Pomeron
would also show up in hadronic scattering processes and
8 that would violate the Froissart bound (an attempt at
describing deep inelastic scattering athigh energies with
43 3.5 4 4.5 5 5.5 6 a logarithmic high-energy behavior consistent with the
W [GeV] Froissart bound is presented in [42]). Actually, the soft
Pomeron is expected to overestimate the physical loga-
FIG. 1: Modulus of the Regge parametrization of the am-
rithmic increase with energy of the amplitude. Since the
plitude TRegge for Q2 = 0, in the region 3 ≤ W ≤ 6GeV.
1 bounds are monotonically decreasing functionals of the
input modulus, the soft Pomeron is expected to lead to
weaker bounds on the subtraction functions compared
to those obtained with the true behavior, preserving the
Q2 = 0, in the region 3 W 6GeV. The small validity of the method.
≤ ≤
difference between the proton and the neutron ampli- Inviewofthesamemonotonicityproperty,theweakest
tudes is due to the different Reggeons contribution and upper and lower bounds are obtained with the modulus
decreases at higher energies, where the Pomeron domi- increasedwithintheerrorchannel. Thelargeuncertainty
nates (for instance, the difference is 8% at 3 GeV and of30%whichweadoptedcoverspossiblevariationsofthe
only2 per mille at100GeV). We emphasize thatthe ex- subasymptotictermsrelatedtothestandardReggepoles
pression obtained by the prescription (34) is used only and also a possible real constant, which can arise due to
on the unitarity cut at fixed Q2 for ν2 ν2, where a fixed pole at J =0 in the angular momentum plane.
νh = (Wh2 −m2 +Q2)/2m with Wh = 3 G≥eVh. No as- As shown in (11), the subtraction function at Q2 = 0
sumptions on the analytic continuation or the analytic is determinedby the magnetic polarizability. The exper-
properties of the Regge parametrization at other points imental values quoted in the recent papers [43] and [44],
oftheν2-complexplanearemade. Asalreadymentioned, expressed in units of 10−4 fm3, are4
anerrorof30%wasattachedtothemodulusoftheRegge
parametrization in our analysis. βp =3.15 0.50, βn =3.65 1.50, (35)
Since we use as input differentmodels atvariousener- M ± M ±
gies, one might ask what are the consequences of match-
which imply βp−n = 0.5 1.6. Using these values in
ing these models for the accuracy of the numerical anal- M − ±
the expression (11), we obtain the subtraction functions
ysis. Actually, the large uncertainties that we adopted
cover to some extent the discrepancies at the matching
Sinel(0)=( 6.18 0.84)GeV−2,
points. It is important to emphasize also that, in con- 1,p − ±
trast to the usual FESR, the present dispersive formal- Sinel(0)=( 7.15 2.51)GeV−2,
ism does not involve the calculation of principal values 1,n − ±
Sinel (0)=(0.96 2.68)GeV−2. (36)
or other quantities sensitive to the discontinuities of the 1,p−n ±
input onthe cut. Therefore,the detailsofthe matchings
haveasmallereffectinourapproachthaninthestandard As discussed above, the method requires the calcula-
dispersion relations. tion of the Fourier coefficients c given in (30) and of
k
Using the expressions (30) and (A.14) of the Fourier the quantity µ given by the norm (A.14) of the Hankel
0
coefficients and the minimal norm µ , it is easy to prove matrix(A.10). Thenumericalalgorithmimpliesactually
0
a monotonicity property already mentioned above. As- the truncationofthe Hankelmatrixatafinite orderand
sumethatwereplacetheinputmodulusatfixedQ2given the calculation of its norm with standard programs. In
intheright-handsideofEq. (9)byanestimateofitfrom practice, we obtained good stability with the first 20-30
above. Then the real and positive values of the outer coefficients.
functionC(ν2,Q2)definedin(14)willincreaseatallreal
points ν2 belowν2. Fromthe definitions (20)and(22)it
h
followsthattherealquantitiesσ(x,Q2)andS(Q2)enter-
ing the expression (30) will decrease if Sinel(Q2) is kept
1
fixed. This will lead to smaller values for all the coeffi- 4 Notealsotheaverage values giveninPDG[45],βp =2.5±0.4
M
cientsck andthereforetoasmallernormµ0. Asaconse- andβMn =3.7±1.2.
7
V. RESULTS Sinel(Q2) in terms of the cross sections and Regge pa-
1
rameters.
As discussed above, the inequality (29) represents a We end this subsection by noting that the formalism
necessary and sufficient condition for the consistency considered in the present paper can be used in princi-
with analyticity of the amplitude Tinel(ν2,Q2). One can ple also for testing the presence or the absence of fixed
1
exploit it either for testing the consistency of the vari- poles in the scattering amplitude. This subject was put
ous pieces ofthe phenomenologicalinput, or forderiving forward a long time ago [12, 13] and received attention
bounds on the subtraction function Sinel(Q2). also recently [14–16]. As is known, a fixed pole at J =0
1
in the angular momentum plane contributes with a real
constant to the high-energy behavior of the amplitude.
A. Consistency of the input at Q2=0 In the presentformalism, the need of an additionalterm
in the high-energy behavior is signaled by values of the
AtQ2 =0,where the subtractionfunctionsareknown minimum norm µ0 larger than 1 when calculated with
only standard Regge poles in the condition (9). The re-
in terms of polarizabilities as shown in (36), we have all
sults given above show that, at Q2 = 0, this does not
the ingredients to calculate the coefficients c defined in
k
seemto be the case: the analyticitytest is passedwithin
(30) and the parameter µ defined in (A.14). Keeping
0
the relatively large uncertainties of the present data. If
fixedthesubtractionfunctionsatthecentralvalues(36),
more precise data in the future will lead to µ > 1 even
we obtain 0
when the input will be varied within errors, it will be
µp =1.099 0.054 0.053 0.099 , possible to conclude that a fixed pole is necessary and,
0 ± MAID± BC± R
µn =1.066 0.053 0.053 0.254 , by adjusting the magnitude of the additional constant
0 ± MAID± BC± R until the value of the quantity µ becomes smaller than
µp0−n =1.407±0.633MAID±0.199BC±0.325R, (37) unity, to find constraints on its re0sidue.
where weindicatedthe effectofthe separatevariationof
the MAID and DMT data [28, 33], Bosted and Christy B. Bounds on the subtraction function Sinel(Q2)
parametrization [35, 36] and the Regge parametrization 1
[37], within the uncertainties specified in the previous
Using arguments presented in [24, 26], one can show
section. Adding in quadrature these effects leads to
that, for given input values of cross sections and Regge
µµnp0 ==11..006969±00..124605,, ptiaornamofetthriezaptairoanms,ettheermS1iinneilm(Qu2m),ndoirsmplaµy0inisgaacsoinnvgelexmfuinnci--
0 ± mum. Therefore,fromthe inequality(29)one canderive
µp0−n =1.407±0.739. (38) exact upper and lower bounds on the subtraction func-
tionSinel(Q2). Theboundsdefineanallowedintervalfor
We recall that the consistency of the input requires val- 1
this quantity. Of course, if the values of µ stay above
uesofµ lessthan1. From(38)onecanseethat, forthe 0
0 unity for all values of the parameter Sinel(Q2), we con-
protonandneutronamplitudes, this conditionis slightly 1
clude that there are no analytic functions satisfying the
violatedbythecentralvaluesoftheinputquantities,but
input conditions, therefore the input data are inconsis-
consistencyisachievedwhentheerrorsaretakenintoac-
tent with analyticity.
count. We note that, for fixed values of the subtraction
We illustrate first the above arguments for the ampli-
functions,inallcasesµ decreasedifthelowenergycross
0 tude corresponding to the proton-neutron difference at
sections or the Regge modulus were increased within er-
Q2 = 0. In Fig. 2, we present µp−n as a function of
rors. 0
Thecentralvalueofµp−n issignificantlylargerthan1, the subtraction function S1i,np−eln(0), showing for conve-
0 nience only the range where the consistency condition
indicating that the central values of the various parts of
(29) is satisfied. From this condition, we obtain the al-
theinputareinconsistentwithanalyticity. Thisisdueto
lowedrange 1.00 Sinel (0) 0.64GeV−2. Bytaking
the fact that the amplitude is obtained as the difference − ≤ 1,p−n ≤
into account the uncertainties of the cross sections and
of the proton and the poorly known neutron amplitude,
the Regge parametrization, this range becomes
and is not very reliable. Nevertheless, consistency with
analyticity is achieved also in this case when the large 1.59 Sinel (0) 1.22GeV−2. (39)
errors discussed in the previous section are taken into − ≤ 1,p−n ≤
account. In this calculation we varied independently the
In the above calculations the subtraction functions MAID/DMT cross sections below 1.3 GeV, the
Sinel(0) have been kept fixed at the central values given parametrization of Bosted and Christy in the range
1
in (36). One may reverse the argument and calculate µ 1.3 < W < 3GeV and the modulus at higher energies,
0
as a function of this parameter, keeping the cross sec- and modified the bounds by the quadratic sum of the
tions and the Regge parameters fixed. We shall apply separate shifts. We note that the central experimental
this procedureinthe nextsubsection,where weshallde- value S1in,pe−l n(0) = 0.96GeV−2 given in (36) is slightly
riveupper andlowerbounds onthe subtractionfunction above the upper bound 0.64GeV−2 obtained with the
8
central input on the unitarity cut, which was reflected 3, where the cyan and turquoise bands are the allowed
by the central value µp−n > 1 given in (38). However, rangesobtainedwithcentralinputandincluding theun-
0
again in accordance with (38), the central experimental certainties. In the same figure we show for comparison
value is within the enlarged range (39) which includes theallowedband(denotedasGHLR)obtainedin[4]from
the uncertainties of the input. We emphasize that this the assumption of “Reggeon dominance”, i.e. by requir-
rangeisnarrowerthantheexperimentalinterval 1.71 ing the absence of constant terms in the asymptotic be-
Sinel (0) 3.64GeV−2 which follows from (36)−, and≤is havioroftheamplitude. Onecanseethatourresultsare
1,p−n ≤
consistent with the value Sinel (0) = 0.3(1.2)GeV−2 considerably weaker, which was to be expected since we
derived in [4]. 1,p−n − work in a less restrictive formalism, where the absence
of the fixed pole is not imposed in a manifest way. It is
importanttonotehoweverthatourresultsareconsistent
1 withinuncertaintieswiththeallowedbandderivedin[4].
Asalreadymentioned,thesubtractionfunctionwasas-
0.8 sumedtobecompletelyindependentofthecrosssections
in Refs. [6, 9, 10], where rather arbitrary parametriza-
0.6 tions of this function have been proposed. It is instruc-
µp-n tive to compare our results also with these parametriza-
0
tions. Asdiscussedin[4],theexpressionproposedin[6]is
0.4
notconsistentwiththeshort-distancepropertiesofQCD.
Therefore, we shall consider only the variant adopted in
0.2
[10], where the inelastic part of the subtraction function
reads
0
-0.8 -0.4 0 0.4
Sinel(0) [GeV-2] m2+cQ2 m2 2 mβ
1, p-n Sinel (Q2)= 1 1 M
1,ESTY −(cid:18) m2+Q2 (cid:19)(cid:18)m2+Q2(cid:19) α
FIG. 2: Dependence of µp−n on the subtraction constant 1 1 em
0 4m2 G (Q2) G (Q2) 2
S1in,pe−l n(0). { E − M } . (40)
− (4m2+Q2)2
We evaluated this expression using the Kelly
parametrizations [46] for the nucleon form factors,
4 the range of βp−n given below (35) and the values c=0
ESTY M
GHLR and m = √3m with m = 0.46 0.1 0.04GeV
1 β β
3 This work (central input) from [10]. The result is represented±in Fig±. 3 as the
This work (including errors)
red band denoted as Erben-Shanahan-Thomas-Young
2 (ESTY). One can see that the allowed band derived in
2Q) the present framework is somewhat narrower than the
inelN S (1, p-n 1 baBndypobertfaoirnmedinwgitthhetshaemEeSaTnYalyasnissaintzt.heprotoncase,we
0 notefirstthattheallowedrangeobtainedatQ2 =0with
central input, namely 30.2 Sinel(0) 7.33GeV−2,
− ≤ 1,p ≤ −
-1 is quite large. This can be explained by the fact that,
due to the Pomeron, the proton amplitude is allowed to
-2 increase more rapidly on the unitarity cut. By invok-
0 0.5 1 1.5 2
Q2 [GeV2] ing the maximummodulus principle,this implies weaker
constraintsatpointsbelowthecut. Weseehoweverthat,
FIG. 3: Bounds on the proton-neutron subtraction function althoughthe intervalis large,the upper bound lies actu-
S1in,pe−l n(Q2) for various Q2. The red band is parametrization ally below the central experimental value given in (36),
(40) proposed in [10]. The band denoted GHLR was derived obtained from the magnetic polarizability (this discrep-
in [4]. Cyan band: allowed range situated between the lower ancy was already signaled actually by the central value
andupperboundsobtainedfromthecondition(29),withthe ofµp largerthan1givenin(38)). Bytakingintoaccount
0
central values of the cross sections and Regge parameters.
the errors of the cross sections and the 30% uncertainty
Turquoise band: enlarged range taking into account the un-
of the Regge modulus, which is quite large in the proton
certainties of the input. As in [4], for an easier comparison
case, the upper bound is pushed up to
the values on the vertical axis are multiplied by the factor
N =(1+Q2/M2)2, with M2 =0.71GeV2.
d d Sinel(0) 3.77GeV−2, (41)
1,p ≤−
By repeating the calculations for other values of Q2, becoming consistent with the experimental range (36).
we obtain the bounds for Sinel (Q2) presented in Fig. This exercise shows that the upper bounds calculated
1,p−n
9
1 1
0.8 0.9
µp0.6 µp0.8
0 0
0.4
0.7
0.2
0.6
0
-25 -20 -15 -10 -5 -0.8 -0.4 0 0.4 0.8 1.2 1.6 2 2.4
Sinel (0.01 GeV2) [GeV-2] Sinel (1 GeV2) [GeV-2]
1, p 1, p
FIG.4: Dependenceofµp onthesubtractionconstantSinel(Q2),fortwovaluesofQ2: Q2=0.01GeV2 (left)andQ2=1GeV2
0 1,p
(right).
with more accurate data on the cross sections might put
a nontrivial constraint on the proton magnetic polariz-
ability. 6
ESTY
At nonzero values of Q2, it turns out that the input is This work (central input)
4
consistentinthesenseexplainedaboveonlyfortheinter- This work (including errors)
vals Q2 0.1GeV2 and Q2 0.5GeV2. For illustration 2
Sowt1iehn,peeslrh(Qoewq2≤)uianfloFrtoitgw1.oG4evtVahl2eu.evsCareoinfatt≥Qrioa2nl,vooafnluµeep0sclwoofsitethhtetohleo0wqa-uneandnetrtightyye inel2S(Q) 1,p-20
cross sections and Regge parameters have been used in
the calculation. In both cases this input is consistent, in -4
thesensethattherearevaluesofthesubtractionconstant
-6
for which the compatibility condition (29) is satisfied.
For convenience, we show only the ranges of Sinel(Q2) -8
1,p 0 0.5 1 1.5 2
thatsatisfythiscondition. Fromthefigure,onecanread 2 2
Q [GeV ]
the upper and lowerbounds 26.6 Sinel(0.01GeV2)
− ≤ 1,p ≤
4.4GeV−2 and 0.84 Sinel(1GeV2) 2.44GeV−2. FIG. 5: Allowed ranges of the proton subtraction function
− − ≤ 1,p ≤
WenotethattheallowedrangesofS1in,pel(Q2)forQ2 close S1in,pel(Q2)forQ2≥0.5GeV2,comparedwiththeparametriza-
to 0 are very large, which shows that the constraining tion (40).
poweroftheformalisminthisdomainisweak. However,
as inthe case Q2 =0 discussedabove,the upper bounds
might lead to nontrivial constraints.
As we mentioned, for 0.1 < Q2 < 0.5GeV2, the pa- withthe behaviorderivedrecentlyin[47]by adispersive
rameter µp was larger than unity for all values of the analysis, assuming the absence of J =0 fixed poles5.
0
subtraction function Sinel(Q2) (more exactly, the mini-
1
mum value of µp in figures similar to Fig. 4 turned out
0
tobelargerthan1,anddidnotdecreasebelowunityeven VI. SUMMARY AND CONCLUSIONS
whenthevariouspiecesoftheinputwerevariedindepen-
dently within errors). Therefore, in Fig. 5 we show the Inthispaperweproposedandinvestigatedanalterna-
allowed band for the subtraction function calculated in tive dispersive framework for the virtual Compton scat-
thisworkonlyforQ2 0.5GeV2. Althoughquitebroad, tering on the nucleon. The work was motivated by the
≥
the band is not consistent with the ESTY ansatz (40) recent interest in the study of this process in connection
near Q2 = 0.5GeV2: the bounds suggest the possibility withtheCottinghamformulafortheproton-neutronelec-
ofpositive valuesfor Sinel(Q2), whicharenotallowedby tromagnetic mass difference, the nucleon polarizabilities
1,p
the ansatz. Of course, in view of the large uncertain- and the proton radius puzzle. We considered in particu-
ties in the data and the fact that for Q2 <0.5GeV2 the lar the inelastic partTinel(ν2,Q2) ofone ofthe invariant
1
structurefunctionsandtheReggeparametersareincon-
flict with analyticity, this result should be taken only as
a very preliminary one. On the other hand, the allowed
band shown in Fig. 5 is consistent, within uncertainties, 5 Ithanktheauthorsof[47]forperformingthiscomparison.
10
amplitudes,whichrequiresasubtractioninthestandard Thecentralvaluesoftheelectroproductioncrosssections,
dispersion relation at fixed Q2. The aim was to investi- the Reggeparametrizationandthe magneticpolarizabil-
gate in a parametrization-free approach the subtraction ities leadtovalues ofµ whichslightlyviolatethe condi-
0
functionandthepossibleexistenceoffixedpolesatJ =0 tion(29), but consistency is achievedwhen uncertainties
in the angular momentum plane. Our formalism does ofthecrosssectionsandReggeparametrizationaretaken
not rely on the hypothesis that the Compton amplitude into account.
is free of fixed poles, but covers that case as well. So, it Theformalismcanbeappliedalsoforinvestigatingthe
is less restrictive than the recent analyses [4, 47], where possible existence of fixed poles at J = 0 in the angular
theabsenceoffixedpolesintheamplitude wasexplicitly momentum plane, which contribute with a real constant
assumed, or in the FESR studies [15], where the contri- to the high-energy behavior of the amplitude. The pres-
butionofafixedJ =0pole wasexplicitly included. The ence of a fixed pole is revealed in principle by a clear
price paid for this generality is the fact that our method inconsistency of the low-energy input with the standard
cannot lead to definite predictions, but gives only upper Regge model, signaled by large values of the parameter
and lower bounds on the subtraction function. Within µ . Of course, in practice inconsistency alone would not
0
the largeuncertainties ofthe experimentalinput, ourre- imply the existence of a fixed pole, it couldsimply mean
sultsfortheproton-neutrondifferenceareconsistentwith thatuncertaintiesintherestoftheinput(crosssections,
those obtainedin [4] and confirm the conclusions of that subasymptotic terms in the Regge amplitude) are larger
work. thanexpected. Therefore,morepreciseandreliabledata
The formalism considered in the present work uses as are necessary in order to establish, from the consistency
inputtheimaginarypartoftheamplitudeobtainedfrom withanalyticityoftheinput,thatafixedpoleofacertain
electron-nucleoncrosssectionsatlowandmoderateener- residue is required or not.
gies, and the modulus of the amplitude at high energies, Finally, we applied the formalism for deriving con-
wherewecanrecovertherealpartoftheamplitudefrom straints on the subtraction function Sinel(Q2) at various
1
itsimaginarypartusingthestandardReggerelation(4). values of Q2, by exploiting the fact that the quantity µ
0
The main mathematical result is the definition of the is a convex function of the value of this parameter. We
quantityµ ,givenbythesolutionofthefunctionalmini- performed this calculation for the proton-neutron differ-
0
mization problem(27), suchthat the inequality (29) is a ence and the proton amplitudes.
necessary and sufficient condition for the consistency of For the proton-neutron difference, the allowed range
the phenomenological input with analyticity and unitar- of Sinel (0) derived by the present method is slightly
1,p−n
ity. The parameter µ0 can be computed by a simple nu- smaller than the experimental range (36) derived from
mericalalgorithmasthenormofaHankelmatrix(A.10) polarizabilities. For Q2 0.5GeV2 our results are con-
constructed from the Fourier coefficients ck, calculable sistent within errors with≥the band derived in [4] by im-
according to (30) in terms of the cross sections and the posing explicitly the absence of the fixed poles. Such an
Regge parameters. For a given phenomenological input assumption is not explicitly adopted in our formalism,
ontheunitaritycut,thequantityµ0 isaconvexfunction which explains the weaker results that we obtain.
ofthe unknownsubtractionfunction. Then, the inequal- For the proton amplitude, we found that for Q2 be-
ity(29)allowsustofindrigorousupperandlowerbounds tween 0.1 and 0.5GeV2 the phenomenological cross sec-
on the subtraction function in terms of this input. tions and the Regge input are mutually inconsistent,
As discussed in Sec. IV, we assumed large uncertain- leading to values of the quantity µ larger than 1. For
0
ties for the input crosssections andthe Reggemodel. In Q2near0,theboundsderivedarequiteweak,howeverat
particular, the 30% uncertainty adopted for the modu- Q2 =0 the upper bound onSinel(0)calculatedwith cen-
1,p
lus of Tinel(ν2,Q2) above 3 GeV covers possible varia- tralinputisinslightconflictwiththeexperimentalvalue
1
tionsofthesubasymptotictermsrelatedto thePomeron obtainedfrom the magnetic polarizability. This suggests
and the Reggeons and can accommodate also constant that with more precise data the formalismmight lead to
terms of a certain magnitude in the asymptotic behav- nontrivial predictions. For Q2 0.5GeV2, the bounds
≥
ior. As proved in Sec. IV, the bounds are monotonically derived in the present work are in agreement within un-
decreasing functionals of the input modulus. This im- certainties with the behavior found in [47] by assuming
portantpropertyallowedustousethephenomenological the absence of J =0 fixed poles.
parametrization by the soft Pomeron adopted from [37] To summarize, the main benefit of the presentformal-
also at higher energies, where it overestimates the loga- ism is that it allows us to make predictions on the sub-
rithmicasymptoticincreaseoftheamplituderequiredby traction function in virtual Compton scattering with no
unitarity. Therefore, the bounds derived in the present explicitassumptionabouttheexistenceortheabsenceof
work are weaker than what would be obtained with the a J = 0 fixed pole. Thus, we obtain model-independent
physical asymptotic increase, which means that our ap- constraintsonthesubtractionfunction,whichcanbecal-
proach is a conservative one. culatedinstandarddispersionrelationsonlybyadopting
InSec. Vweappliedfirsttheformalismfortestingthe a specific assumption about the fixed poles. The numer-
consistency withanalyticity ofthe availablephenomeno- ical results reported in this paper depend on the phe-
logical input on the amplitude Tinel(ν2,Q2) at Q2 = 0. nomenologicalinputavailableatpresent,whichhaslarge
1