Spin-dependent sum rules connecting real and virtual Compton scattering verified

0.65 MB·

by  Vadim Lensky

MITP/16-119
Spin-dependent sum rules connecting real and virtual Compton scattering verified
Vadim Lensky
Institut für Kernphysik, Cluster of Excellence PRISMA,
Johannes Gutenberg Universität, Mainz D-55099, Germany
Institute for Theoretical and Experimental Physics, 117218 Moscow, Russia and
National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), 115409 Moscow, Russia
Vladimir Pascalutsa and Marc Vanderhaeghen
Institut für Kernphysik, Cluster of Excellence PRISMA,
Johannes Gutenberg Universität, Mainz D-55099, Germany
Chung Wen Kao
7 Department of Physics and Center for High Energy Physics,
1 Chung-Yuan Christian University, Chung-Li 32023, Taiwan
0
(Dated: April4,2017)
2
r We present a detailed derivation of the two sum rules relating the spin polarizabilities
p
measured in real, virtual, and doubly-virtual Compton scattering. For example, the polar-
A
izability δ , accessed in inclusive electron scattering, is related to the spin polarizability
LT
3 γE1E1 and the slope of generalized polarizabilities P(M1,M1)1−P(L1,L1)1, measured in, re-
spectively,therealandthevirtualComptonscattering. Weverifythesesumrulesindifferent

] variantsofchiralperturbationtheory,discusstheirempiricalverificationfortheproton,and
h
prospecttheiruseinstudiesofthenucleonspinstructure.
p
-
p
e
h
[
Contents

2
v
I. Introduction 2
7
4
9 II. Sum rule derivation 3
1 A. Forward double virtual Compton scattering 3
0
B. Low-energy expansions 4
.
1
C. VVCS sum rules 6
0
7 1. Spin-independent amplitude T1 6
1 2. Spin-independent amplitude T 7
2
:
v 3. Spin-dependent amplitude S 8
1
Xi 4. Spin-dependent amplitude S2 9
r
a III. Verification in heavy-baryon chiral perturbation theory 10
IV. Verification in covariant χPT 12
V. Empirical verification 14
VI. Predictions for the VCS response function P at low Q2 16
TT
VII. Summary and Conclusion 19
Acknowledgments 19
References 20
2
I. INTRODUCTION
The low-energy nucleon structure is presently at the forefront of many precision studies of the Standard
Model and beyond. Given the complexity of low-energy QCD, a popular method of calculating the nucleon-
structure effects is the data-driven approach based on model-independent relations, such as sum rules.
Perhaps the best known sum rule for the electromagnetic structure of the nucleon is the Gerasimov-Drell-
Hearn (GDH) sum rule [1–3], relating the anomalous magnetic moment of the nucleon to a weighted
integral over the polarized photo-absorption cross section. An even older sum rule is the one of Baldin [4],
which gives the sum of the electric α and magnetic β dipole polarizabilities in terms the total cross
E M
section σ(ν) as follows:
∞
(cid:90)
1 dν
α +β = σ(ν), (1)
E M 2π2 ν2
ν0
where ν is the photon energy in the laboratory frame, and ν is the inelastic threshold. This sum rule has
0
proventobeveryusefulforanaccurateextractionofthenucleonpolarizabilities,seeRefs.[5–8]forreviews.
The Baldin sum rule, derived from general considerations of the forward Compton scattering ampli-
tude,iseasilygeneralizedtothecaseofvirtualphotons,i.e.,theforwarddouble-virtualComptonscattering
(VVCS). The sum of the two polarizabilities becomes dependent on the photon virtuality Q2, and is con-
nected with the unpolarized nucleon structure function F , measured in electron-nucleon scattering, via
1
(cid:90)x0
8α M
α (Q2)+β (Q2) = em dxxF (x,Q2), (2)
E M Q4 1
0
where M is the nucleon mass, α is the fine structure constant, and x = Q2/2Mν is the Bjorken scal-
em
ing variable with x corresponding to the inelastic threshold. Other forward sum rules involving the spin
0
dependentnucleonstructurefunctionsallowforsuchageneralization[9,10],too. Inthisway,onecanchar-
acterizethespin-dependentVVCSprocessthroughQ2-dependenttransverse(γ )ortransverse-longitudinal
0
(δ )spinpolarizabilitiesofthenucleon[5,11,12]. Thesepolarizabilitiesarethusrelatedwiththenucleon
LT
spin-dependent structure functions, and have been the subject of dedicated experimental activities at the
Jefferson Lab, see [13, 14] for reviews.
Thenucleon’sresponsetoexternalelectromagneticfieldscanalsobeprobedthroughthevirtualCompton
scattering (VCS) process, in which the initial photon has finite virtuality Q2 whereas the final one is real.
The linear response of a nucleon in the low-energy VCS process can be expressed through generalized
polarizabilities(GPs)[15],seeRef.[16]foradetailedreview. TheGPs,whichencodethespatialdistribution
of the polarization densities in a nucleon [17], have been the subject of several dedicated experiments at
MAMI [18–22], MIT-Bates [23, 24], and JLab [25, 26].
As the VCS process is a non-forward process and apparently asymmetric under photon crossing, it has
precludedanimmediateconnection,viasumrules,ofGPswithphotoabsorptioncrosssections. Nonetheless,
a new type of relation, presented recently in Ref. [27], allows to relate two of the spin-dependent GPs with
quantitiesmeasuredinRCSandVVCS.ThenewrelationsprovideanextensionoftheGDHsumruletofinite
virtuality, and as a result involve new quantities which are accessible in independent experiments.
Inthisworkweprovideadetailedderivationofthetwonewsumrules. Wefirstdiscusstheforwarddou-
blevirtualComptonscattering(SectionIIA),itslow-energyexpansions(SectionIIB),andthederivationof
forward sum rules for the four amplitudes characterizing the VVCS on the nucleon (Section IIC). We then
show how these sum rules are satisfied in heavy baryon chiral perturbation theory (Section III) as well as
in its covariant counterpart — baryon chiral perturbation theory (Section IV). We discuss the phenomeno-
logical status of these sum rules and further experimental opportunities in Section V. In particular, by using
the sum rules, we will obtain an empirical prediction for the slope of one of the VCS response functions,
denoted by P , and will compare it with the dispersive evaluations and with the predictions of baryon
TT
chiral perturbation theory calculations (Section VI). Finally, we will present our conclusions in Section VII.
3
II. SUMRULEDERIVATION
A. ForwarddoublevirtualComptonscattering
Our starting point is the double-virtual Compton on a nucleon
γ∗(q,λ)+N(p,s) → γ∗(q(cid:48),λ(cid:48))+N(p(cid:48),s(cid:48)), (3)
where λ,λ(cid:48) denote the photon helicities, and s,s(cid:48) are the nucleon helicities. This process is described by 18
helicity amplitudes introduced as:
T ≡ e2ε (q,λ)ε∗(q(cid:48),λ(cid:48))u¯(p(cid:48),s(cid:48))Mµνu(p,s), (4)
λ(cid:48)s(cid:48),λs µ ν
where e is the proton electric charge, and ε (ε∗) stands for the initial (final) nuclear polarization vector.
µ ν
The double virtual Compton tensor Mµν can be Lorentz-decomposed as (in the notation of Ref. [28]):
Mµν = (cid:88)B (q2,q(cid:48)2,q·q(cid:48),q·P)Tµν,
i i
i∈J
J = {1,...,21}\{5,15,16}, (5)
whereP = 1(p+p(cid:48)). The18independenttensorsTµν canbeconstructedtobegaugeinvariant,andfreeof
2 i
kinematical singularities as shown by Tarrach [29]. The invariant amplitudes B have definite transforma-
i
tion properties with respect to the photon crossing, as well as charge conjugation combined with nucleon
crossing[28]. Thelatterreferencealsocontainsthelow-energyexpansionsofB ’suptoO(k3)(k = {q,q(cid:48)}),
i
which will be useful in the following.
To derive the sum rules one considers the forward double VCS process (VVCS), which is a special case
of the process (3), with q(cid:48) = q and p(cid:48) = p. The VVCS process is described by the four invariant amplitudes,
denoted by T ,T ,S ,S , which are functions of Q2 ≡ −q2 and ν ≡ p·q/M. Its covariant tensor structure
1 2 1 2
can be written as:
(cid:26)(cid:18) qµqν(cid:19) 1 (cid:18) p·q (cid:19)(cid:18) p·q (cid:19)
α Mµν(ν,Q2) = − −gµν + T (ν,Q2)+ pµ− qµ pν − qν T (ν,Q2)
em q2 1 M2 q2 q2 2
(cid:27)
i i
+ (cid:15)νµαβq s S (ν,Q2)+ (cid:15)νµαβq (p·q s −s·q p )S (ν,Q2) , (6)
M α β 1 M3 α β β 2
wherethefine-structureconstantα ≡ e2/4π (cid:39) 1/137isconventionallyintroducedindefiningtheforward
em
amplitudes T , T , S , and S . Furthermore, (cid:15) = +1, and sα is the nucleon covariant spin vector
1 2 1 2 0123
satisfying s · p = 0, s2 = −1. The optical theorem relates the imaginary parts of the four amplitudes
appearing in Eq. (6) to the four structure functions of inclusive electron-nucleon scattering as:
e2 e2
Im T (ν, Q2) = F (x, Q2) , Im T (ν, Q2) = F (x, Q2),
1 1 2 2
4M 4ν
e2 e2M
Im S (ν, Q2) = g (x, Q2) , Im S (ν, Q2) = g (x, Q2), (7)
1 4ν 1 2 4 ν2 2
where x ≡ Q2/2Mν, and where F ,F ,g ,g are the conventionally defined structure functions which
1 2 1 2
parametrizeinclusiveelectron-nucleonscattering. Theimaginarypartsoftheforwardscatteringamplitudes,
Eqs. (7), get contributions from both elastic scattering at ν = ν ≡ Q2/(2M) or equivalently x = 1, as well
B
as from inelastic processes above the pion threshold, corresponding with ν > ν ≡ m +(Q2 +m2)/(2M)
0 π π
or equivalently x < x ≡ Q2/(2Mν ). The elastic contributions are obtained as pole parts of the direct and
0 0
crossed nucleon Born diagrams. The latter are conventionally separated off the Compton scattering tensor
in order to define structure-dependent constants, such as polarizabilities. The Born terms are defined by
using the electromagnetic vertex for the transition γ∗(q)+N(p) → N(p+q) as given by
q
Γµ = F (Q2)γµ + F (Q2)iσµν ν , (8)
D P
2M
withF andF theDiracandPauliformfactorsofnucleonN,normalizedtoF (0) = e andF (0) = κ ,
D P D N P N
where e is the charge in units of e, and where κ is the anomalous magnetic moment in units of e/2M;
N N
4
σµν = (i/2)[γµ,γν]. ThischoiceoftheelectromagneticvertexensuresthattheBorncontributionsaregauge
invariant and leads to the following contributions:
α ν2 α Q2
TBorn = − em(F2 + B G2 ), TBorn = − em (F2 +τ F2),
1 M D ν2−ν2 +iε M 2 M ν2−ν2 +iε D P
B B
α Q2 α ν
SBorn = − em(F2 + F G ), SBorn = em F G , (9)
1 2M P ν2−ν2 +iε D M 2 2 ν2−ν2 +iε P M
B B
with G (Q2) = F (Q2)+F (Q2), and τ ≡ Q2/4M2. The Born contributions of Eq. (9) can be split into
M D P
non-pole and pole contributions in a dispersion relation framework. The pole contributions (also called
elastic contributions) can be immediately read off Eqs. (9). Their real parts are given by
α ν2 α Q2
ReTpole = − em B G2 , ReTpole = − em (F2 +τ F2),
1 M ν2−ν2 M 2 M ν2−ν2 D P
B B
α Q2 (cid:16) (cid:17) α ν2
ReSpole = − em F G , Re νSpole = em B F G , (10)
1 2M ν2−ν2 D M 2 2 ν2−ν2 P M
B B
B. Low-energyexpansions
Following Ref. [28], in order to obtain a low-energy expansion (LEX) in k = {ν,Q} for the forward VCS
amplitudes T ,T ,S , and S , we express them in terms of the B of Eq. (5):
1 2 1 2 i
T (ν,Q2) = α (cid:8)Q2B −4M2ν2B +Q4B −4MνQ2B (cid:9), (11a)
1 em 1 2 3 4
T (ν,Q2) = α 4M2Q2(cid:8)−B −Q2B (cid:9), (11b)
2 em 2 19
S (ν,Q2) = α M (cid:8)−4MνB +Q2[B +M(4B +2B )+4B ](cid:9), (11c)
1 em 7 8 10 21 18
(cid:26) Q2 (cid:27)
S (ν,Q2) = α M2 − B −2B +Mν(4B +2B )−Q2B , (11d)
2 em 6 17 10 21 12
2
wheretheB alsodependonν andQ2 forforwardkinematics. Wecannextusetheexpansionsink = {ν,Q}
i
established in [28]:
B = b +O(k2), (i = 1,2,3,8,10,18,19,21), (12)
i i,0
B = 2Mν(cid:2)b +O(k2)(cid:3), (i = 4,6,7,12,17), (13)
i i,1
where b and b are low-energy constants. As we are only interested in the lowest-order terms in k =
i,0 i,1
{ν,Q}, we obtain the following LEXs for Eqs. (11a)–(11d):
T (ν,Q2) = α (cid:8)Q2b −4M2ν2b +O(k4)(cid:9), (14a)
1 em 1,0 2,0
T (ν,Q2) = −α (cid:8)4M2Q2b +O(k4)(cid:9), (14b)
2 em 2,0
S (ν,Q2) = α M (cid:8)−8M2ν2b +Q2[b +M(4b +2b )+4b ]+O(k4)(cid:9), (14c)
1 em 7,1 8,0 10,0 21,0 18,0
S (ν,Q2) = α M3ν(cid:8)−4b +4b +2b +O(k2)(cid:9). (14d)
2 em 17,1 10,0 21,0
Of the eight coefficients appearing in Eqs. (14a)–(14d), six can be related to the scalar and spin dipole
polarizabilitiesasmeasuredinrealComptonscattering(RCS).Aspolarizabilitiesareconventionallydefined
by separating off the Born parts of the amplitudes, one splits the amplitudes into Born and non-Born parts
as T = TBorn+TnB, and analogously for the other three amplitudes. The Born parts are given by Eqs. (9).
1 1 1
5
The non-Born (nB) parts of six of the low-energy constants are then expressed in terms of polarizabilities:
1
bnB = β , (15a)
1,0 α M
em
1 1
bnB = − (α +β ), (15b)
2,0 α 4M2 E M
em
1 1
bnB = − γ , (15c)
7,1 α 8M2 0
em
1 1
bnB = (γ +γ ), (15d)
10,0 α 4M M1E2 E1M2
em
1 1
bnB = (γ −γ ), (15e)
17,1 α 4M M1E2 M1M1
em
1 1
bnB = − γ , (15f)
18,0 α 2 M1E2
em
where α (β ) are the electric (magnetic) dipole polarizabilities respectively, and γ , γ , γ ,
E M M1E2 E1M2 M1M1
γ are the lowest-order spin polarizabilities of the nucleon, which are related to the forward spin polar-
E1E1
izability γ as:
0
γ = −(γ +γ +γ +γ ). (16)
0 M1E2 E1M2 M1M1 E1E1
WenoticefromEqs.(14a)–(14b)andEqs.(15a)–(15b)thattheelectricandmagneticdipolepolarizabil-
ities measured in RCS fully determine the terms of order ν2 and Q2 in the LEXs of both VVCS amplitudes
T and T . In order to fully specify the LEXs for the spin-dependent forward VCS amplitudes S , and S ,
1 2 1 2
we need in addition the coefficients b and b . We next show how they can be related to two of the
8,0 21,0
generalized polarizabilities (GPs), determined from the (non-forward) VCS process
γ∗(q)+N(p) → γ(q(cid:48))+N(p(cid:48)), (17)
where the outgoing photon is real and carries a low momentum, i.e. q(cid:48)2 = 0 and q(cid:48) → 0.
TheVCSexperimentsatlowoutgoingphotonenergiescanalsobeanalyzedintermsofLEXs,asproposed
in Ref. [15]. The VCS tensor describing the process (17) has been split in Ref. [15] into a Born part, which
is defined as the nucleon intermediate state contribution using the γ∗γN vertex of Eq. (8), and a non-
Born part. The latter describes the response of the nucleon to the quasi-static electromagnetic field, due to
the nucleon’s internal structure. For the lowest-order nucleon-structure terms, one considers the response
linear in the energy of the produced real photon. The VCS tensor describing the process (17) can generally
be parametrized in terms of 12 independent amplitudes. In Ref. [28], a gauge-invariant tensor basis was
constructedsuchthatthenon-Borninvariantamplitudesarefreeofkinematicalsingularitiesandconstraints:
12
Mµν = (cid:88)f (q2,q·q(cid:48),q·P)ρµν, (18)
i i
i=1
wheretheexplicitexpressionforthetensorsρµν canbefoundinRef.[28]. Furthermoreinthelimitq(cid:48)2 = 0,
i
the 12 invariant amplitudes f are related with the invariants B of Eq. (5), describing the doubly-virtual
i i
Compton scattering process:
f = B , f = B , f = B ,
1 1 2 2 3 4
f = B , f = B −B , f = B ,
4 7 5 8 9 6 10
f = B , f = B +B , f = B ,
7 11 8 12 13 9 14
f = B , f = B , f = B +B , (19)
10 17 11 18 12 20 21
where the limit q(cid:48)2 = 0 is taken in the argument of the B .
i
The behavior of the non-Born VCS tensor at low energy (q(cid:48) → 0) but at arbitrary three-momentum q¯ of
thevirtualphoton,whichisconvenientlydefinedinthec.m.systemoftheγ∗N system,canbeparametrized
by six independent GPs [15, 28]. The GPs can be accessed in experiment through the eN → eNγ process;
6
see the reviews [5, 16] for more details. At lowest order in the outgoing photon energy, there are two spin-
independent GPs, denoted by P(L1,L1)0, P(M1,M1)0, and four spin GPs, denoted by P(L1,M2)1, P(M1,L2)1,
P(L1,L1)1, and P(M1,M1)1, which are all functions of Q2.1 In this notation, L stands for the longitudinal (or
electric) and M for the magnetic nature of the transition respectively. At Q2 = 0, four of the six GPs are
related to the polarizabilities from RCS as
√ √
3 3
α = −α √ P(L1,L1)0(0), β = −α √ P(M1,M1)0(0),
E em M em
2 2 2
√
3 3 3
γ = −α √ P(L1,M2)1(0), γ = −α √ P(M1,L2)1(0), (20)
E1M2 em M1E2 em
2 2 2
whereastheremainingtwoGPsvanishintherealphotonlimit,i.e. P(L1,L1)1(0) = 0,andP(M1,M1)1(0) = 0.
The GPs can be expressed through the non-Born (nB) parts of the invariant amplitudes f . Using the
i
shorthand notation,
f¯(Q2) ≡ fnB(q2 = −Q2,0,0), (21)
i i
together with Eq. (19), these expressions are [28]:
bnB = f¯(0) = −6MP(cid:48)(M1,M1)1(0), (22)
8,0 5
3 (cid:104) (cid:105) 1 1
bnB = f¯ (0) = P(cid:48)(M1,M1)1(0)−P(cid:48)(L1,L1)1(0) + γ . (23)
21,0 12 2 α 2M M1E2
em
In Eqs. (22, 23) we have introduced the notations for the slopes at Q2 = 0 of two GPs as:
(cid:12)
P(cid:48)(L1,L1)1(0) ≡ d P(L1,L1)1(Q2)(cid:12)(cid:12) , (24)
dQ2 (cid:12)
Q2=0
(cid:12)
P(cid:48)(M1,M1)1(0) ≡ d P(M1,M1)1(Q2)(cid:12)(cid:12) . (25)
dQ2 (cid:12)
Q2=0
We note that the lowest-order polarizabilities as measured through RCS together with the slopes at
Q2 = 0 of the two lowest-order GPs which themselves vanish at Q2 = 0, and thus require a measurement
through the VCS process, specify all low-energy constants appearing in the VVCS amplitudes of Eqs. (14a-
14d).
C. VVCSsumrules
HavingestablishedtheLEXsoftheforwarddoubleVCSamplitudesT ,T ,S andS ,wearereadytouse
1 2 1 2
the analyticity in ν, for fixed spacelike photon virtuality, i.e. Q2 ≥ 0. We distinguish two cases depending
on their symmetry under s ↔ u crossing, which flips the sign of ν: the amplitudes T ,T and S are even
1 2 1
functions of ν whereas S is odd. We will present the relations for the non-pole parts of the amplitudes,
2
Tnp(ν, Q2) = T (ν, Q2) − Tpole(ν, Q2) etc., i.e., the well-known pole amplitudes given by Eq. (10) are
1 1 1
subtracted from the full amplitudes.
1. Spin-independentamplitudeT
1
The dispersion relation for T requires one subtraction, which we take at ν = 0, in order to ensure
1
high-energy convergence :
ν2 (cid:90) ∞ 1 e2
ReTnp(ν, Q2) = ReTnp(0, Q2)+ P dν(cid:48) F (x(cid:48),Q2), (26)
1 1 2π ν(cid:48)(ν(cid:48)2−ν2) M 1
ν0
1Equivalently,theycanbeconsideredasfunctionsofq¯2 =Q2(1+τ);thisdefinitionisusedinRef.[15].
7
with x(cid:48) ≡ Q2/(2Mν(cid:48)). Because the non-pole amplitudes are analytic functions of ν, they can be expanded
inaTaylorseriesaboutν = 0withaconvergenceradiusdeterminedbythelowestsingularity,thethreshold
of pion production at ν = ν . Analogous to the low-energy expansion of RCS, the series in ν, at fixed value
0
of Q2, for forward double VCS takes the following form [5]:
Tnp(ν, Q2) = Tnp(0, Q2) + (cid:0)α (Q2)+β (Q2)(cid:1) ν2 + O(ν4), (27)
1 1 E M
The coefficients of the Taylor series of Eq. (27) follow by expanding the dispersion integrals as function of
ν. This yields a generalization of Baldin’s sum rule for the forward dipole polarizabilities [5]:
e2M (cid:90) x0
α (Q2)+β (Q2) = dx2xF (x, Q2), (28)
E M πQ4 1
0
wherex correspondswiththepionproductionthreshold. Wenextdiscussthesubtractionfunctionatν = 0,
0
Tnp(0, Q2),enteringthedispersionrelationofEq.(26). AlthoughingeneraltheQ2 behaviorofthisfunction
1
is unknown, one can express its behavior at low Q2 in terms of polarizabilities, see, e.g., Ref. [30]. We like
to emphasize that polarizabilities are conventionally defined by separating the Compton amplitudes into
Born and non-Born parts, with Born parts given by Eqs. (9). The non-Born part of T can then be read off
1
Eqs. (14a), (15a), (15b) as
TnB(ν,Q2) = (α +β )ν2+β Q2 +O(k4), (29)
1 E M M
with k = {ν,Q}. To obtain the low-energy expansion in k of the non-pole part Tnp entering Eq. (26), we
1
alsoneedtoaccountforthedifferencebetweentheBornandpoleparts,whichcanbeeasilyreadoffEq.(9)
as
α α α
TBorn(ν,Q2)−Tpole(ν,Q2) = − emF2 = − eme2 + eme (cid:104)r2(cid:105)Q2+O(Q4), (30)
1 1 M D M N 3M N 1
where (cid:104)r2(cid:105) is the squared Dirac radius of the nucleon. Combining Eqs. (29) and (30), one then obtains the
1
low-energy expansion of Tnp in both ν2 and Q2 as
1
α (cid:16)α (cid:17)
Tnp(ν,Q2) = − eme2 +(α +β )ν2+ eme (cid:104)r2(cid:105)+β Q2 +O(k4). (31)
1 M N E M 3M N 1 M
Consequently, the subtraction function at ν = 0, which enters the dispersion relations of Eq. (26), is given
up to terms of order O(Q4) by
α (cid:16)α (cid:17)
Tnp(0,Q2) = − eme2 + eme (cid:104)r2(cid:105)+β Q2 +O(Q4). (32)
1 M N 3M N 1 M
2. Spin-independentamplitudeT
2
For the amplitude T , which is even in ν, one can write down an unsubtracted DR in ν:
2
1 (cid:90) ∞ 1
ReTnp(ν, Q2) = P dν(cid:48) e2F (x(cid:48), Q2). (33)
2 2π ν(cid:48)2−ν2 2
ν0
The expansion of the amplitude T at small k = {ν,Q} can be read off Eqs. (14a, 15b) as 2
2
Tnp(ν,Q2) = (α +β )Q2+O(k4). (34)
2 E M
By evaluating Eq. (33) at ν = 0, taking its derivative with respect to Q2 at Q2 = 0, and using the relation
(cid:20) (cid:21) (cid:20) (cid:21)
1 1 1
F (x,Q2) = 2xF (x,Q2) = σ , (35)
Q2 2 Q2 1 e2π T
Q2=0 Q2=0
with σ the total (real) photon absorption cross section, one recovers the Baldin sum rule [4], i.e. Eq. (28)
T
evaluated at Q2 = 0, for (α +β ).
E M
2FortheamplitudeT ,thereisnodifferencebetweentheBornandpolecontributions,asseenfromEq.(9).
2
8
3. Spin-dependentamplitudeS
1
We next discuss the DR for the spin-dependent amplitude S . The amplitude S is even in ν, and the
1 1
unsubtracted DR for its non-pole part reads
1 (cid:90) ∞ ν(cid:48) e2
ReSnp(ν, Q2) = P dν(cid:48) g (x(cid:48), Q2). (36)
1 2π ν(cid:48)2−ν2 ν(cid:48) 1
ν0
The low-energy expansion in ν, at fixed value of Q2, for Snp takes the form [5]
1
(cid:20) (cid:18) (cid:19) (cid:21)
2α 2α
Snp(ν, Q2) = em I (Q2)+ em I (Q2)−I (Q2) +Mδ (Q2) ν2 + O(ν4), (37)
1 M 1 M Q2 TT 1 LT
where the leading term of O(ν0) follows from Eq. (36) as
2M2 (cid:90) x0
I (Q2) = dxg (x, Q2). (38)
1 Q2 1
0
Using Eqs. (9) and (14c) one obtains the low-energy theorem result : Snp(0, 0) = −α κ2 /(2M), which
1 em N
yieldstheGDHsumruleforrealphotons[2,3],I (0) = −κ2 /4. Theν2-dependenttermintheexpansionof
1 N
Eq.(37)involves,besidesI ,alsothemomentI ofthehelicitydifferencecrosssectionsandalongitudinal-
1 TT
transverse polarizability δ , which are expressed through moments of spin structure functions as [5]
LT
2M2 (cid:90) x0 (cid:26) 4M2 (cid:27)
I (Q2) = dx g (x, Q2) − x2g (x, Q2) , (39)
TT Q2 1 Q2 2
0
δ (Q2) = 4e2M2 (cid:90) x0 dxx2 (cid:8)g (x, Q2) + g (x, Q2)(cid:9). (40)
LT πQ6 1 2
0
At Q2 = 0, the ν2 term in the low-energy expansion of Snp can be read off Eqs. (14c) and (15c), yielding 3
1
2α (cid:18) κ2 (cid:19)
Snp(ν, 0) = em − N +M γ ν2 + O(ν4), (41)
1 M 4 0
whereγ istheforwardspinpolarizabilityasaccessedinRCS,whichcanbeobtainedastheQ2 → 0limitof
0
the integral obtained in Ref. [5]:
4M2e2 (cid:90) x0 (cid:26) 4M2 (cid:27)
γ (Q2) = dxx2 g (x, Q2) − x2g (x, Q2) . (42)
0 πQ6 1 Q2 2
0
WecanderiveanewsumrulebyperformingaTaylorseriesinQ2 atν = 0forSnp. ByexpandingI (Q2)
1 1
in Eq. (37), we obtain
2α (cid:26) κ2 (cid:27)
Snp(0, Q2) = em − N +Q2I(cid:48)(0)+O(Q4) , (43)
1 M 4 1
(cid:12)
where I1(cid:48)(0) ≡ dQd2I1(Q2)(cid:12)(cid:12)(cid:12) is the Q2 slope at Q2 = 0 of the first moment of the structure function g1.
Q2=0
Using the low-energy expansion of Eq. (14c), we can identify the Q2-dependent term of the non-Born part
SnB at ν = 0 as
1
SnB(0,Q2) = α MQ2(cid:8)4M bnB +4bnB +bnB +2M bnB (cid:9)+O(Q4),
1 em 10,0 18,0 8,0 21,0
(cid:26) (cid:27)
1 (cid:104) (cid:105)
= α MQ2 γ −3M P(cid:48)(M1,M1)1(0)+P(cid:48)(L1,L1)1(0) +O(Q4), (44)
em E1M2
α
em
(cid:12)
3Notethatthisimpliestherelation[5]: IT(cid:48)T(0)−I1(cid:48)(0)= 2Mαe2m (γ0−δLT),withIi(cid:48)(0)≡ dQd2Ii(Q2)(cid:12)(cid:12)(cid:12) ,andδLT ≡δLT(0).
Q2=0
9
where in the last line we have used Eqs. (15d), (15f), (22), (23) for the corresponding low-energy coeffi-
cients. To relate I(cid:48)(0) with the expression in Eq. (44), we need to account for the difference between Born
1
and pole parts, which can be read off Eq. (9) as
α 2α (cid:26) κ2 κ2 (cid:27)
SBorn(ν,Q2)−Spole(ν,Q2) = − emF2 = em − N + N(cid:104)r2(cid:105)Q2+O(Q4) , (45)
1 1 2M P M 4 12 2
where (cid:104)r2(cid:105) is the nucleon mean squared Pauli radius. Combining Eqs. (43), (44), and (45) then allows
2
us to derive a new sum rule relating the slope at Q2 = 0 of the GDH sum rule to the Pauli radius and
polarizabilities as measured in RCS and VCS:
κ2 M2 (cid:26) 1 (cid:104) (cid:105)(cid:27)
I(cid:48)(0) = N(cid:104)r2(cid:105)+ γ −3M P(cid:48)(M1,M1)1(0)+P(cid:48)(L1,L1)1(0) . (46)
1 12 2 2 α E1M2
em
WeliketoemphasizethatallquantitiesenteringEq.(46)areobservablequantities: thelhsisobtainedfrom
thefirstmomentofthespinstructurefunctiong [13,14],whereastherhsinvolvesthePauliradiusaswell
1
as spin polarizabilities measured through the RCS and VCS processes. Therefore the sum rule of Eq. (46)
provides us with a model-independent and predictive relation. In the next sections, we will test this new
GDHsumruleforfinitephotonvirtualityusingheavy-baryonaswellascovariantbaryonchiralperturbation
theory. We will also provide a phenomenological evaluation based on available data.
4. Spin-dependentamplitudeS
2
Finally, for the second spin-dependent forward double VCS amplitude S , which is odd in ν, an unsub-
2
tracted DR takes the form
ν (cid:90) ∞ 1 e2M
ReS (ν, Q2) = ReSpole(ν, Q2) + P dν(cid:48) g (x(cid:48), Q2). (47)
2 2 2π ν(cid:48)2−ν2 ν(cid:48)2 2
ν0
If we further assume that the amplitude S converges faster than 1/ν for ν → ∞, we may write an unsub-
2
tracted dispersion relation for the amplitude νS , which is even in ν,
2
Re(cid:2)νS (ν, Q2)(cid:3) = Re(cid:2)νS (ν, Q2)(cid:3)pole + 1 P (cid:90) ∞ dν(cid:48) 1 e2M g (x(cid:48), Q2). (48)
2 2 2π ν(cid:48)2−ν2 2
ν0
If we now multiply Eq. (47) by ν and subtract it from Eq. (48), we obtain the Burkhardt-Cottingham (BC)
“superconvergence sum rule” [31], valid for any value of Q2:
(cid:90) 1
g (x, Q2)dx = 0, (49)
2
0
provided that the integral converges for x → 0. Notice that the upper integration limit in the integral of
Eq.(49)extendsto1,andthusincludestheelasticcontribution. Byseparatingtheelasticandinelasticparts
in the integral of Eq. (49), the BC sum rule can be cast into the equivalent form
2M2 (cid:90) x0 1
I (Q2) ≡ g (x, Q2)dx = F (Q2)G (Q2). (50)
2 Q2 2 4 P M
0
The BC sum rule was shown to be satisfied in quantum electrodynamics by an explicit calculation at lowest
order in the coupling constant α [32]. In perturbative QCD, the BC sum rule was verified for a quark
em
target to first order in α [33]. Furthermore, in the non-perturbative domain of low Q2, the BC sum rule
s
was also verified within heavy-baryon chiral perturbation theory [34, 35].
The LEX of (νS )np can be expressed as [5]
2
(cid:2)νS (ν, Q2)(cid:3)np = 2α I (Q2) + 2αem I(3)(Q2)ν2 + O(ν4), (51)
2 em 2 Q2 2
10
where the observable I(3)(Q2) is defined through the third moment of the spin structure function g as
2 2
8M4 (cid:90) x0
I(3)(Q2) ≡ dxx2g (x, Q2), (52)
2 Q4 2
0
= I (Q2)−I (Q2), (53)
1 TT
and where the last line has been obtained by using Eqs. (38) and (39). Note that the slope at Q2 = 0 of
I(3)(Q2) follows from footnote 3 as
2
M2
(3)(cid:48)
I (0) = (δ −γ ). (54)
2 2α LT 0
em
Using the low-energy expansion of Eq. (14d), we can identify the ν2 dependent term of the non-Born part
of νSnB as
2
νSnB(ν,Q2) = α M3ν2(cid:8)4bnB −4bnB +2bnB (cid:9)+O(ν4,ν2Q2), (55)
2 em 10,0 17,1 21,0
TorelateEqs.(51)and(55),weneedtoaccountforthedifferencebetweenBornandpoleparts,whichcan
be read off Eq. (9) as
νSBorn(ν,Q2)−(cid:2)νS (ν,Q2)(cid:3)pole = αemF (Q2)G (Q2), (56)
2 2 2 P M
andpreciselyaccountsfortheleadingtermofO(ν0)inEq.(51),asgivenbytheBCsumrule,Eq.(50). The
terms of O(ν2) in Eqs. (51) and (55) can then be identified to yield the new sum rule:
I(3)(cid:48)(0) = M3(cid:8)2bnB −2bnB +bnB (cid:9). (57)
2 10,0 17,1 21,0
On the rhs of Eq. (57), the low-energy quantities bnB , bnB , and bnB are related to polarizabilities as
10,0 17,1 21,0
measured in RCS and VCS through Eqs. (15d), (15e), and (23). In this way we obtain the following sum
rule:
M2 (cid:26) 1 (cid:104) (cid:105)(cid:27)
I(3)(cid:48)(0) = − [γ +γ ]+3M P(cid:48)(M1,M1)1(0)−P(cid:48)(L1,L1)1(0) . (58)
2 2 α 0 E1E1
em
Using Eq. (54), the sum rule of Eq. (58) can be expressed equivalently as
(cid:104) (cid:105)
δ = −γ +3Mα P(cid:48)(M1,M1)1(0)−P(cid:48)(L1,L1)1(0) . (59)
LT E1E1 em
NotethatsimilartoitscounterpartofEq.(46),allquantitieswhichenterEq.(59)areobservables. Therefore
the new sum rule of Eq. (59) provides us with a second model-independent and predictive relation among
low-energy spin structure constants of the nucleon.
III. VERIFICATIONINHEAVY-BARYONCHIRALPERTURBATIONTHEORY
In this section we verify the new GDH sum rule of Eq. (46) for finite photon virtuality as well as the
sumruleofEq.(59)forδ withinthecontextofheavy-baryonchiralperturbationtheory(HBχPT).Forthe
LT
purposeofthisverification,wewillexpressthetwosumrulesofEqs.(46)and(59)equivalentlyasrelations
for the GPs as
1 (cid:26) 2 (cid:18)κ2 (cid:19) 1 (cid:27)
P(cid:48)(M1,M1)1(0) = N(cid:104)r2(cid:105)−I(cid:48)(0) + (γ +γ +δ ) , (60)
6M M2 12 2 1 α E1M2 E1E1 LT
em
1 (cid:26) 2 (cid:18)κ2 (cid:19) 1 (cid:27)
P(cid:48)(L1,L1)1(0) = N(cid:104)r2(cid:105)−I(cid:48)(0) + (γ −γ −δ ) . (61)
6M M2 12 2 1 α E1M2 E1E1 LT
em
In HBχPT, the leading order (LO) contribution in the chiral expansion of both of these sum rules corre-
spondswithtermswhichareproportionalto1/m2,withm thepionmass. Thenext-to-leadingorder(NLO)
π π