The constraint on the spin dependent structure function $g_1$ at low $Q^2$ through the sum rule corresponding to the moment at $n=0$

0.03 MB·English

by  Susumu Koretune

The constraint on the spin dependent structure
2
function g at low Q through the sum rule
1
corresponding to the moment at n = 0
Susumu Koretune
7
0 DepartmentofPhysics,ShimaneUniversity,Matsue,Shimane,690-8504,Japan
0
2
Abstract. Thesumrulesforthespindependentstructurefunctiongab inthenull-planeformalism
n 1
correspondingtothemomentatn=0hasbeentransformedtothe sumrulewhichrelatesthegab
a 1
J withthecrosssectionoftheisovectorphotonortherealphoton.Basedonthesesumrules,weargue
thatthereisadeepconnectionamongtheelastic,theresonance,andthenon-resonantcontributions,
8
2 and that it explainswhy the sign of the generalized Gerasimov-Drell-Hearnsum changesat very
smallQ2.

1
Keywords: sumrule,polarizedstructurefunction
v
PACS: 24.85.+p.11.55.Hx,13.60.Hb,24.70.+s
6
3
2
1 INTRODUCTION
0
7
The fact that the sign of the Gerasimov-Drell-Hearn(GDH) sum rule[1, 2] and that of
0
/ theEllis-Jaffesum rule[3] was different had motivatedthe studyof thesesum rules and
h
p thespinstructurefunctionsg1 and g2 at lowQ2.Experimentally,thesignoftheG P(Q2)
- defined as
p
2m2
he Ip(Q2)= pG P(Q2), (1)
Q2
:
v
i 1
X G P(Q2)= dxgp(x,Q2), (2)
Z 1
r 0
a
where the I (0) is known to be negative through the GDH sum rule was studied at
P
CLAS[4], andwasshownthatitchanged asignintheverysmallQ2 region.Furtherthe
largenegativecontributioninthisregionwasshowntobecomesmallquicklyaswegoto
the Q2 near 1 (GeV/c)2. In this talk we show that this rapid change is tightly connected
totherapidchangeoftheelasticcontributionin thisregion[5,6].
THE SUM RULE IN THE ISOVECTOR REACTION
The sum rule for the gab is derived from the current commutation relation on the null-
1
planeforthegood-badcomponent[J+(x),Ji(0)] [7], and givenas
a b |x+=0
1 dx 1 ¥
g[ab](x,Q2)= f da [A5(a ,0)+a A¯5(a ,0)], (3)
Z0 x 1 −16 abcZ ¥ c c
−
where A5(a ,0) and A¯5(a ,0) is the matrix element of the bilocal current, and gab is
c c 1
defined as
1
Wab = d4xexp(iqx)< p,s Ja(x) Jb(0) p,s> (4)
mn |spin 4p Z | m · n | c |spin
= ie mnls ql ss Ga1b+ie mnls ql (n ss q sps )Ga2b, (5)
− ·
withgab=n Gab andgab=n 2Gab.SincetherighthandsideofEq.(3)isQ2 independent,
1 1 2 2
weobtainfortheanti-symmetriccombinationwith respect toa,b
1 dx 1dx
g[ab](x,Q2)= g[ab](x,Q2). (6)
Z x 1 Z x 1 0
0 0
Now the Regge theory predicts as g[ab] b x a (0) with a (0) 0, and hence the sum
1 ∼ − ≤
rule is convergent. However, the perturbative behavior like the DGLAP is divergent.
The double logarithmic (log(1/x))2 resummation gives more singular behavior than
the Regge theory[8] and the sum rule is also divergent. Though, whether the sum rule
diverges or not can not be judged rigorously by these discussions, it is desirable to
discuss the regularization of the sum rule and gives it a physical meaning even when
the sum rule is divergent. Now, the regularization of the divergent sum rule has been
known to be done by the analytical continuation from the nonforward direction[9]. We
first derive the finite sum rule in the small but sufficiently large t region by assuming
| |
themovingpoleorcut.Thenwesubtractthesingularpieceswhichwemeetaswegoto
the smaller t from both hand sides of the sum rule by obtaining the condition for the
| |
coefficient of the singular piece. After taking out all singular pieces we take the limit
t 0. Because of the kinematical structure in the course to derive the sum rule, we
| |→
can mimic this procedure in the forward direction by introducing the parameter which
reflects the t in the non-forward direction. The sum rule obtained in this way can be
transformed to the form where the high energy behavior from both hand sides of the
sum rule is subtracted away. Practically, if the cancellation at high energy is effective,
sincetheconditionisneededonlyinthehighenergylimit,weconsiderthatthesumrule
holds irrespective of the condition. In this way, we subtract the high energy behavior
frombothhand sidesofEq.(6).
NowwetakeQ2 =0 and usetherelation
0
1 1
Gab(n ,0)= s ab (n ) s ab (n ) = D s ab(n ). (7)
1 −8p 2a { 3/2 − 1/2 } −8p 2a
em em
Then we define n Q = m E where E = E +Q2/2m and E is a energy of the
c p Q Q c p
real(virtual) photon in the laboratory frame. By setting a = (1+i2)/√2,b = a†, and
usingthesamemethodasintheCabibbo-Radicati sumrule[10],weobtainthesumrule
which relates the g and the cross section of the isovector photon by separating out the
1
elasticcontributionas
1 dx
[2g1/2(x,Q2) g3/2(x,Q2)] (8)
Zxc(Q) x 1 − 1
= B(Q2) mp EcdE[2D s 1/2 D s 3/2]+K(E ,Q2),
−8p 2a em ZE0 − c
wherex (Q2)= Q2 and
c 2n cQ
1 1 Q2
B(Q2)= (m m ) G+(Q2)[G+(Q2)+ G+(Q2)] , (9)
4{ p− n −1+Q2/4m2 M E 4m2 M }
p p
G+(Q2)=Gp(Q2) Gn(Q2), G+(Q2)=Gp (Q2) Gn (Q2), (10)
E E − E M M − M
¥ dE m ¥
K(E ,Q2)= [2g1/2(x,Q2) g3/2(x,Q2)] p dE[2D s 1/2 D s 3/2],
c −ZEQ E 1 − 1 −8p 2a em ZEc −
(11)
withgI and D s I beingthequantitiesin thereaction
1
“isovectorphoton + proton statewithisospinI.“
−→
It should be noted that, because of the reguralization of the sum rule, we are allowed to
take the integral over E in K(E ,Q2) after the subtraction. Since K(E ,Q2) is expected
c c
tobesmall,wehavetherelationamongtheelastic,theresonance,and thenon-resonant
contributions.
THE SUM RULE IN THE ELCTROPRODUCTION
The current anticommutation relation on the null-plane was derived using the DGS
representation[11, 12, 13, 14]. It should be noted that this relation is not the operator
relation ant that it holds only for the matrix element between the stable one particle
hadronicstate.Thesumrulein thiscase takestheform
1dx 1 ¥
gab(x,Q2)= d da ln a S5(a ,0)+a S¯5(a ,0)) , (12)
Z0 x 1 −8p abcZ ¥ | |{ c c }
−
1 dx 1dx
gp(x,Q2)= gp(x,Q2), (13)
Z x 1 Z x 1 0
0 0
whereS5(a ,0)andS¯5(a ,0)isdefinedsimilarlyasA5(a ,0)andA¯5(a ,0).Byseparating
c c c c
out the Born term, and using the same method as in the current commutator case we
obtain
1 dxgp(x,Q2)=B(Q2) mp EcdE s g p s g p +K(E ,Q2), (14)
Zxc(Q2) x 1 −8p 2a em ZE0 { 3/2− 1/2} c
where
1 1 Q2
B(Q2)= m Gp (Q2)[Gp(Q2)+ Gp (Q2)] , (15)
2{ p−1+Q2/4m2 M E 4m2 M }
p p
K(E ,Q2)= mp ¥ dE s g p s g p ¥ dEgp(x,Q2). (16)
c 8p 2a em ZEc { 1/2− 3/2}−ZEcQ E 1
Similar sum rule can be derived in case of the neutron target. However it should be
noted that, in this case, the corresponding term of the m in the Born term becomes
p
0 m =0sinceGn(0)=0.
n E
×
Now, usingparameter in [15], we find that K(2,Q2) below Q2 =0.5(GeV/c)2 is very
small.Further, by theexperimentaldatafrom GDHcollaboration[16], wefind
mp 2dE s g p s g p 0.45, (17)
8p 2a em ZE0 { 3/2− 1/2}∼
with the possible error about 20%. We use the standard dipole fit for the Sachs form
factor and find that, in the small Q2 region near Q2 0.1(GeV/c)2, the integral
∼
1 dx
gp(x,Q2) changes a sign. The integral is effectively cut off by the same
Zxc(Q2) x 1
W2 = (p+q)2 both in the real and the virtual photon. This means that the same
1 dx
resonances contribute to the integral gp(x,Q2) both in the real and the virtual
Zxc(Q2) x 1
photon and hence this integral measures the change of the resonances and the back-
groundin thevery smallQ2 region.Thusitschange istightlyrelated tothesignchange
oftheGDH suminthisregion.
CONCLUSION
We find, in the small Q2 region near Q2 0.1(GeV/c)2, that the integral
∼
1 dx
gp(x,Q2) becomes zero and that it changes a sign from the negative to
Zxc(Q2) x 1
thepositive.Thisbehavioriscausedbytherapidchangeoftheresonancestogetherwith
the continuum to compensate the rapid change of the elastic to satisfy the sum rule. It
is this rapid change of the resonances which gives the sign change of the GDH sum.
Henceweseewhyitoccurs in thevery smallQ2 region.
REFERENCES
1. S.D.DrellandA.C.Hearn,Phys.Rev.Lett.16,908(1966).
2. S.B.Gerasimov,YadernFiz.2,598(1965)[Sov.J.Nucl.Phys.2,430(1966)]
3. J.EllisandR.L.Jaffe,Phys.Rev.D9,1444(1974);10,1669(E)(1974).
4. R.Fatemietal,Phys.Rev.Lett.91,222002(2003).
5. S.Koretune,Phys.Rev.C73,058201(2006);74,049902(E)(2006).
6. S.Koretune,Phys.Rev.C72,045205(2005);74,059901(E)(2006).
7. D.A.Dicus,R.Jackiw,andV.L.Teplitze,Phys.Rev.D4,1733(1971).
8. B.Badelek,ActaPhys.Polon.B34,2943(2003).
9. S.P.deAlwis,Nucl.Phys.B43,579(1972).
10. N.CabibboandL.A.Radicati,Phys.Lett.19,697(1966).
11. S.Deser,W.Gilbert,andE.C.G.Sudarshan,Phys.Rev.115,731(1959).
12. S.Koretune,Phys.Rev.21,820(1980).
13. S.Koretune,Prog.Theor.Phys.72,821(1984).
14. S.Koretune,Phys.Rev.D47,2690(1993).
15. S.Simula,M.Osipenko,G.RiccoandM.Taiuti,Phys.Rev.D65,034017(2002).
16. DutzH,etal.(GDHCollaboration).Phys.Rev.Lett.91,192001(2003).