The new elastic scattering measurements of TOTEM --- are there hints for asymptotics?

0.07 MB·

by  S. M. Troshin

The new elastic scattering measurements of
TOTEM — are there hints for asymptotics?
7 Sergey M. Troshin, Nikolai E. Tyurin
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NRC“KurchatovInstitute”–IHEP
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r Protvino, 142281, RussianFederation
a
M [email protected]

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] Abstract
h
p
We point out to another indication of the black-disk limit exceeding in
-
p
hadroninteractionsfoundintherecentimpactparameteranalysisperformed
e
h bytheTOTEMCollaborationat√s=8TeVandemphasizethatthisobserva-
[ tionmightbeinterpreted asaconfirmation ofthereflectivescattering mode

2 appearance attheLHCenergies.
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The geometrical picture of hadron interactions is often based on the impact–
parameter dependence of the inelastic overlap function. However, such approach
isnotquitecomplete,i.e. elasticandtotaloverlapfunctionsaretobeconsideredin
line. Onthegroundofsuchconsiderationaslowgradualtransitiontotheemerging
at theLHC picture, where the interaction region starts to become reflective at the
center (b = 0) and simultaneously becomes relatively edgier, larger and black
at its periphery (we are using an acronym REL to denote this picture) has been
discussed in [1], where the references for the earlier papers can be found. The
transitiontothismodeseemstobeobservedbytheTOTEMexperimentunderthe
measurements ofthe dσ/dtin elasticpp–scattering. Thisis based on theanalysis
of the impact parameter dependences of the overlap functions performed in [2].
Thoseoverlapfunctionsenterunitarityrelation:
Imf(s,b) = h (s,b)+h (s,b). (1)
el inel
In Eq. (1) the function f(s,b) is the elastic scattering amplitude in the impact
parameter representationwhileh (s,b) istheelasticoverlapfunction,
el
2
h (s,b) = f(s,b) .
el
| |
The inelastic overlap function h (s,b) corresponds to the total contribution of
inel
all the inelasticprocesses. From Eq. (1) the following inequality for the real part
ofthescatteringamplitudeRef(s,b) isobtained[3]:
1 1
p1 4hinel(s,b) Ref(s,b) p1 4hinel(s,b). (2)
− 2 − ≤ ≤ 2 −
It should be noted that Eq. (1) is an approximate one due to the kinematical
constraints existing at finite energies. It is valid with an accuracy of (1/s) [4]
O
being a result of the Fourier-Bessel transform of the unitarity equation written in
s and tvariables:
ImF(s,t) = H (s,t)+H (s,t). (3)
el inel
Considering the limit s , the question on the limiting value for the scat-
→ ∞
teringamplitudecan beposed,isittheblackdisklimitortheunitaritylimit? The
well-known black-disk limit for the scattering amplitude f is reached when the
maximal absorption, h = 1/4, takes place. It is definition of this limit, which
inel
corresponds to the values Imf = 1/2 and Ref = 0 (cf. Eqs. (1) and (2)). The
impact-parameter analysis performed in [2] implies that the black-disk limit has
been overcomeat √s = 7 TeV.
In this comment we would like to point out that the most recent impact-
parameter analysis performed by TOTEM at √s = 8 TeV [5] is in favor of this
conclusionon exceedingtheblack-disklimit. Wedo notdiscusshere theparticu-
lar schemes oftheelasticscattering amplitudeunitarization,but it couldbe noted
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that the usual eikonal unitarization scheme meets problems with the black–disk
limit exceeding. Possible way to accommodate the situation is consideration of
thequasi-eikonalunitarizationscheme[6, 7].
Indeed, as it followsfrom Fig. 19 in[5], thevalueofh at b = 0 and √s = 8
el
TeV is 0.31 while the black-disk limiting value is 0.25. It is true for the central
b-dependence ofh withmaximumat b = 0.
el
Unfortunately, the impact parameter analysis performed in the paper [5] does
not account for therespectiveexperimentaldata error bars ofthedσ/dtmeasure-
menrs, but, the analysis [2] does and its error bars do not include the black disk
limitatb = 0.
As it was noted in [5], the peripheral dependence with maximalvalue of 0.05
atb = 1.2fmisalsoconsistentwiththedataatthisenergy. Theexistenceofthese
two rather different forms, central and peripheral, is due to uncertainty in the nu-
clearphasechoicefortheelasticscatteringamplitude(cf. e.g. [5]). Theformand
role of the nuclear phase is essential in the CNI (Coulomb-Nuclear Interference)
region of very small values of t. However, the peripheral form is at variance
−
withReggeand geometricalmodelsfortheelasticscattering.
Moreover, the further observations can be made. The slope parameter, an
experimentallyobservedquantity,B(s),
d dσ
B(s) ln −t=0,
≡ dt dt|
is determinedbytheaveragevalue b2 , where[8]:
tot
h i
∞
2 2
b totσtot(s) = X b n(s)σn(s), (4)
h i h i
n=2
hereσ (s)isthen–particleproductioncross–section. Thisrelationshowshowthe
n
slope B(s) is constructed from the individual elastic and inelastic contributions.
Theexplicitfunctionalenergydependenciesoftheupperboundsforthefunctions
∞ 3
b dbh (s,b)
2 R0 el,inel
b (s) =
el,inel ∞
h i bdbh (s,b)
R0 el,inel
have been obtained in [9]. They follow from the bound on b2 and have the
tot
h i
forms:
4
C s
2 4
b (s) 32π ln , (5)
el
h i ≤ σel(s) s0
4
C s
2 4
b (s) 8π ln . (6)
inel
h i ≤ σinel(s) s0
3
The two above bounds assume similar energy dependence in case when both
2
σ (s) and σ (s) have also similar s-dependence, say ln s. It corresponds
el inel
to the case of the black-disc limit saturation and the uppe∝r bounds for b2 and
el
b2 are bothproportionalto ln2s underthisscenario. h i
inel
h i
However, it is not the case when the elastic and inelastic cross-sections have
2
different energy dependencies at s , e.g. σ ln s, while σ lns.
el inel
→ ∞ ∝ ∝
Such dependencies of elastic and inelastic cross-sections are typical for the uni-
tarity limit saturation when Imf 1 and Ref 0 at s . In this case
the upper bounds for b2 and b→2 would als→o have diff→eren∞t energy depen-
el inel
h i h i 2
dencies, the former one would be ln s while the latter one is proportional to
3 ∝
ln s. Suchfunctionaldifferencecan beconsideredasananotherqualitativeissue
in favorofacentral form forh and aperipheral oneforh .
el inel
It is worth to note here that the above two scenarios, corresponding to the
black-disk limit or unitarity limit saturation have been discussed, in particular, in
[10]and thenewdataseemsto behelpfulintheirdiscriminating.
Conclusion on central or peripheral forms of the overlap functions can also
be made on the grounds of unitarity and analyticity of the scattering amplitude
in the Lehmann–Martin ellipse. It is a straightforward consequence of the above
general properties of the elastic scattering that the ratio of the overlap functions
h (s,b)/h (s,b) is decreasing with b like a linear exponent e−µb at large
el inel
∼
values of b and fixed high energy value. It follows from a similar decreasing
behaviorofanelasticscatteringamplitude[11]. Thus,ratherexoticsituationwith
acentral formofh andaperipheraloneofh isat variancewiththeresultsof
inel el
unitarityandanalyticity(underanaturalassumptionofamonotonousdependence
oftheoverlapfunctionsin theregionoflargevaluesofb).
The existence of the more central character of elastic scattering compared to
theimpactparameterdistributionofthetotalprobabilityoftheinelasticprocesses
isknownforalongtimeandisinagreement withCERN ISRdata[12], inpartic-
ular, itwas predicted at smallb valuesat theLHCenergies (cf. e.g. [13, 14]).
It should be noted that peripheral b–dependence of h with maximum at
inel
b = 0 (i.e. with the fall down at small b) was depicted and discussed in [12] for
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the energy √s = 3 TeV. Currently, various interpretations ([15, 16, 17]) of this
effect havebeen proposed.
So, reasonably leaving aside the peripheral option, one can conclude that the
twoindependentimpactparameteranalysisoftheTOTEMdata[2,5]at theLHC
energies √s = 7 TeV and 8 TeV indicate an existence of the transition to the
reflectivescatteringmode[18]relevantto theasymptoticpicture.
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Acknowledgements
WearegratefultoM.Deile,J.Kaspar,E.MartynovandV.Petrovfortheinterest-
ing discussionsand correspondence.
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