Asymptotic High Energy Total Cross Sections and Theories with Extra Dimensions

0.33 MB·English

by  J. Swain

Asymptotic High Energy Total Cross Sections
and Theories with Extra Dimensions
J. Swain and A. Widom
Physics Department, Northeastern University, Boston MA USA
Y.N. Srivastava
Physics Department & INFN, University of Perugia, Perugia IT
Therateatwhichcrosssectionsgrowwithenergyissensitivetothepresenceofextradimensions
inarathermodel-independentfashion. Weexaminehowrateswouldbeexpectedtogrowifthereare
more spatial dimensions than 3 which appear at some energy scale, making connections with black
holephysicsandstringtheory. Wealsoreviewwhatisknownaboutthecorrespondinggeneralization
of the Froissart-Martin bound and the experimental status of high energy hadronic cross sections
which appear to saturate it up to the experimentally accessible limit of 100 TeV. We discuss how
3 extra dimensions can be searched for in high energy cross section data and find no room for large
1 extra dimensions in present data. Any apparent signatures of extra dimensions at the LHC would
0 seem to have to be interpreted as due to some other form of new physics.
2

n PACSnumbers: 11.10.Jj,11.10.Kk,11.25.-w,11.25.Wx,11.55.-m
a
J
2 I. INTRODUCTION of order unity at a distance
2
(cid:126)S(E)
Thereisawealthofpossibleconstraintsonnewphysics R= . (2)
h] which can be obtained without the need to look for spe- 2µckB
p cific exclusive processes which may be difficult to detect
(iii) The total cross section for shadow scattering is then
- and distinguish from backgrounds simply by looking at
p given by σ =2πR2 yielding
how cross sections grow with energy. We review simple total
e
h arguments for how cross sections are expected to grow π (cid:20) (cid:126) (cid:21)2(cid:20)S(E)(cid:21)2
[ with energy, and connect them with black hole thermo- σ (E)= . (3)
dynamics and string-theoretical models. Allowing for total 2 µc kB
3
a number of space dimensions greater than 3 we find,
v
3 in agreement with earlier discussions of the Froissart- The asymptotic E → ∞ total cross section is thereby
5 Martin bound in higher dimensions, much faster growth determined by the entropy S(E).
5 ofcrosssectionswithenergythanisallowedbyunitarity A typical entropy estimate may be made via the fol-
2 in 3+1 dimensions. Since the experimental results satu- lowing reasoning: The equipartition theorem for a gas
4. rate the unitarity bounds, we find there is no room for of ultra-relativistic particles implies a mean particle en-
0 extra dimensions[1] at scales below 100 TeV and point ergyvaryinglinearlywithtemperature;(cid:15)≈c|p|≈3kBT.
1 out that the best way to search for such dimensions may A Boltzmann gas of such particles has a constant heat
1 well be via measurements of the energy dependence of capacity. A system with a constant heat capacity C∞
v: cross sections rather than via searches for exclusive pro- obeys
i cesses.
X dE dE
E =C T =C ⇒ dS =C , (4)
r ∞ ∞dS ∞ E
a
II. HIGH ENERGY TOTAL CROSS SECTION yielding an entropy logarithm
(cid:18) (cid:19)
A clear physical argument[2] for predicting the high E
S(E)=C ln . (5)
center of mass energy E total cross section is the follow- ∞ E
0
ing: (i)Iftheelasticscatteringamplitudeathighenergy
is dominated by the exchange of the lightest mass µ par- By virtue of Eqs.(3) and (5), the total cross section for a
ticle, then the probability of the exchange in space reads constant heat capacity system,
(cid:20) 2µcr S(E)(cid:21) π (cid:20)(cid:126)C (cid:21)2 (cid:18) E (cid:19)
P(r,E)∼exp − + , (1) σ (E)= ∞ ln2 , (6)
(cid:126) k total 2 µck E
B B 0
where 2µr/(cid:126) is the WKB factor for a particle to move saturates the Froissart-Martin bound[3, 4]. In a more
a space-like distance r and the entropy S(E) determines general thermodynamically stable situation, the entropy
the density of final states. (ii) The probability becomes S(E) is determined parametrically by the heat capacity
2
as a function of temperature Note that the energy,
C(T)= dE(T) =TdS(T) , E = MPc2 , (14)
dT dT MPc2 8πkBT
(cid:90) T
E = C(T(cid:48))dT(cid:48), decreasesasthetemperatureincreasesinviolationofthe
second law of thermodynamics, i.e. the black hole heat
0
(cid:90) T dT(cid:48) capacity C = dE/dT < 0. This instability is a common
S = C(T(cid:48)) . (7) feature of classical self-gravitationally bound systems.
T(cid:48)
0
The saturation Eqs.(5) and (6) will then hold true only
inthehighanergyandhightemperaturelimitofastable IV. HAGEDORN-STRING ENTROPY
heat capacity C(T →∞)=C .
∞
To compute the total high energy cross section for Theorieswithextradimensionsgrewoutofstringtheo-
models with extra dimensions, the central theoretical riesofinteractionswhichwouldincludeinprinciplegrav-
problem is to understand the entropy implicit in such itational theories. A classical picture of a rotating string
models. In so far as extra dimensional theories are can be developed. If one considers a string of length L,
thought to describe the gravitational interaction, the re- then the lowest resonance frequency obeys
sulting entropy may be expected to exhibit a second law
πc
violation. Let us first review this second law instability ω = . (15)
L
for the well known case of the black hole entropy.
Treatingtheresonantfrequencyω asanangularvelocity
conjugate to the string angular momentum J one finds
III. BLACK HOLE ENTROPY
dE
ω = (16)
dJ
The gravitational coupling strength of a black hole
with a mass M may be defined via where the energy is related to the string tension τ via
S
GM2 S E =τSL. (17)
α (M2)= = , (8)
G (cid:126)c 4πk
B Eqs.(15), (16) and (17) imply the differential equation,
whereinS istheblackholeentropy,E =Mc2istheblack EdE
hole energy and dJ = , (18)
πcτ
S
(cid:18) (cid:19)
G whose solution is the linear Regge trajectory,
S =4πk E2. (9)
B (cid:126)c5
J =(cid:126)(cid:0)α +α(cid:48)E2(cid:1). (19)
0
The black hole temperature is then
In Eq.(19), α is the Pomeranchuck intercept[5, 6] and
0
1 dS (cid:18) (cid:126)c (cid:19)
= ⇒ E =2πτ , (10) 1
T dE G kBT 2πτS = (cid:126)cα(cid:48) (20)
wherethegravitationalvacuumtensionτ isdetermined is the generalized string tension analog of the gravita-
G
by tional tension Eq.(11) which is now determined by the
Reggetrajectoryslopeparameterα(cid:48). Letusconsiderthe
2πτ = c4 ≈4.816×1042 Newton. (11) entropy consequences of these types of models.
G 8πG
The physical meaning of the vacuum gravitational ten-
A. Hagedorn Temperature
sion becomes evident if the Einstein field equations for
the energy-pressure tensor are written in the form
The canonical partition function for a system in an
(cid:20) 1 (cid:21) environmental temperature T˜ has the form
T =2πτ R − g R . (12)
µν G µν 2 µν Z(T˜)=(cid:88)Ω(E)exp(−E/k T˜), (21)
B
The Planck mass M is defined via Eq.(8) employing E
P
αG(MP2)=1, whereΩ(E)isthenumberofquantumstateswithenergy
E. In terms of the entropy
(cid:114)
(cid:126)c
M = ≈6.88×10−5gm≈3.86×1028eV/c2. (13)
P G S(E)=kBlnΩ(E) (22)
3
and the canonical free energy C. String Theory
F(T˜)=−k T˜lnZ(T˜), (23)
B The extra dimension value n for space-time is related
to the full dimension of space-time D via
one finds from Eqs.(21), (22) and (23) that
(cid:34) F(T˜)(cid:35) (cid:88) (cid:20)S(E) E (cid:21) D =3+1+n=4+n. (33)
exp − = exp − . . (24)
kBT˜ E kB kBT˜ The Hagedorn entropy for string theories is illustrated
by bosonic string example where the entropy can be
The Hagedorn string entropy has the form[7, 8] computed from the number theory integer partition
functions[9]. It is
(cid:18) (cid:19)
E E
S(E)= −k ηln +k γ, (25) (cid:115)
T B k T B S(E) (cid:20)D−2(cid:21)
H B H
=2πE α(cid:48)
k 6
wherein η and γ are dimensionless constants and T is B
the Hagedorn temperature. H (cid:18)D+1(cid:19) √
− ln( α(cid:48)E)
2
(cid:18) (cid:19) (cid:18) (cid:19)
D−1 D−2 ln2
B. Statistical Thermodynamics ln − , (34)
4 24 2
In order that the partition function in Eq. (21) con- which is clearly a special case of the general Hagedorn
verge, the environmental temperature T˜ must be less string entropy Eq.(25). For the supersymmetric string
than the Hagedorn temperature T . whichincludesfermions,thefunctionalformoftheHage-
H
dorn entropy in Eq.(25) is expected to remain intact.
T˜ <T , (26) However, the detailed dependence of the parameters
H
T ,η and γ on D is expected to depend on the specific
H
On the other hand, the string temperature T given by, string model chosen.
1 dS
= , (27)
T dE
V. COMPACT EXTRA DIMENSIONS
yields a positive energy
Let us suppose that n extra dimensions are compact
(cid:20) (cid:21) (cid:20) (cid:21)
T T with a gravitational length scale Λ. For distance r large
E =ηk T ≡Φ >0, (28)
B H T −T T −T on the length scale Λ, the Newtonian attraction is deter-
H H
mined by the gravitational string tension in Eq.(11);
Eq.(28) implies a string temperature larger than the
m m
Hagedorn temperature, −U =G 1 2
r
T >T , (29) 1 (cid:20)(m c2)(m c2)(cid:21)
H = 1 2
16π2 τ r
G
as well as a threshold for the string excitation energy 1 (cid:20)(m c2)(m c2)(cid:21)
= 1 2 (r (cid:29)Λ). (35)
E >Φ=ηkBTH, (30) 16π2 σnΛnr
Note the inequalities Thenatureofaddingncompactextradimensionsisillus-
trated schematically in FIG.1. For short distances, the
T˜ <T <T, (31) power law energy has an exponent
H
which imply an environmental temperature T˜ which is −U ∝(cid:20)(m1c2)(m2c2)(cid:21) (r (cid:28)Λ). (36)
lessthathestringtemperatureT. Thatathermodynam- σ r1+n
n
ically stable environment cannot come into equilibrium
with a string is due to the latter having negative heat Similar considerations apply for the string entropy
capacity with n extra dimansions each of length Λ. For a grav-
itational string one may begin with an entropy of the
(cid:20) (cid:21)
dE T Hagedorn form as given in Eq.(25); i.e. as E →∞
C = =−Φ H <0 (32)
dT (T −T )2
H (cid:20) (cid:21)
S (E) E
0 =(cid:36) √ +··· , (37)
which is a thermodynamic second law violation. k 0 (cid:126)cτ
B G
4
section in the high energy limit
π (cid:20) (cid:126) (cid:21)2
σ (s)= ×
total 2 µc
(cid:12)(cid:12) C (cid:18) s (cid:19) (cid:114)(cid:126)c (cid:12)(cid:12)2
(cid:12) ln +(cid:36) s(n+2)/4+···(cid:12) (39)
(cid:12)k s n σ (cid:12)
(cid:12) B 0 n (cid:12)
√
wherein the center of mass energy E =(cid:126)c s→∞. Be-
lowthethresholdfortheobservationofextradimensions
(cid:20) (cid:16)σ (cid:17)2/(2+n)(cid:21)
s (cid:28)s(cid:28) n ,
0 (cid:126)c
π (cid:20) C(cid:126) (cid:21)2 (cid:18) s (cid:19)
σ (s)= ln2 . (40)
total 2 k µc s
B 0
Above the threshold for the observation of extra dimen-
sions,
(cid:20)(cid:16)σ (cid:17)2/(2+n) (cid:21)
n (cid:28)s ,
FIG.1: Foraonedimenisionalstringoflengthr undergrav- (cid:126)c
itational tension, τ ≡ σ , the energy of stretching τ r en-
ters into Newton’sGgravit0ational law as in the denomGinator σ (s)= π(cid:36)n2(cid:126)c(cid:20) (cid:126) (cid:21)2s(n+2)/2. (41)
of Eq.(35). If the string has one extra dimension of length total 2σ µc
n
Λ(cid:28)r, then the stretching two dimensional membrane is de-
scribed by a surface tension σ = τ /Λ. If the string has The transition from Eq.(40) to (41) at the energy
1 G
twoextradimensionseachoflengthΛ(cid:28)r,thenthestretch-
ingthreedimensionalsquareprismisdescribedbyanegative (cid:16)σ (cid:17)1/(2+n)
pressureσ2 =τG/Λ2. Forthecaseofn≥1addeddimensions, Etransition ∼(cid:126)c (cid:126)nc
one finds σn =τG/Λn as in Eq.(35). (cid:16)τ (cid:17)1/(2+n) (cid:126)c
∼ G (42)
(cid:126)c [Λn]1/(n+2)
wherein (cid:36) is a dimensionless constant of order unity.
0 whichdependsonboththenumbernofextradimensions
With n extra compact dimensions
and the compact length scale Λ.
S (E) (cid:20) E (cid:21)(cid:18)E(cid:19)n/2
n =(cid:36) √ +···
k n (cid:126)cσ (cid:126)c
B n VII. THE FROISSART BOUND IN HIGHER
(cid:114)(cid:126)c (cid:18)E(cid:19)1+(n/2) DIMENSIONS
=(cid:36) +··· . (38)
n σ (cid:126)c
n
The discussions above have been quite physical, but
Thiscanbecomparedwiththestringscatteringresults one can also directly generalize the Froissart bound by
ofAmati,CiafaloniandVenezianowhofindforthetotal, repeating the original argument but now in more than 3
diffractiveandinclusivecrosssectionsσtot,σdif andσinel spatialdimensions. ThishasbeendonebyChaichianand
respectively Fisher[10] who find that cross sections must be bounded
D−2
σtot ∼sD−4 byhigherpowersofln(s)orevenaspowersofsmultiplied
σ ∼s by logarithms.
dif
σ ∼(ln(s))D−2
inel
athighenergies,whereD isthenumberofdimensionsof VIII. GENERIC EFFECTS OF HIGHER
spacetime. DIMENSIONS
We now discuss how extra compact dimensions have
an effect on the total hadron-hadron cross section in the A key observation of this paper is that if one assumes
linit of high center of mass energy. that there are extra dimensions of space, one generically
expects a qualitative change in how cross sections rise
with energy. That change is that they rise significantly
VI. TOTAL HADRONIC CROSS SECTION faster than one would expect in 3 spatial dimensions
as soon energies rise high enough that those extra di-
From the relationship between the total cross section mensions are accessible. This is an essentially model-
entropyinEqs.(3), (5)and(38),onefindsthetotalcross independent statement, and due physically to the fact
5
the adding dimensions opens up phase space for decays. asymptotic domain) may be written as
This is true whether or not the Standard Model mat-
s
ter particles are restricted to lie on some 3-dimensional σ(pp,pp¯)(s)=C [ln( )]2
“brane”[1]. As long as one can excite degrees of free- total 1 GeV2
s
domwhichseemorethan3spacedimensions,therateat + C [ln( )]
which cross sections grow will be higher than one would 2 GeV2
expect in 3. Note that such behavior could even be ex- +C3+..., (44)
pectedsimplyondimensionalgrounds,withthematerial
presented above being a clarification of this point. where the constants Ci, i = 1,2,3 are determined
from the data.
Nowitiswell-knownthathadroniccrosssectionssatu-
ratetheunitarityboundssoonecouldsearchforevidence
• GGPS[14]: Thisisanindependenttheoreticalanal-
of extra dimensions simply by search for energies above
ysis -developed over two decades- which is based
which cross sections grow faster than with energy than
on mini-jets, incorporating aspects of confinement
at lower dimensions. A detailed fit with two functional
and soft-gluon resummation. It has been quite
formsaboveandbelowE asdefinedabovecould
transition successful in obtaining a rather complete descrip-
be used to clarify the nature and number of such extra
tionoftotalcross-sectionsforpp,pp¯,πp,γp,γγ pro-
dimensions, should any evidence for them at all arise. cesses. This group finds the rise to be (ln(s/s ))q
o
However, as we show in the following section, extra di-
where 1 < q < 2, with a phenomenological value:
mensions seem to be ruled out up to scales of about 100
4/3<q <3/2.
TeV already from the present data.
Thus,allviablephenomenologicalanalysesoftotaland
√
inelastic cross-sections up to s = 100 TeV conclude
that
IX. CONSTRAINTS ON COMPACT
DIMENSIONS FROM HIGH ENERGY σ(s) → σ [ln(s/s )]q, with q ≤ 2. (45)
o o
CROSS-SECTION DATA
Eq.(45)isatcompleteoddswithEq.(41)evenforn = 1.
Hence,thereappearstobenoroomforanycompactified
The experimental situation on high-energy hadronic
dimensions whose thresholds are below 100 TeV.
cross-sections may be summarized as follows. High qual-
√
ity data exist up to s = 2 TeV for σ(pp¯)(s) from the
√ total
Tevatron and up to s = 0.2 TeV for σ(pp)(s). Less
total X. CONCLUSIONS
precise data from cosmic rays are available for σ(pp)(s)
√ total
up to s = 100 TeV. Recently, the ATLAS group We argue on general grounds, and via concrete calcu-
at LHC has released preliminary data on σ(pp) (s) at lations,thatifextraspacedimensionsareaccessibletoat
√ inelastic
s = 7 TeV. least some excitations produced in high energy hadronic
Therearethreereasonablycompletephenomenological collisions above some energy scale Ecritical one should
analyses of the above cross-section data which may be see an increase in the rate at which total cross sections
summarized as follows: rise. This suggests a general means to search for such
extra dimensions by looking at cross sections as a func-
tion of energy. Such searches are particularly attractive
• PDG[12]: They assume that σ (s) rises as since they are highly model-independent and largely un-
total
(ln(s/s ))2,themaximumriseallowedbytheFrois- affected by single events which might, due to statistical
o
sartbound. Theirparametrizationforalltheavail- fluctuations,seemtosupportsomemodelwithhigherdi-
able data reads mensions. Since the data so far apparently saturate the
Froissart-Martinboundin3+1dimensions,thereappears
σ(pp,pp¯)(s)=(0.308mb) [ln( s )]2 to be no room for any compactified dimensions whose
total GeV2 thresholdsarebelow100TeV.Thislimitcanbeextended
+(35.45mb)+..., (43) as data from ever higher energy collisions are obtained,
butforthenearfuturethiswillhavetocomefromcosmic
ray data. Any events which might seem to be signals of
the dots meaning non-leading terms vanishing at
extraspacetimedimensionsattheLHCwouldhavetobe
high energy. A χ2/d.o.f. = 1.05 -for all data
attributed to some other sort of new physics.
considered- is given.
Since this paper was written, Block and Halzen
[15](who have accepted our work with an aim to extend-
• BH[13]: BlockandHalzenalsofindstrongevidence ing it) have suggested that higher dimensions might be
thattheFroissartboundissaturated. Thatis,their ruled out to arbitrarily high energies via the same argu-
result for a generic hadronic cross-section (in the ment, but they use a Froissart bound which is tied to
6
3+1 dimensions, which as discussed above, is dimension- None of these publications changes the conclusions of
dependent. FurtherrecentdiscussionsincludethatofFa- our paper.
gundes, Menon and Silva [16, 17] (see also the commen-
tary by Block and Halzen[18]) who suggest that energy
dependence of hadronic total cross section at high ener-
XI. ACKNOWLEDGEMENTS
gies may be an open problem still). Block and Halzen
have also recently argued that the Auger cross section
datasupportstheideathatprotonasymptoticallydevel- J. S. thanks National Science Foundation for its sup-
ops into a disk[19]. port via NSF grants PHY-0855388 and PHY-1205845.
[1] SeeK.AgasheandA.Pomarol,NewJ.Phys.12075010 Shekhovtsova and Y.N. Srivastava, Total and inelastic
√
doi: 10.1088/1367-2630/12/7/075010 (2010) for a very cross-sections at LHC at s = 7 TeV and beyond,
short,butveryrecentreviewandreferencestherein.The arXiv:1102.1949/hep-ph(2011); ibid, Phys. Lett., B659,
literature is very large by now and growing rapidly. 137(2008)[arXiv:0708.3626];R.M.Godbole,A.Grau,G.
[2] R.J. Eden, High Energy Collisions of Elementary Parti- Pancheri,andY.N.Srivastava,Phys.Rev.,D72,076001
cles, Cambridge University Press, Cambridge (1967). (2005) , [hep-ph/0408355]; A. Grau, G. Pancheri, O.
[3] M. Froissart, Phys. Rev. 123, 1053 (1961). Shekhovtsova, and Y. N. Srivastava, Modeling pion and
[4] A. Martin, Phys. Rev. 129, 1432 (1963). proton total cross-sections at LHC, Phys.Lett., B693,
[5] I.Ya. Pomeranchuck, Sov. Phys. JETP 3, 306 (1956). 456 (2010), [arXiv:1008.4119]; GGPS, Total photopro-
[6] I.Ya. Pomeranchuck, Sov. Phys. JETP 7, 499 (1958). duction cross-section at very high energy, Eur.Phys.J.
[7] R. Hagedorn. Nuovo Cim. Suppl. 3, 147 (1965). C63, 69 (2009), [arXiv:0812.1065]; ibid, Minijets, soft
[8] R. Hagedorn. Nuovo Cim. 56A, 1027 (1968). gluon resummation and photon cross- sections, Nucl.
[9] G.H. Hardy and E.M. Wright, An Introduction to the Phys.Proc.Suppl.184,85(2008),[arXiv:0802.3367];ibid,
Theory of Numbers, Oxford University Press, London Soft Gluon k(t)-Resummation and the Froissart bound,
(1960). Phys.Lett., B682, 55 (2009) [arXiv:0908.1426]; A. Grau,
[10] M. Chaichian and J. Fischer, Nucl. Phys B303 557 G. Pancheri, and Y. Srivastava, Hadronic total cross-
(1988) and M. Chaichian and J. Fischer in “Gauge sections through soft gluon summation in impact pa-
Theories of Fundamental Interactions”, Proceedings of rameter space, Phys.Rev., D60, 114020 (1999), [hep-
the XXXII Semester in the Stefan Banach Interna- ph/9905228].
tional Mathematical Center, ed. M. Paw(cid:32)loweski and R. [15] M. M. Block and F. Halzen, arXiv:1201.0960.
Ra¸kczka, World Scientific, 1988. See also [11]. [16] D.A. Fagundes, M.J. Menon and P.V.R.G. Silva, Braz.
[11] D. Amati, M. Ciafaloni, and G. Veneziano, preprint J. Phys. 42, 452 (2012), arXiv:1112.4704 v5 [hep-ph].
S.I.S.S.A. 121 EP, Sep. 1988. [17] D.A. Fagundes, M.J. Menon and P.V.R.G. Silva,
[12] K. Nakamura et al. [Particle Data Group], J. Phys., G arXiv:1208.3456 v2 [hep-ph].
37, 075021 (2010). [18] M. M. Block and F. Halzen, arXiv:1208.3008 [hep-ph].
[13] M. Block, arXiv:1009.0313v1 [hep-ph]; M. Block and F. [19] M. M. Block and F. Halzen, arXiv:1208.4086 [hep-ph].
Halzen, arXiv:1102.3163v1 [hep-ph].
[14] A. Achilli, R.M. Godbole, A. Grau, G. Pancheri, O.