Elastic pp scattering at the LHC from an empirical...
Transcript of Elastic pp scattering at the LHC from an empirical...
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Elastic pp scattering at the LHC from an empiricalstandpoint
Daniel A. Fagundes∗
Universidade Estadual de Campinas - IFGW
XXV RETINHA
05 de Fevereiro de 2014
∗ in collaboration with A. Grau, S. Pacetti, G. Pancheri and Y.N. Srivastava
1 / 23
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Outline
Differential elastic cross section at LHC - data vs theory
Barger-Phillips empirical amplitude
Modified BP amplitudes - correcting the small t behaviour
Model predictions for LHC8 and LHC14, asymptotic behaviourand geometric scaling
2 / 23
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Elastic pp scattering at LHCDifferential elastic cross section at
√s = 7 TeV∗
Diffractive events measured by TOTEM
processes mediated by colorless exchange → quantum numbers preserved in the final state∗
Left: G. Antchev, et al., Europhys.Lett. 101 (2013) 21002. Right: Jan Kaspar, Doctoral Thesis, CERN-Thesis-2011-214.3 / 23
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Elastic pp scatteringfrom ISR to LHC
Diffractive pattern at high-energies revealed with scanned t
)2
|t| (GeV
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
)-2
/d|t|
(mbG
eVel
σd
-710
-610
-510
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-310
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-110
1
10
210
310
AMBROSIO, PL B 115 (1982) 495
BREAKSTONE, NP B (1984) 248
BREAKSTONE, PRL 54 (1985) 2180
AMALDI, NP B 166 (1980) 301
ANTCHEV, EPL 95 (2012) 41001
ANTCHEV, CERN-PH-EP-2012-239
AMBROSIO, PL B 115 (1982) 495
BREAKSTONE, NP B (1984) 248
BREAKSTONE, PRL 54 (1985) 2180
AMALDI, NP B 166 (1980) 301
ANTCHEV, EPL 95 (2012) 41001
ANTCHEV, CERN-PH-EP-2012-239
pp
(s)totσ| ~ 1/dip|t
)2
|t| (GeV
0 0.5 1 1.5 2 2.5
)-2
/d|t|
(mbG
eVel
σd
-510
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1
10
210
310
CERN SPS - UA4
TEVATRON - CDF
TEVATRON - E710
TEVATRON - D0
pp
one dip present in the pp channel and a shoulder in p̄p
4 / 23
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Elastic differential cross sectionat LHC7 and model predictions
Failure of representative models†
Transition from soft to (semi) hard domain unclear
†Left: G. Antchev et al., Europhys.Lett. 95 (2011) 41001. Right: A.A. Godizov, PoS (IHEP-LHC-2011) 005.
5 / 23
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Physics of the ‘dip-shoulder’ regiondelicate cancelation between C = ±1 amplitudes
The dip/shoulder occur through cancellations in elastic amplitude due to t−channel processes:
App,p̄p(s, t) =A+(s, t)± A−(s, t)
2
A±(s, t) are even/odd amplitudes corresponding to C = ±1 exchanges in the t-channel. InRegge Phenomenology, they are called “Pomeron” and “Odderon” terms, which can betranslated into QCD (LO) language as 2g -exchange‡ and 3g -exchange§. Eventually, thenonleading contribution of secondary Reggeons and Pomeron cuts make their relative phaseφ 6= π.
‡F.E. Low, Phys.Rev. D12 (1975) 163 and S. Nussinov, Phys.Rev.Lett. 34 (1975) 1286§
A. Donnachie and P.V. Landshoff, Z.Phys. C2 (1979) 55
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Parametrizing the elastic differential cross-sectionBarger-Phillips amplitude and observables
ABPel (s, t) = i[√
A(s)eB(s)t/2︸ ︷︷ ︸leading : C=+1
+ eiφ(s)√
C(s)eD(s)t/2︸ ︷︷ ︸non−leading : C=±1
]
↓ at t = 0
σtot (s) = 4√π(√
A(s) +√
C(s) cosφ)
σel (s) =A(s)B(s)
+ C(s)D(s)
+ 4
√A(s)C(s)
B(s)+D(s)cosφ
dσeldt
∣∣∣t=0
= A(s) + C(s) + 2√
A(s)C(s) cosφ
↓ at t 6 0
dσeldt
= A(s)eB(s)t + C(s)eD(s)t + 2√
A(s)C(s)e(B(s)+D(s))t/2 cosφ
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A simple Regge-model for the non-leading termfor constant average φ
A standard C = +1 state contribution to the elastic amplitude follows
A(+)R (s, t) = iC+(1
s)[(
se−iπ/2
so)]α+(t),
with α+(t) is the positive signature trajectory and for a C = −1 state,
A(−)R (s, t) = C−(1
s)[(
se−iπ/2
so)]α−(t),
with α−(t) a negative signature trajectory. Having α+(t) = α−(t), degenerate trajectories,linear and assuming that α±(0) = 1 - the critical Pomeron - their total contribution follows
AR (s, t) = iC+ − iC−
soetα′
(ln(s/so )−iπ/2).
DefiningC+ = so Ccosφ; C− = −so Csinφ,
one gets
AR (s, t) = iCetα′
(ln(s/so )−iπ/2)e iφ.
Apart from the “extra” phase e−itα′π/2, this corresponds to the second term of the
Barger-Phillips amplitude - the “C-term” - with D(s) = 2α′ln(s/so ). Notice that this phase is
present in any Regge amplitude.8 / 23
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Parametrizing the elastic differential cross-sectionthe original Barger-Phillips model¶
We have applied the old Barger-Phillips parametrization to LHC7 data
ABPel (s, t) = i [√
A(s)e−B(s)|t|/2 +√
C(s)e iφ(s)e−D(s)|t|/2]
)2|t| (GeV
0 0.5 1 1.5 2 2.5
)-2
/d|t
| (m
bG
eV
el
σd
-510
-410
-310
-210
-110
1
10
210
ANTCHEV, EPL 101 (2013) 21002
ANTCHEV, EPL 101 (2013) 21002
BP FIT - 5 parameters
-2 15 mbGeV±A = 237
-2 0.21 GeV±B = 15.19
-2 0.084 mbGeV±C = 1.364
-2 0.049 GeV±D = 4.687
0.0084 rad± = 2.7599 φ
/DOF = 106/73 = 1.42χ
2 = 0.38 GeV
min|t|
1.5 1.6 1.7 1.8 1.9 2-4
10
-310
BP fit-8
/d|t| ~ |t|el
σTOTEM fit: d
excellent description for |t| > 0.4 GeV2 → need to correct small −t behaviour¶
Phillips and Barger, Phys.Lett. B46 (1973) 412 ; Grau et al., Phys.Lett. B714 (2012) 70
9 / 23
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Parametrizing the elastic differential cross-sectionmodified BP model - mBP1
Our first attempt - introduction of a square root threshold at small |t| (normalized)‖:
AmBP1el (s, t) = i [√
A(s)e−B(s)|t|/2e−γ(s)(√
4m2π+|t|−2mπ) +√
C(s)e iφ(s)e−D(s)|t|/2]
)2|t| (GeV
0 0.5 1 1.5 2 2.5
)-2
/dt (
mb
GeV
el
σd
-510
-410
-310
-210
-110
1
10
210
310
ANTCHEV, EPL 101 (2013) 21002
ANTCHEV, EPL 101 (2013) 21002
-2 2.0 mbGeV±A = 565.3
-2 0.16 GeV±B = 13.69
-2 0.036 mbGeV±C = 0.969
-2 0.033 GeV±D = 4.425
-1 0.060 GeV± = 2.005 γ
0.0068 rad± = 2.7030 φ
/DOF = 502/155 = 3.22χ
0 0.05 0.1 0.15 0.2 0.25 0.31
10
210
310
2 = 0.0117 GeV
min|t|
= 101.2 mbtot
σ
-2 = 523.9 mbGeV
t=0/dt|
elσd
)2|t| (GeV
0 1 2 3 4 5 6 7 8 9 10
)-2
/d|t
| (m
bG
eV
el
σd
-1910
-1710
-1510
-1310
-1110
-910
-710
-510
-310
-110
10
210
AMALDI, NP B 166 (1980) 301
AMBROSIO, PL B 115 (1982) 495
AMOS, NP B 262 (1985) 689
BREAKSTONE, NP B 248 (1984) 253
BREAKSTONE, PRL 54 (1985) 2180
FFBP fits
24 GeV
31 GeV
-2x10
45 GeV
-4x1053 GeV
-6x1063 GeV
-8x10
‖the two − pion loop insertion in the Pomeron trajectory: A.A. Anselm and V.N. Gribov, Phys.Lett. B 40 (1972); Fiore et
al. Int.J.Mod.Phys. A24 (2009) 2551
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Parametrizing the elastic differential cross-sectionmodified BP model - mBP2
We correct the small −t behaviour with the proton’s FF at the BP amplitude ∗∗
AmBP2el (s, t) = i [√
A(s)e−B(s)|t|/21(
1 + |t|t0
)4 +√C(s)e iφ(s)e−D(s)|t|/2]††
)2|t| (GeV
0 0.5 1 1.5 2 2.5
)-2
/dt
(mb
Ge
Vel
σd
-510
-410
-310
-210
-110
1
10
210
310
ANTCHEV, EPL 95 (2012) 41001
ANTCHEV, CERN-PH-EP-2012-239
-7.8/d|t| ~ |t|
elσTOTEM FIT: d
OBP fit
FFBP fit
-2 2.3 mbGeV±A = 565.3 -2 0.19 GeV±B = 8.24
-2 0.072 mbGeV±C = 1.372 -2 0.042 GeV±D = 4.661
0.0081 rad± = 2.7552 φ2 0.013 GeV± = 0.689
0t
/DOF = 383/155 = 2.52χ
0 0.05 0.1 0.15 0.2 0.25 0.31
10
210
310
2 = 0.0117 GeV
min|t|
= 100.3 mbtot
σ
-2 = 515.1 mbGeV
t=0/dt|
elσd
)2|t| (GeV
0 1 2 3 4 5 6 7 8 9 10
)-2
/d|t
| (m
bG
eV
el
σd
-1910
-1710
-1510
-1310
-1110
-910
-710
-510
-310
-110
10
210
AMALDI, NP B 166 (1980) 301
AMBROSIO, PL B 115 (1982) 495
AMOS, NP B 262 (1985) 689
BREAKSTONE, NP B 248 (1984) 253
BREAKSTONE, PRL 54 (1985) 2180
FFBP fits
24 GeV
31 GeV
-2x10
45 GeV
-4x10
53 GeV
-6x1063 GeV
-8x10
Fp(t) to account for elastic rescatterings as |t| increases ↔ proton does not break up∗∗
D.A. Fagundes et al., Phys.Rev. D88 (2013) 094019††
for√
s > 7TeV we make the ansatz t0 → 0.71 GeV2 (the EM FF scale) 11 / 23
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Asymptotic sum rulesand impact parameter structure
Asymptotic sum rules - total absorption of partial waves (η(s, b)→ 0)
SR1 =1
2√π
∫ 0−∞
dtAIel (s, t)→ 1 SR0 =1
2√π
∫ 0−∞
dtARel (s, t)→ 0
b (fm)
0 0.5 1 1.5 2 2.5 3
(s,b
)e
lA
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(LHC7)el
Re A
(ISR53)el
Re A
(LHC7)el
Im A
(ISR53)el
Im A
s(LHC7) = 0.953
1SR
(ISR53) = 0.7171
SR
(LHC7) = 0.0480
SR(ISR53) = 0.049
0SR
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Energy dependendence of fit parametersamplitudes and slopes with t0 → 0.71 GeV2
(GeV)s
210
310
410
510
(m
b)
Aπ
4
40
60
80
100
120
140
160
180
s [mb]2 = 47.8 -3.81lns+0.398lnA(s)π4
pp: ISR and LHC7
p: SPS and TEVATRONp
(LHC7) = 105.2 mbAπ4
(LHC8) = 108.0 mbAπ4
(LHC14) = 120.2 mbAπ4
(AUGER57) = 155.4 mbAπ4
(GeV)s
210
310
410
510
)-2
B (
Ge
V
0
2
4
6
8
10
12
14 s2B(s) = -0.23+0.028ln
in B(s) at lower energies0
Effect of t
pp: ISR and LHC7
p: SPS and TEVATRONp
-2B(LHC7) = 8.55 GeV
-2B(LHC8) = 8.82 GeV
-2B(LHC14) = 10.0 GeV
-2B(AUGER57) = 13.2 GeV
(GeV)s
210
310
410
510
(m
b)
Cπ
4
-110
1
10
[mb] s
31.18 + 0.001ln
s3
9.56 - 1.81lns + 0.0103ln = C(s)π4
pp: ISR and LHC7
p: SPS and TEVATRONp
(LHC7) = 5.18 mbCπ4
(LHC8) = 5.30 mbCπ4
(LHC14) = 5.76 mbCπ4
(AUGER57) = 6.71 mbCπ4
(GeV)s
210
310
410
)-2
D (
GeV
1.5
2
2.5
3
3.5
4
4.5
5
5.5
D(s) = -0.41 + 0.29lns
pp: ISR and LHC7
p: SPS and TEVATRONp
-2D(LHC7) = 4.66 GeV
-2D(LHC8) = 4.74 GeV
-2D(LHC14) = 5.06 GeV
-2D(AUGER57) = 5.85 GeV
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Energy evolution of the dip positionusing Geometric Scaling (GS)
We assume GS is valid asymptotically, thus
−tdipσtot ∼ constant −→ tdip ' −a
1 + b ln2 s
(GeV)s
210
310
410
510
)2
(G
eV
dip
t
-1.4
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
; b = 0.00972 s] - a = 2.15 GeV2 = -a/[1+b*lndip
t
= 1.47α from PLB 718 (2013) 1571 - dip
t
= 1.52α from PLB 718 (2013) 1571 - dip
t
ISR31
ISR45
ISR53
ISR63
SPS546
FNAL1960
LHC7
2(LHC8) = -0.51 GeVdip
t
2(LHC14) = -0.45 GeVdip
t
14 / 23
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This model predictionsfor LHC8 and LHC14
Asymptotic SR dictate the energy behaviour of fit parameters, allowing to make predictions
)2
|t| (GeV
0 0.5 1 1.5 2 2.5
)-2
/d
|t| (m
bG
eV
el
σd
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-310
-210
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1
10
210
310
-2 = 543.7 mbGeV
t=0/d|t||
elσd
-2(t=0) = 20.8 GeVeffB = 103.1 mb
totσ
= 26.6 mbel
σ
2 = 0.511 GeVdip
|t|
-2 = 541.8 mbGeV
t=0/d|t||
elσd
-2(t=0) = 20.8 GeVeffB
= 103.0 mbtot
σ
= 26.4 mbel
σ
2 = 0.495 GeVdip
|t|
= 2.72 rad]φLHC8 - FFBP [
= 2.81 rad]φLHC8 - FFBP [
)2
|t| (GeV
0 0.5 1 1.5 2 2.5
)-2
/d
|t| (m
bG
eV
el
σd
-510
-410
-310
-210
-110
1
10
210
310
-2 = 674.5 mbGeV
t=0/d|t||
elσd
-2(t=0) = 22.0 GeVeffB = 114.9 mb
totσ
= 31.0 mbel
σ
2 = 0.471 GeVdip
|t|
-2 = 671.1 mbGeV
t=0/d|t||
elσd
-2(t=0) = 22.0 GeVeffB
= 114.6 mbtot
σ
= 30.8 mbel
σ
2 = 0.452 GeVdip
|t|
= 2.76 rad]φLHC14 - FFBP [
= 2.92 rad]φLHC14 - FFBP [
Uncertainty in φ specifies the band of predictions
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This model predictionsfrom ISR to AUGER and asymptotia
(GeV)s10
210
310
410
510
to
tσ
/
el
σ
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
ppAccelerator data:
ppAccelerator data:
AUGER57 - PRL 109 (2012) 062002
= 2.92 rad] φ(s) - FFBP prediction [ tot
σ/el
σ
Interpolating line from ISR to LHC7
ISR
TO
TE
M
E710,C
DF
UA
4,U
A4/2
Black Disk Limit
(GeV)s
1 102
103
104
105
106
107
108
109
1010
10
to
tσ
/
el
σ
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
σel
σtot−→ 1/2 at
√s ' 1010GeV (Elab ∼ 1020GeV)
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Dip position, the Black Disk limit and Geometrical scalingtwo scales at non-asymptotic energy
No scaling with σtot (s) and central opacity, Rel = σel/σtot , plays a role in GS at LHC energies
(GeV)s
210
310
410
510
)2
(m
bG
eV
to
tσ
dip
= -
tto
tτ
20
30
40
50
60
70
80
90
100
Black Disk Limit
2 = 35.92 mbGeVBD
τ → tot
τ
(GeV)s
210
310
410
510
)2
(m
bG
eV
to
tσ
el
σd
ip =
-t
new
el
τ
0
10
20
30
40
50
60
Black Disk Limit
2/2 = 25.4 mbGeV2BD
τ → newel
τ
2 5.0 mbGeV±(LHC8) = 26.6 newel
τ
2 5.0 mbGeV±(LHC14) = 26.7 newel
τ
Rel 6= 1/2 at present energies ⇒ influence of two scales with different energy behaviour, σel (s)and σtot (s), through the geometric average cross section σ̄ =
√σel (s)σtot (s)
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Summary
What have we learned from this simple model?
1. the Barger-Phillips amplitude dissects the differential cross section in building blocks: diff.peak, dip region and tail
2. when augmented by the proton FF, the BP amplitude reproduce data from ISR to LHC →giving σtot (s), σel (s),Bel (s)
3. the dip structure arising from the interference of two terms with a relative phase (mixingof C = ±1 processes)
4. the first term (leading) is well understood, A(s) giving σtot (s) and B(s)+t0(s) giving theforward slope;
5. the second term (nonleading) carries an energy dependece through the slope D(s), whichrequires deeper understanding
6. sum rules in impact parameter space and asymptotic theorems → hints towards energydependence of parameters → predictions for LHC8 and LHC14
7. Geometric scaling with σtot achieved at asymptotic energies and at LHC two scales, σeland σtot still present
18 / 23
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Acknowledgements
THANK YOU!!!
19 / 23
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BackupDifferential elastic cross section at
√s = 8 TeV‡‡
‡‡from Jan Kašpar talk “Total, elastic and diffractive cross sections with TOTEM”, CERN, December 4th, 2012
20 / 23
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Backupparameters of the modified BP model mBP1
However, the new term does not behave as expected, with γ(s) ∼ ln s...
(GeV)s
210
310
410
0
0.2
0.4
0.6
0.8
1
1.2
B2π4m
D2π4mγπ2m
...instead it ‘swings’ with increasing c.m. energy → interpretation fails21 / 23
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Backupmodified BP model mBP2 applied to p̄p data
As an empirical formula, it can be applied for the crossed channel just as well
)2|t| (GeV
0 0.5 1 1.5 2 2.5
)-2
/dt
(mb
GeV
el
σd
-510
-410
-310
-210
-110
1
10
210
310
-2 0.84 mbGeV±A = 217.32 -2 0.058 GeV±B = 5.073
-2 0.023 mbGeV±C = 0.134 -2 0.13 GeV±D = 3.42
0.015 rad± = 2.650 φ
(fixed)2 = 0.76 GeV0
t
/DOF = 524/181 = 2.92χ
Bernard - PLB 198 (1987) 583
Battiston - PLB 127 (1983) 472
Bozzo - PLB 155 (1985) 197
0 0.05 0.1 0.15 0.2 0.25 0.31
10
210
2 = 0.01014 GeV
min|t|
0.11 mb± = 63.76 tot
σ
-2 0.69 mbGeV± = 207.92
t=0/dt|
elσd
)2|t| (GeV
0 0.5 1 1.5 2 2.5
)-2
/dt
(mb
GeV
el
σd
-510
-410
-310
-210
-110
1
10
210
310
Abe - PRD 50 (1994) 5518
Amos - PLB 247 (1990) 127
Abazov - PRD 86 (2012) 012009
-2 2.1 mbGeV±A = 319.4 -2 0.058 GeV±B = 5.684
-2 0.10 mbGeV±C = 1.00 -2 0.065 GeV±D = 4.308
0.019 rad± = 2.704 φ
(fixed)2 = 0.73 GeV0
t
/DOF = 226/88 = 2.62χ
0.05 0.1 0.15 0.2 0.25 0.31
10
210
2 = 0.034 GeV
min|t|
0.55 mb± = 75.02 tot
σ
-2 4.2 mbGeV± = 288.0
t=0/dt|
elσd
22 / 23
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Backuplocal and forward slopess
Beff (s) =d
dtln
(dσel
dt
)∣∣∣∣t=0
)2|t| (GeV0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Lo
cal S
lop
es
-5
0
5
10
15
20
25 (LHC7)FFBPeffB(LHC7)SQRTBPeffB
(ISR53)FFBPeffB(ISR53)SQRTBPeffB
Amaldi 1971Barbiellini 1972Baksay 1978Ambrosio 1982Antchev 2012
(GeV)s
210
310
410
)-2
(G
eV
eff
B
12
14
16
18
20
22
-2
(LHC7) = 19.8 GeVeffB-2
(LHC8) = 20.1 GeVeffB-2
(LHC14) = 21.3 GeVeffB-2
(AUGER57) = 24.5 GeVeffB
s 2(s) = 11.04 + 0.028lneffB
Forward slope data from ISR to LHC7
pp: ISR and LHC7
p: SPS and TEVATRONp
interaction radius “speed up” at LHC - Beff (s) ∼ ln2 s
23 / 23