Elastic pp scattering at the LHC from an empirical...

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

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

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

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

-410

-310

-210

-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


AMALDI, NP B 166 (1980) 301

ANTCHEV, EPL 95 (2012) 41001


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

-410

-310

-210

-110

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

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

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

6 / 23

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

7 / 23

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

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

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


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

10 / 23

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


-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


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

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

12 / 23

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


-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


(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


-2D(LHC7) = 4.66 GeV

-2D(LHC8) = 4.74 GeV

-2D(LHC14) = 5.06 GeV

-2D(AUGER57) = 5.85 GeV

13 / 23

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

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

-510

-410

-310

-210

-110

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

|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|



Uncertainty in φ specifies the band of predictions

15 / 23

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)

16 / 23

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)

17 / 23

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

Acknowledgements

THANK YOU!!!

19 / 23

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

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

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

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



(AUGER57) = 24.5 GeVeffB

s 2(s) = 11.04 + 0.028lneffB

Forward slope data from ISR to LHC7

pp: ISR and LHC7


interaction radius “speed up” at LHC - Beff (s) ∼ ln2 s

23 / 23

Elastic pp scattering at the LHC from an empirical...

Documents

Transcript of Elastic pp scattering at the LHC from an empirical...