compton scattering from the proton, deuteron, and neutroncompton scattering from the proton,...

Compton Scattering from the proton,deuteron, and neutron

DANIEL PHILLIPS

Department of Physics and Astronomy

Ohio University, Athens, Ohio

INT Mini-workshop on NN/NNN systems, October 2003 – p.1/39

Outline

Compton scattering on the proton in chiralperturbation theory for

An EFT for p scattering in the Delta region

Compton scattering on deuterium and the extractionof nucleon polarizabilities

Conclusions, Future Work, and ShamelessAdvertising


Chiral perturbation theory

constrained by (approximate)

of QCD.

Chiral perturbation theory is the most general consistentwith the symmetries of QCD and the pattern of their breaking, up toa given order in the small expansion parameter:

Unknown coefficients at a given order need to be determined.expansion employed: (usually) useful, not essential.

PT is:Model-independent;

Systematically improvable.

Many successful applications to A=1 at low energy. (Also .)

The : to include or not to include? ?Jenkins, Manohar, Hemmert, Holstein, . . .




of QCD.

Chiral perturbation theory is the most general

consistentwith the symmetries of QCD and the pattern of their breaking, up toa given order in the small expansion parameter:

Unknown coefficients at a given order need to be determined.

expansion employed: (usually) useful, not essential.



Many successful applications to A=1 at low energy. (Also .)





of QCD.







Many successful applications to A=1 at low energy. (Also

.)





of QCD.







Many successful applications to A=1 at low energy. (Also

.)

The

: to include or not to include?

?Jenkins, Manohar, Hemmert, Holstein, . . .


Power counting in PT

Rules:

for a vertex with powers of or :

;

for each pion propagator:

;

for each nucleon propagator:

;

for each loop:

;

Power counting for loops as well as for

,

: “naturalness”.


Nucleon Compton Scattering in PT

Powell X-Sn +

non-analyticity

from loops

Small expansion:

.Bernard, Kaiser, Meißner (1992)

PDG average:;

.


Nucleon Compton Scattering in PT

Powell X-Sn +

non-analyticity

from loops

Small expansion:

"! .

Bernard, Kaiser, Meißner (1992)

PDG average: # $ ;

% # % .


N

LO:

amplitude at

J. McGovern, Phys. Rev. C 63, 064608 (2001)

Short-distance physics via contact terms , with coefficients whichshould be fit to data:

Experiments: SAL/Illinois, LEGS, MAMI, . . . .Kinematic restriction ( -less PT): MeV.

Fits with explicit : R. Hildebrandt et al., nucl-th/0307070.


N

LO:

amplitude at

J. McGovern, Phys. Rev. C 63, 064608 (2001)

Short-distance physics via contact terms , with coefficients whichshould be fit to data:

Experiments: SAL/Illinois, LEGS, MAMI, . . . .Kinematic restriction (

-less PT):

MeV.

Fits with explicit

: R. Hildebrandt et al., nucl-th/0307070.


ResultsdΣdW lab

Ωlab

50 100 150 200

12

16

20

24Θlab=107ë

50 100 150 200

1520253035 Θlab=135ë

50 100 150 200

5101520 Θlab=60ë

50 100 150 200

12

16

20

24Θlab=85ë

/d.o.f.=170/131

S. R. Beane, J. McGovern, M. Malheiro, D. P., U. van Kolck, Phys. Lett. B, 567, 200 (2003).


ResultsdΣdW lab

Ωlab

50 100 150 200

12

16

20

24Θlab=107ë

50 100 150 200

1520253035 Θlab=135ë

50 100 150 200

5101520 Θlab=60ë

50 100 150 200

12

16

20

24Θlab=85ë

/d.o.f.=170/131

#

#

S. R. Beane, J. McGovern, M. Malheiro, D. P., U. van Kolck, Phys. Lett. B, 567, 200 (2003).


What’s in a name?

Washington-Arizona-Rio-Manchester-Ohio: WARMO

Fluminense-Ohio-Arizona-Manchester-INT: FOAMI

COMPTONComputing One Much-discussed Process That Occurs in Nature

Calculating the Outcome of Many Photons Targetted On a NucleusCollaboration Of Many Physicists: Theorists or Nincompoops?

Manchester-Athens-Fluminense-

INT-Arizona: MAFIA


What’s in a name?

Washington-Arizona-Rio-Manchester-Ohio: WARMOFluminense-Ohio-Arizona-Manchester-INT: FOAMI




INT-Arizona: MAFIA


What’s in a name?


COMPTON

Computing One Much-discussed Process That Occurs in Nature



INT-Arizona: MAFIA


What’s in a name?





INT-Arizona: MAFIA


What’s in a name?



Calculating the Outcome of Many Photons Targetted On a Nucleus

Collaboration Of Many Physicists: Theorists or Nincompoops?


INT-Arizona: MAFIA


What’s in a name?





INT-Arizona: MAFIA


Baldin Sum Rule

Βp

Αp

10.5 11 11.5 12 12.5 13

2.5

3

3.5

4

4.5

5

With Baldin Sum Rule constraint:


Cutoff dependence of p fit

200 MeV 180 MeV 165 MeV

140 MeV 120 MeV 100 MeV

1Σ contour example with HΧ2Lmin=1

9 10 11 12 13 14 15

0

1

2

3

4

5

6

9 10 11 12 13 14 15

0

1

2

3

4

5

6

9 10 11 12 13 14 15

0

1

2

3

4

5

6

9 10 11 12 13 14 15

0

1

2

3

4

5

6

9 10 11 12 13 14 15

0

1

2

3

4

5

6

9 10 11 12 13 14 15

0

1

2

3

4

5

6

9 10 11 12 13 14 15

0

1

2

3

4

5

6

9 10 11 12 13 14 15

0

1

2

3

4

5

6


Going higher for p

Breakdown scale set by first omitted degree of freedom:

isobar.

But how to count c.f. ?

“Small-scale expansion” (Hemmert, Holstein, et al.), count:

“ -expansion” (Pascalutsa and D.P.):

We should consider two distinct kinematic regions for p

scattering;

’s and ’s must be kept track of separately, then to get

overall counting index of graph set , .


Going higher for p


isobar.

But how to count c.f.

?


“ -expansion” (Pascalutsa and D.P.):


scattering;




Going higher for p


isobar.


?


“

-expansion” (Pascalutsa and D.P.):


scattering;




Going higher for p


isobar.


?


“

-expansion” (Pascalutsa and D.P.):


scattering;

’s and

’s must be kept track of separately, then to get

overall counting index of graph set

,

.


Power counting for

if

.

Diagrams with no ’s: count as in HB PT but with .

if .: loops with insertions from , loops too.

Counterterms: , .


Power counting for

if

.

Diagrams with no

’s: count as in HB PT but with

.

if .: loops with insertions from , loops too.

Counterterms: , .


Power counting for

if

.

Diagrams with no


.

if

.

: loops with insertions from , loops too.

Counterterms: , .


Power counting for

if

.

Diagrams with no


.

if

.

:

loops with insertions from

,

loops too.

Counterterms:

,

.


: O R diagrams

k k’

p

k k’

p

Diverges for

. Problem with all O

R diagrams.

Solution: Dyson equation

begins with

all terms Use dressed propagator.


: O R diagrams

k k’

p

k k’

p

Diverges for

. Problem with all O

R diagrams.

Solution: Dyson equation

begins with

all terms Use dressed

propagator.


Dressing the propagator

Treat

etc. in perturbation theory

Consistent couplings

Resum renormalized third-order self-energy


Consistent couplings

invariant under

has correct number of spin degrees of freedom

Spin-3/2 gauge invariance

k k’

p

k k’

p

Unphysical spin-1/2 degrees of freedom do not enter anyphysical amplitude


: power counting

LO and NLO O

R diagrams:

These + Thomson term define NLO calculation:

+

N LO, :

V. Pascalutsa and D. R. Phillips, Phys. Rev. C 67, 0552002 (2003).


: power counting

LO and NLO O

R diagrams:

These + Thomson term define NLO calculation:

+

N

LO,

:

V. Pascalutsa and D. R. Phillips, Phys. Rev. C 67, 0552002 (2003).


-expansion for p: Summary

V. Pascalutsa and D. R. Phillips Phys. Rev. C 67, 0552002 (2003).

, as in PT, pole graphs + pion loops LETs;

, dominated by dressed with energy-dependent width:

To describe amplitude includes all mechanisms

which are NLO in either region;

Tools of field theory: Ward-Takahashi identities, consistent

number of spin d.o.f., . . .

Power counting error estimates





, dominated by dressed

with energy-dependent width:

To describe amplitude includes all mechanisms









, dominated by dressed

with energy-dependent width:

To describe

amplitude includes all mechanisms






Results: I

Fit to p data from threshold to

MeVFree parameters:

MeV =2.81

Errors: estimate of N

LO effect

c.f. large- , , ;

Reference

NLO HB PT 12.2 1.2

NLO

NLO SSE 16.4 9.1

PDG average

Beane et al.

Large corrections to spin polarizabilities (especially ), Pascalutsa and D.P., PRC, in press.


Results: I


MeVFree parameters:

MeV =2.81


LO effectc.f. large-

,

,

;

Reference

NLO HB PT 12.2 1.2

NLO

NLO SSE 16.4 9.1

PDG average

Beane et al.

Large corrections to spin polarizabilities (especially ), Pascalutsa and D.P., PRC, in press.


Results: I


MeVFree parameters:

MeV =2.81


LO effectc.f. large-

,

,

;

Reference

NLO HB PT 12.2 1.2

NLO

NLO SSE 16.4 9.1

PDG average

Beane et al.

Large

corrections to spin polarizabilities (especially ), Pascalutsa and D.P., PRC, in press.INT Mini-workshop on NN/NNN systems, October 2003 – p.18/39

Results: II

0

10

20

30

dσ/d

Ωcm

s [n

b/sr

]

MAMI’01Born+π0

+Loop+Loop+∆

0

10

20

30

0

10

20

30

dσ/d

Ωcm

s [n

b/sr

]

0

10

20

30

0 90 180θcms [deg]

0

10

20

30

dσ/d

Ωcm

s [n

b/sr

]

0 90 180θcms [deg]

0

10

20

30

ωlab=59 MeV

89 MeV

99 MeV 108 MeV

118 MeV 126 MeV

0 90 180θcms [deg]

0

20

40

60

80

100

120dσ

/dΩ

cms [n

b/sr

]

0 90 180θcms [deg]

0

100

200

300

0

10

20

30

40

dσ/d

Ωcm

s [n

b/sr

]

MAMI’01SAL’93LEGS’01

0

10

20

30

40

0

10

20

30

40

50

dσ/d

Ωcm

s [n

b/sr

]0

10

20

30

40

50

ωlab=135 MeV

149 MeV

164 MeV 182 MeV

230 MeV 286 MeV


Results: III

0

50

100

150

200

250

300

dσ/d

Ωc.

m. [

nb/s

r]

SAL 93LEGS 97LEGS 01Born+π0

+πN loops+ ∆

0 100 200 300ωlab [MeV]

−1

−0.5

0

0.5

Σ

0

50

100

150

200

250

300

0 90 180θc.m. [deg]

−0.6

−0.3

0

0.3

θc.m.= 90o ωlab=286 MeV


Reactions on deuterium

: calculated from chiral NN potential, or from a potential model.

: also has a PT expansion. (Weinberg, van Kolck)

Description of observables which should be: model independent, sys-

tematically improvable, and accurate at low momentum/energy trans-

fer.



: calculated from chiral NN potential,

or from a potential model.




fer.







fer.







fer.INT Mini-workshop on NN/NNN systems, October 2003 – p.21/39

Naive dimensional analysis for

Rules:


;


;

for each

propagator:

;

for each loop:

;

!

for a two-body diagram:

! " #absent.

Loops, many-body effects, and vertices from etc.

suppressed by powers of .


Naive dimensional analysis for

Rules:


;


;

for each

propagator:

;

for each loop:

;

!

for a two-body diagram:

! " #absent.

Loops, many-body effects, and vertices from

etc.

suppressed by powers of .


Deuteron wave functions

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0r (fm)

0.00

0.05

0.10

0.15

0.20

w(r

) (f

m-1

/2)

Nijm93

OPEP (R=2.0 fm)

V (2)

of χPT (Λ=600 MeV)

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.00.0

0.1

0.2

0.3

0.4

0.5

0.6

u(r)

(fm

-1/2

)

Same at long distances:

,

,

,

, .

Some differences at

two-pion range.


Compton scattering on deuterium

Want to determine and . Naive idea:

INCORRECT

Possible to extract and from d d data, but

need to treat 2B effects SYSTEMATICALLY.


d experiments

Illinois (1994): M. Lucas, Ph.D. thesis,

MeV;

SAL (2000): D. Hornidge et al., PRL 84, 2334 (2000), MeV;

Lund (2003): M. Lundin et al., PRL 90, 192501 (2003),

MeV.

0

50

100

70 80 90 100 110

Detected photon energy (MeV)

Yie

ld


d in PT to

S. R. Beane, M. Malheiro, D. P., U. van Kolck, Nucl. Phys. A656, 367 (1999)

No free parameters at this order PREDICTION


Results

0 45 90 135 180θ

cm (deg.)

0

5

10

15

20

25

30

35

dσ/d

Ωcm

(nb

/sr)

Lucas data at Eγ, lab=69 MeV, lab. dcs

O(P3), Λ=600 MeV

O(P3), Λ=600 MeV, NN contribution included

O(P3), Λ=540 MeV

γ-d scattering, data from LundEγ cm

=66 MeV

Wave-function de-

pendence gives es-

timate of theory er-

ror.


d scattering at

S. R. Beane, M. Malheiro, J. McGovern, D. P., U. van Kolck, Phys. Lett. B, in press

Ingredients:

1. N amplitude at ; boosted to d centre-of-mass frame.

2. d two-body pieces at .


Calculable in terms of , , , , and .

4. Resummation to deal with very-low-energy region(relevant only for lower-energy data sets).

Only free parameters are and .


d scattering at


Ingredients:1. N amplitude at

;

boosted to d centre-of-mass frame.







d scattering at



; boosted to d centre-of-mass frame.







d scattering at




2. d two-body pieces at

.






d scattering at





.


.

Calculable in terms of

, , , , and .




d scattering at





.


.


, , , , and .

4. Resummation to deal with very-low-energy region

(relevant only for lower-energy data sets).



d scattering at





.


.


, , , , and .

4. Resummation to deal with very-low-energy region

(relevant only for lower-energy data sets).

Only free parameters are and .INT Mini-workshop on NN/NNN systems, October 2003 – p.28/39

Convergence

0 45 90 135 180θ

cm (deg.)

0

5

10

15

20

25dσ

/dΩ

cm (

nb/s

r)

Lund data (stat. errors only)Illinois data (stat. errors only)

O(Q4): α

N=11.1, β

N=1.3

O(Q3), no free parameters

O(Q2)

γ-d scattering: χPT convergenceω=66 MeV, χPT Λ=600 MeV wave function


Very-low-energy resummation

Diagrams (b) and (c) crucial for recovery of d Thomson limit:

Also crucial is to use: NOT

Modification of power-counting necessary for ;

Leading effect [in EFT ] comes from diagrams (b) and (c);

Significant effects at 49 and 55 MeV. Negligible at 95 MeV.




Also crucial is to use:

NOT

Modification of power-counting necessary for ;

Leading effect [in EFT ] comes from diagrams (b) and (c);





Also crucial is to use:

NOT

Modification of power-counting necessary for

;

Leading effect [in EFT

] comes from diagrams (b) and (c);



Very-low-energy resummation effect

0.0 45.0 90.0 135.0 180.0 θlab (deg.)

0.0

5.0

10.0

15.0

20.0

25.0

30.0

35.0

dσ/d

Ω (

nb/s

r)

Eγ=49 MeV

0.0 45.0 90.0 135.0 180.0 θlab (deg.)

0.0

5.0

10.0

15.0

20.0

25.0

30.0

dσ/d

Ω (

nb/s

r)

Eγ=69 MeV

0.0 45.0 90.0 135.0 180.0 θlab (deg.)

0.0

5.0

10.0

15.0

20.0

25.0

dσ/d

Ω (

nb/s

r)

Eγ=95 MeV

O(Q2)

O(Q2) + NN contribution

O(Q3)

O(Q3) + NN contribution


Dependence of cross section on

0 45 90 135 180θ

cm (deg.)

0

5

10

15

20

25dσ

/dΩ

cm (

nb/s

r)

Lund dataIllinois data

O(Q4), NLO χPT

O(Q4), OBEPQ

O(Q4), Nijm93

O(Q4), OPEP + R=1.5 fm

O(Q3), NLO χPT


Polarizability extractions

Wave function

! # ! # /d.o.f.

NLO PT < 160 MeV 9.0 1.7 1.48

NLO PT < 200 MeV 8.2 3.1 1.58

Nijm93 < 160 MeV 12.6 1.1 2.95

Including backward-angle SAL points increasesmarkedly. ? dynamics.

Results for and rather different with Nijm93 wave function:differential cross section larger by about 10%, shape similar.

Isoscalar polarizabilities from low-energy d d:



Wave function

! # ! # /d.o.f.

NLO PT < 160 MeV 9.0 1.7 1.48

NLO PT < 200 MeV 8.2 3.1 1.58

Nijm93 < 160 MeV 12.6 1.1 2.95

Including backward-angle SAL points

increasesmarkedly. ?

dynamics.

Results for and rather different with Nijm93 wave function:differential cross section larger by about 10%, shape similar.




Wave function

! # ! # /d.o.f.

NLO PT < 160 MeV 9.0 1.7 1.48

NLO PT < 200 MeV 8.2 3.1 1.58

Nijm93 < 160 MeV 12.6 1.1 2.95

Including backward-angle SAL points

increasesmarkedly. ?

dynamics.

Results for and

rather different with Nijm93 wave function:differential cross section larger by about 10%, shape similar.



Results

0 90 180θ

lab (deg.)

0

10

20

30

dσ/d

Ω

Eγ=49 MeV

0 90 180θ

cm (deg.)

0

10

20

30ω=55 MeV

0 90 180θ

cm (deg.)

0

5

10

15

20

dσ/d

Ω

Eγ=69 MeV

ω=66 MeV

0 90 180θ

cm (deg.)

0

5

10

15

20

ω=95 MeV

Fit to data with

MeV

shown.


Conclusions and Future Work

p scattering:N

LO HB PT calculation yields

MeV

-expansion:

dressed

MeV

d scattering:[NLO]: Parameter-free predictions: MeV.[N LO]: Extraction of and

Future work:Other processes in -expansion

degrees of freedom in d (in progress with R. Hildebrandt, T. Hemmert, H. Grießhammer);

Better understanding of dependence;

More data on d d!

(Kossert et al. Phys. Rev. Lett. 88, 162301 (2002)).

Thanks to the U.S. Department of Energy for financial support.



p scattering:N


MeV

-expansion:

dressed

MeV

d scattering:

[NLO]: Parameter-free predictions:

MeV.

[N

LO]: Extraction of and

Future work:Other processes in -expansion


Better understanding of dependence;

More data on d d!


Thanks to the U.S. Department of Energy for financial support.



p scattering:N


MeV

-expansion:

dressed

MeV

d scattering:

[NLO]: Parameter-free predictions:

MeV.

[N

LO]: Extraction of and

Future work:Other processes in

-expansion


Better understanding of

dependence;

More data on d d!


Thanks to the U.S. Department of Energy for financial support.INT Mini-workshop on NN/NNN systems, October 2003 – p.35/39

d with explicit ’s

R. Hildebrandt, H. Grießhammer, T. Hemmert, D.P.

Calculation to N

LO—

—in

-counting;

and

promoted by one order, and (here) fixed at

values extracted from fit to p data:

25 50 75 100 125 150 175Θlab @degD

5

10

15

20

25

HdΣ-dWL la

b@nbD

Ωlab =69 MeV

25 50 75 100 125 150 175Θlab @degD

2.5

5

7.5

10

12.5

15

17.5

20

HdΣ-dWL la

b@nbD

Ωlab =90 MeV


2004 Photonuclear GRC

MARK YOUR CALENDARS!

2004 Gordon Research Conferenceon Photonuclear Reactions

August 1–6, 2004Tilton School, Tilton, NH

Organizers: D. Phillips (Ohio), A. Sarty (St. Mary’s), B. Krusche (Basel)

Topics:

Chiral dynamics and the Delta

Nucleon resonances

Theory: present and future

Z

-nuclear physics

The transition to the scaling regime

Strong interactions at the Standard Model frontier

Photonuclear interactions in the nuclear medium

Electromagnetic probes of few-nucleon systems


compton scattering from the proton, deuteron, and neutroncompton scattering from the proton,...

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