Compton scattering from the proton in an effective field theory with explicit Delta degrees of freedom

0.98 MB·English

by  Judith A. McGovern

INT-PUB-12-048
Compton scattering from the proton in an effective field theory
with explicit Delta degrees of freedom
J. A. McGovern∗
School of Physics and Astronomy, The University of Manchester, Manchester M13 9PL, UK
D. R. Phillips†
Department of Physics and Astronomy and Institute of Nuclear and Particle Physics,
2
1 Ohio University, Athens, Ohio 45701, USA‡ and
0
School of Physics and Astronomy, The University of Manchester, Manchester M13 9PL, UK
2

t
c H. W. Grießhammer§
O

5 Institute for Nuclear Studies, Department of Physics,
1
The George Washington University, Washington DC 20052, USA¶ and

] Ju¨lich Centre for Hadron Physics and Institut fu¨r Kernphysik (IKP-3),
h
t Forschungszentrum Ju¨lich, D-52428 Ju¨lich, Germany
-
l
c Abstract
u
We analyse the proton Compton-scattering differential cross section for photon energies up to
n
[ 325 MeV using Chiral Effective Field Theory (χEFT) and extract new values for the electric and

1 magnetic polarisabilities of the proton. Our EFT treatment builds in the key physics in two different
v regimes: photon energies ω < m (“low energy”) andthehigher energies wherethe∆(1232) resonance
4 π
0 plays akey role. TheCompt∼onamplitudeis completeatN4LO, (e2δ4), inthelow-energy region, and
O
1 at NLO, (e2δ0), in the resonance region. Throughout, the Delta-pole graphs are dressed with πN
4 O
loops and γN∆ vertex corrections. A statistically consistent database of proton Compton experiments
.
0
is used to constrain the free parameters in our amplitude: the M1 γN∆ transition strength b (which
1 1
2 is fixedin theresonance region) andthepolarisabilities α andβ (which arefixedfrom databelow
E1 M1
1
170 MeV). Inorder toobtain areasonable fitwe findit necessary to addthespinpolarisability γ
: M1M1
v
as a free parameter, even though it is, strictly speaking, predicted in χEFT at the order to which we
i
X work. We show that the fit is consistent with the Baldin sum rule, and then use that sum rule to con-
r strainα +β . Inthiswayweobtainα = [10.65 0.35(stat) 0.2(Baldin) 0.3(theory)] 10−4 fm3,
a E1 M1 E1
± ± ± ×
and β = [3.15 0.35(stat) 0.2(Baldin) 0.3(theory)] 10−4 fm3, with χ2 = 113.2 for 135 degrees
M1
∓ ± ∓ ×
of freedom. A detailed rationale for the theoretical uncertainties assigned to this result is provided.
‡ Permanent address
¶ Permanent address
∗Electronic address: [email protected]
†Electronic address: [email protected]
§Electronic address: [email protected]
1
I. INTRODUCTION
Compton scattering from the proton, γp γp, at photon energies up to a few hundred
→
MeV, hasproven anexcellent tooltostudy thesubtle interplay ofeffective low-energydegrees of
freedom of hadrons, and of the symmetries anddynamics which govern them; see recent reviews
for details [1–3]. It reveals the chiral symmetry of QCD and the pattern of its breaking by
probing pion-cloud effects, and the properties of the ∆(1232) as the lowest nucleonic excitation.
Since it tests the (real and virtual) excitation spectrum of the target, probing the two-photon
response of a proton also complements the information available in the one-photon response
(e.g. form factors).
Since the earliest experiments on the proton [4–15], a particular goal has been an extraction
of the electric and magnetic polarisabilities, α and β , which parametrise the stiffness of the
E1 M1
proton against deformation in uniform, static electric and magnetic fields. At photon energies
ω below 50 MeV the Compton cross section is completely dominated by the Born terms, with
lab
the Thomson cross section as the low-energy limit. The polarisabilities manifest themselves as
corrections to this point-like cross section which grow as ω2 , but there is only a narrow range
lab
of energy before faster-varying terms contribute significantly. At ω 150 MeV non-analytic
lab
∼
structure associated with pion photoproduction appears, and around the same energy the tail
of the Delta resonance also becomes apparent. In order to extract the polarisabilities, therefore,
we need a reliable description of these effects. In this paper we analyze Compton scattering
data in such a framework in order to extract α and β in a model-independent way. Our
E1 M1
tool is Chiral Effective Field Theory, χEFT, the low-energy version of QCD [16–25]. In it,
theoretical uncertainties can be estimated using its power counting, a systematic expansion
scheme for physical observables. As the baryonic sector of Chiral Perturbation Theory (χPT)
including nucleons and the ∆(1232), it orders all interactions consistent with the symmetries of
QCD (and in particular the pattern of its chiral-symmetry breaking) by an expansion in powers
of “low” momentum scales in units of a “high” scale Λ at which new degrees of freedom become
relevant. Calculating both tree and loop diagrams to a given order in this small parameter
results in an amplitude that is truncated at a certain, prescribed level of accuracy.
Indeed, Compton scattering provided oneoftheearly successes ofχEFT appliedto nucleons.
Bernard et al. [26, 27] calculated the polarisabilities at leading one-loop order, without explicit
∆(1232) degrees of freedom, to be
10e2g2
αLO = 10βLO = A = 12.6 10−4 fm3, (I.1)
E1 M1 192πm f2 ×
π π
in remarkably good agreement with both the extractions of these parameters extant at that
time, and those obtained in subsequent experiments. Furthermore the cross sections obtained
from that theory were in qualitative agreement with the then-available data below about
150 MeV [21, 28]. Subsequently, a large number of experiments [29–40] have measured un-
polarised differential cross sections at photon energies between about 60 and 400 MeV; see
the thorough discussions in the reviews [1–3]. However, we shall argue in this article that the
database remains quite sparse in important regions, in particular between 150 and 250 MeV.
In addition, there are unresolved consistency issues between data sets from different experi-
ments. Therefore, plans to measure Compton cross sections at MAMI and HIγS [41–43] are
very welcome.
2
Results obtained for α and β using χEFT may be compared to direct determinations
E1 M1
from fully dynamical lattice QCD which appear imminent [44–48]. Such simulations, in turn,
take advantage of the fact that χEFT reliably predicts the strong dependence of the polaris-
abilities on the pion mass, so that results can be extrapolated from unphysical quark masses.
Thus, Compton scattering provides an excellent example of how χEFT serves as a bridge be-
tween cross sections measured in experiments and the non-perturbative quark-gluon dynamics
underlying the physics of hadrons.
Apart from its relevance in Compton scattering, the magnetic polarisability β also con-
M1
tributes the largest error on the two-photon-exchange contribution to the Lamb shift in muonic
hydrogen[49–51]. And, whilewefocushereonthecaseoftheproton,thisstudyalsoprovidesin-
put to extractions of neutron polarisabilities from data onelastic Compton scattering data from
deuterium; see e.g. Refs [1, 52]. Small proton-neutron differences stem from chiral-symmetry-
breaking pion-nucleon interactions, probing details of QCD. For example, Walker-Loud et
al. [53] recently found that the biggest uncertainty in theoretical determinations of the elec-
tromagnetic proton-neutron mass difference now comes from the contribution of 0.47 0.47
≈ ±
MeV from this effect [1]. However, such effects can be probed with confidence only if both
experimental and theoretical uncertainties are well under control.
As already mentioned, there has been a significant increase in the data available on γp
scattering over the last twenty years. Over the same period the EFT description of this process
has been refined in several ways. The pion-nucleon-loopamplitudes arenow known to one order
higher [54–56], where two short-distance contributions to α and β enter the Compton
E1 M1
amplitude. The polarisabilities are then no longer predicted, but can be determined from
low-energy Compton data, and the fit quality indicates the extent to which χEFT correctly
captures the energy-dependence of the Compton amplitudes. This has been done in Refs. [57,
58]. In another line of attack, the ∆(1232) was included as an explicit degree of freedom,
applying Lagrangians developed in Refs. [59–62]. This allows for a description which applies
from zero photon energy into the Delta resonance region, and thus also for using data at
intermediate energies for alternative extractions of the polarisabilities [62, 63]. In both these
variants, baryons have traditionally been included as non-relativistic degrees of freedom in a
version called “Heavy-Baryon Chiral Perturbation Theory” (HBχPT) [19, 20, 26]. A good
qualitative reproduction of Compton data in this regime has also been obtained in a χEFT
calculation which does not invoke the heavy-baryon expansion [64–67], and the agreement
between the cross sections with and without the heavy-baryon expansion is very good—even at
leading one-loop order—once the polarisabilities are fixed [68]. In line with the power-counting
philosophy, it is encouraging that the extractions or predictions of α and β have not varied
E1 M1
dramatically from the original predictions, as demonstrated by the values advocated in a recent
review [1].
In this paper, we merge these three refinements: higher-orders in the chiral sector, explicit
Delta degrees of freedom, and partially covariant formulations. In low-energy Compton scat-
tering, one identifies the expansion coefficients
m M M ω
π ∆ N
P , ǫ − , and . (I.2)
≡ Λ ≡ Λ Λ
where Λ 700 MeV is a typical large scale. Eq. (I.2) indicates both that different power
≈
countings involving choices about the relative size of P and ǫ are possible, and that different
kinematic regimes exist, depending on whether ω m or ω M M . In order to account
π ∆ N
∼ ∼ −
3
for the hierarchy of mass scales m < M M , we follow Pascalutsa and Phillips [63] and
π ∆ N
−
identify an expansion parameter δ ǫ while counting P δ2. We differentiate between two
∼ ∼
photon regimes: at low energies, we count ω m and obtain an amplitude which is complete
π
∼
at (e2δ4), where theleading (Thomson) amplitude is (e2), whilearoundtheDeltaresonance,
O O
we count ω δ and the amplitude is complete at order δ relative to leading order.
∼
Wedetermine α andβ by fittingto theprotonCompton-scattering databaseestablished
E1 M1
in Ref. [1]. To that end, we constrain the parameters of the M1 γN∆ coupling by the data
around the Delta-resonance, and then extract polarisabilities from the data at lower energy. We
findthatanexcellent fitcanbeobtainedbyincluding onecontactoperatorwhichcorrespondsto
a short-distance contribution to the spin-polarisability γ —even though, strictly speaking,
M1M1
this effect only appears at one order beyond that to which our calculation is complete. In doing
this we are departing from the strict EFT expansion, but the same strategy is necessary at
lower order where, once the Delta is included, α and β must be promoted [62].
E1 M1
This article is structured as follows. First, we define the theoretical ingredients of our
calculation: the Lagrangians with both a covariant and nonrelativistic ∆(1232) in Sec. IIA;
the Compton amplitudes and the definition of nucleon polarisabilities in Sec. IIB; the power
counting in the different kinematic regimes and how to arrive at an amplitude which is valid
across regimes in Sec. IIC; explicit results for the γN∆ vertex corrections in Sec. IID; the
prescription for the kinematically-correct position of the pion production threshold in Sec. IIE;
andconstraintsfromsumrulesandotherprocessesinSec.IIF.Sec.IIIisdevotedtothefititself.
After discussing our fitting strategy in Sec. IIIA, we present our results in Sec. IIIB, followed
by a discussion of details, alternative scenarios and convergence issues as well as, importantly,
an estimate of the residual theoretical uncertainties in Sec. IIIC. After the Summary and
Outlook of Sec. IV, an Appendix gives references for the explicit forms of the components of
the amplitudes. Preliminary versions of our findings appeared in Refs. [1, 69, 70].
II. THEORETICAL INGREDIENTS
A. Lagrangian
Since the χEFT Lagrangian is well-established, see e.g. Refs. [21, 25]; we quote here only
the terms relevant to Compton scattering at the order to which we work, so as to establish
notation and conventions. The leading meson-meson terms are:
1 1 1
(2) = (∂ πa)(∂µπ )+eA ǫ π ∂µπ + e2A Aµ(π2 +π2) m2π2 +... (II.1)
Lπ 2 µ a µ 3ij i j 2 µ 1 2 − 2 π
Here, πa is the pion isotriplet with isospin index a = 1,2,3, f the pion-decay constant, Aµ the
π
photon field, and e = e the electron charge.
−| |
In heavy-baryon χPT, the relativistic nucleon Dirac spinor and isospinor ψ is re-expressed
in terms of a “heavy-baryon” spinor H that has the straight-line propagation of a nucleon of
mass M and four-velocity v removed from its dynamics, so that v/H = H. In the rest-frame
N
of the nucleus its velocity is vµ = g0µ and its spin is Sµ = (0,~σ/2), where σ are the Pauli spin
i
matrices [19, 20]. At leading order, the pion and nucleon couple via the axial coupling g :
A
(1) = H†(iv D+g u S)H, (II.2)
LπN · A ·
4
where Dµ ∂µ ie Aµ is the gauge-covariant derivative, = 1(1 + τ ) the nucleon charge
≡ − Z Z 2 3
operator in isospin space, τa the Pauli matrices in isospin space and
1
u = (τ ∂ π +eǫa3bτ π A +...). (II.3)
µ a µ a a b µ
−f
π
Lorentz invariance generates the first three terms of the next-order Lagrangian:
1 1
(2) =H† (v D)2 D2 ig S D,v u +4c m2 1 π2 (II.4)
LπN (cid:26)2M · − − { · · } 1 π(cid:18) − 2f2 (cid:19)
N(cid:16) (cid:17) π
g2 i
+ c A (v u)2 +c u u [Sµ,Sν]eF (1+κ(s))+(1+κ(v))τ H +...
2 3 µν 3
(cid:18) − 8M (cid:19) · · − 4M (cid:27)
N N (cid:0) (cid:1)
with Fµν the electromagnetic field tensor. The coefficients of the other five operators are low-
energy constants (LECs): the isoscalar and isovector quantities κ(s) and κ(v) are the sum and
difference of the proton and neutron anomalous magnetic moments, and c to c contribute to
1 3
πN scattering. The latter in particular encode the effects of higher-energy baryon resonances,
andso have different values in theories with andwithout anexplicit Delta. Weuse thefollowing
values:
c = 0.9, c˜ = 0.2, c˜ = 1.6, (II.5)
1 2 3
− −
in units of GeV−1. These numbers are obtained from fits in Ref. [24] to pion-nucleon and NN
scattering in the theory without an explicit Delta,
c = 0.9+0.2, c = 3.3 0.2, c = 4.7+1.2, (II.6)
1 − −0.5 2 ± 3 − −1.0
by subtracting the Delta-pole contribution of 4g2 /9(M M ) from c and c respectively.
± πN∆ ∆− N 2 3
The (e2P) Compton amplitude also includes a tree-level 1/M2 piece from (3) which has
O N LπN
a fixed coefficient due to the low-energy consequences of Lorentz invariance, i.e. due to the
1/M expansion, and is required for the correct low-energy behavior of the forward scattering
N
amplitude:
1
(3) = H† [Sµ,Sν]eF vσD (1+2κ(s))+(1+2κ(v))τ H +h.c.. (II.7)
LπN − 8M2 µσ ν 3
N (cid:16) (cid:17)
There are additional terms in (3) that contribute to the (e2P2) Born amplitude. We do not
LπN O
list them here, but they are discussed in the Appendix. In addition, the pion-pole graph enters
(4)
which involves the anomalous part of (ǫ = +1):
ππ 0123
L
e2
(4),a = π ǫ FµνFρσ. (II.8)
Lππ −32π2f 3 µνρσ
π
At (e2P2), divergent loops need to be renormalised by operators that occur in (4). Those
O LπN
relevant for Compton scattering are:
(4),CT = 2πe2H† 1(δβ(s) +δβ(v)τ )g (δα(s) +δβ(s) )+(δα(v) +δβ(v))τ v v FµρFν H.
LπN 2 M1 M1 3 µν − E1 M1 E1 M1 3 µ ν ρ
h h i i
(II.9)
5
Theytranslateintoadditional,energy-independent contributions, δα andδβ , totheelectric
E1 M1
and magnetic polarisabilities which are isoscalar or isovector. This defines the πN sector of the
calculation.
For the ∆(1232) resonance, our calculation uses two variants: a manifestly covariant form,
and its heavy-baryon reduction. In the latter, the field ∆a of mass M is an iso-quadruplet
µ ∆
heavy-baryon reduction of a Rarita-Schwinger field [60]. Since we work to NLO in the domain
where the photon energy and the Delta-nucleon mass splitting ∆ M M are comparable,
M ≡ ∆− N
ω ∆ , we need selected terms from the Lagrangian up to third order [71]:
∼ M
HB,(1) = (∆a)†( iv D +∆ )∆aν (II.10)
L∆ ν − · M
g
HB,(1) = πN∆(H†∂νπa∆a +(∆a)†∂νπaH +...) (II.11)
LπN∆ − f ν ν
π
ieb
HB,(2) = 1 H†S Fµρ∆3 (∆3)†S FµρH (II.12)
LγN∆ −M ρ µ − µ ρ
N(cid:16) (cid:17)
eb ← eD
HB,(3) = 2 H† S D Fµρv +Fµρv S D ∆3 + 1 H†vαSβ τ3[D ,f+ ] ∆3 +H.c. ,
LγN∆ −2M2 · ρ ρ · µ 4M2 h µ αβ i µ
N (cid:16) (cid:17) N (cid:2) (cid:3)
(II.13)
where a (µ) is the index of the isovector (vector) coupled to the isospinor (spinor), and the
angled brackets indicate isospin traces. The choice of the parameters g , b , b and M will
πN∆ 1 2 ∆
be discussed in Sec. III. The term of the axial coupling g is of order ω, and the b interaction
πN∆ 1
of order eω. Both the b and D terms give rise to couplings which are of order eω2. Terms
2 1
generated by the heavy-baryon reduction are omitted in Eq. (II.13). Unfortunately, early works
on the systematic heavy-baryon reductions of χEFT with an explicit ∆(1232) [59–61, 71, 72]
were not consistent in their definition of b ; signs and factors of 2 come and go. The convention
1
used here is in accord with that of Refs. [1, 62, 72, 73]. In Eq. (II.13) we have identified
Ref. [71]’s b 2b .
6 2
≡ −
These are the pertinent structures in the heavy-baryon Lagrangian. As we discuss in
Secs. IIC2 and IID, we perform a fully Lorentz covariant calculation of the Delta-pole di-
agrams. To do this, we employ the relativistic Lagrangian used in the δ-expansion [63]. With
Ψa the relativistic Rarita-Schwinger spinor, the free, πN∆ and γN∆ pieces are [63]:
ν
ig
PP,(1) = Ψ¯a(iγµνα∂ M γµν)Ψa + πN∆ ψ¯γµνλ∂ πa(∂ Ψa)+H.c. , (II.14)
L∆ µ α − ∆ ν 4M f λ µ ν
∆ π (cid:2) (cid:3)
3e
PP,(2) = ψ¯(ig F˜µν g γ Fµν)∂ Ψ3 Ψ¯3←∂− (ig F˜µνg γ Fµν)ψ , (II.15)
LγN∆ 2M (M +M ) M − E 5 µ ν − ν µ M E 5
N N ∆ h i
where γµν = 1[γµ,γν], γµνα = 1 γµν,γα , γ = iγ0γ1γ2γ3, and F˜µν = 1ǫµνρσF . The (leading)
2 2{ } 5 2 ρσ
terms involving the electromagnetic field that emerge upon heavy-baryon reduction of this La-
grangian are equivalent to those in Eqs. (II.12) and (II.13), provided we make the identification
g = b (1+M /M )/3, g = b (1+M /M )/3. (II.16)
M 1 ∆ N E 2 ∆ N
In this presentation, we use the relativistic variant of the γN∆ couplings, but quote results in
terms of b and b , using this conversion formula throughout, since these are more widely used.
1 2
The s- and u-channel Delta-pole diagrams then reproduce the results given in the Appendices
of Ref. [63]. However, the Lagrangians of Refs. [63, 74] do not include a structure that reduces
to that with coefficient D in Eq. (II.13); this point will play a role in Sec. IID.
1
6
B. Compton amplitudes and polarisabilities
The standard decomposition for the proton Compton amplitude involves six non-relativistic
invariants:
ˆ ˆ ˆ ˆ
T = A (~ǫ′∗ ~ǫ)+A (~ǫ′∗ ~k) (~ǫ ~k′)+iA ~σ (~ǫ′∗ ~ǫ)+iA ~σ (~k′ ~k)(~ǫ′∗ ~ǫ)
1 2 3 4
· · · · × · × ·
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ
+iA ~σ [(~ǫ′∗ ~k)(~ǫ ~k′) ~ǫ ~k′)(~ǫ′∗ ~k)]+iA ~σ [(~ǫ′∗ ~k′)(~ǫ ~k′) (~ǫ ~k)(~ǫ′∗ ~k)] ,
5 6
· × · − × · · × · − × ·
(II.17)
where ~k and ~ǫ (~k′ and ~ǫ′) are the three-momentum and polarisation vector of the incoming
(outgoing) photon, and ~kˆ (k~ˆ′) unit vectors pointing in the direction of ~k (~k′). The A ’s, i =
i
1,...,6, are scalar functions of photon energy ω and scattering angle θ, and are real below the
πN threshold.
All data is reported in either the centre-of-mass (cm) or laboratory (lab) frames. The
corresponding differential cross sections are
dσ
= Φ 2 T 2, (II.18)
dΩ(cid:12) | | | |
(cid:12)cm,lab
(cid:12)
(cid:12)
where the spin-summed and averaged, squared amplitude is:
1 1
T 2 = A 2(1+cos2θ)+ A 2(3 cos2θ)
1 3
| | 2| | 2| | −
+sin2θRe 4A∗A +(A∗A +2A∗A A∗A )cosθ
3 6 3 4 3 5 − 1 2
1(cid:2) 1 (cid:3)
+sin2θ A 2sin2θ+ A 2(1+cos2θ)+ A 2(1+2cos2θ)+3 A 2
2 4 5 6
2| | 2| | | | | |
(cid:2)
+2Re[A∗(A +3A )]cosθ +2Re[A∗A ]cos2θ . (II.19)
6 4 5 4 5
(cid:3)
The frame-dependent phase-space factor Φ in Eq. (II.18) is given by
1 M 1 ω′
Φ = N; Φ = lab. (II.20)
cm lab
4π √s 4πω
lab
Following Ref. [56], we use Compton amplitudes A –A in the Breit frame, in which the
1 6
photon does not transfer energy to the proton and the amplitudes are either even or odd in the
photon energy. The Breit frame variables ω and θ are then:
2M ν t
N
ω = , cosθ = 1 . (II.21)
4M2 t − 2ω2
N −
p
where
s u 1
t = 2ω ω′ (cosθ 1), ν − = (ω +ω′ ) (II.22)
lab lab lab − ≡ 4M 2 lab lab
N
s and t being the usual Mandelstam variables. See Refs. [1, 75] for more details.
The most famous manifestation of chiral dynamics in Compton scattering occurs in the
nucleon scalar polarisabilities, α and β . These parametrise the global stiffness of the
E1 M1
nucleon’s internal degrees of freedom against displacement in an electric or magnetic field;
7
see our recent review [1] for details. They affect a Compton-scattering reaction in which the
photon makes E1 E1 and M1 M1 transitions, respectively. In consequence, the low-
→ →
energy expansion of the A ’s can be used to define these electric and magnetic polarisabilities
i
α and β . They occur as coefficients of the ω2 terms in A and A . For the Breit frame,
E1 M1 1 2
such an ω-expansion gives (keeping terms up to (1/M3)) [56, 58]:
O N
e2 e2
A (ω,θ) = + (1+κ(p))2(1+cosθ) 1 (1 cosθ)ω2
1 −M 4M3 − −
N N(cid:16) (cid:17)
+4π(α +β cosθ)ω2 + (ω4),
E1 M1
O
e2
A (ω,θ) = κ(p)(2+κ(p))ω2cosθ 4πβ ω2 + (ω4),
2 4M3 − M1 O
N
e2ω
A (ω,θ) = (1+2κ(p)) (1+κ(p))2cosθ +Aπ0
3 2M2 − 3
N(cid:16) (cid:17)
4πω3(γ +γ +(γ +γ ) cosθ)+ (ω5),
E1E1 E1M2 M1E2 M1M1
− O
e2ω
A (ω,θ) = (1+κ(p))2 +Aπ0 4πω3(γ γ )+ (ω5),
4 −2M2 4 − M1M1 − M1E2 O
N
e2ω
A (ω,θ) = (1+κ(p))2 +Aπ0 +4πω3γ + (ω5),
5 2M2 5 M1M1 O
N
e2ω
A (ω,θ) = (1+κ(p))+Aπ0 +4πω3γ + (ω5), (II.23)
6 −2M2 6 E1M2 O
N
with κ(p) the proton anomalous magnetic moment, and
e2g ω3(cosθ 1) e2g ω3
Aπ0 = A − , Aπ0 = Aπ0 = A . (II.24)
3 4π2f2(m2 t) 6 − 5 8π2f2(m2 t)
π π0 − π π0 −
the contributions from the pion-pole graph (see Fig. II.1(ii) below). In Eq. (II.23) α and
E1
βM1 are supplemented by 4 spin-dependent dipole polarisabilities γXlYl′ which describe photon
transitions of definite multipolarities Xl Yl′, with l′ = l+ 0; 1 ; the relation to Ragusa’s
↔ { ± }
spin polarisabilities γ [55] is given for example in Ref. [1].
1−4
The polarisabilities used in Eq. (II.23) can be generalised to define dynamical, i.e. energy-
dependent, polarisabilities [62, 76]. (These should not be confused with the generalised polaris-
abilities that can be accessed in virtual Compton scattering.) This is equivalent to a multipole
decomposition of the structure (i.e. non-Born) part of the Compton amplitude. As in all mul-
tiple decompositions, a complete set of dynamical multipole polarisabilities contains the same
information as the Compton amplitudes, A , but examining predictions for these functions
i
channel-by-channel can provide insight into how physical mechanisms may manifest themselves
in data. For further discussion of the definition and usefulness of energy-dependent/dynamical
polarisabilities, see Ref. [1]. Here, however, we work with the full amplitudes rather than a
truncated multipole decomposition.
C. Power counting regimes
As discussed in the Introduction, three typical low-energy scales exist in Compton scattering
with a dynamical ∆(1232): the pion mass m 140 MeV as the typical chiral scale; the Delta-
π
≈
8
nucleon mass splitting ∆ Re[M ] M 290 MeV; and the photon energy ω. Each
M ≡ ∆ − N ≈
provides a small, dimensionless expansion parameter when it is measured in units of a natural
“high” scale Λ ∆ ,m ,ω at which χEFT with explicit ∆(1232) degrees of freedom can be
≫ M π
expected to break down because new degrees of freedom enter. Typical values of Λ are the
masses of the ω and ρ as the next-lightest exchange mesons (about 700 MeV), so that
m M M
π ∆ N
P 0.2, ǫ − 0.4. (II.25)
≡ Λ ≈ ≡ Λ ≈
Whileathreefoldexpansionofallinteractionsispossible, itismoreconvenient toapproximately
identify some scales so that only one dimensionless parameter is left. In order to assign a unique
index to each graph, we therefore follow the δ-expansion of Pascalutsa and Phillips [63]. It
notices that m M M and takes advantage of the numerical proximity P ǫ2 in the
π ∆ N
≪ − ≈
real world to define
M M m 1/2
∆ N π
δ − = , (II.26)
≡ Λ Λ
(cid:16) (cid:17)
1
i.e. numerically δ ǫ P2. The approach then separates the calculation into two different
regimes, dependin≈g on≈the photon energy. In regime I, ω < m 140 MeV is “low”, so
π
that one counts ω m δ2Λ like a chiral scale. At higher e∼nergies≈(regime II, ω ∆
∼ π ∼ ∼ M ≈
300 MeV), one counts ω ∆ δΛ.
∼ M ∼
This power counting accommodates the fact that Compton scattering changes qualitatively
with increasing energy. In regime II, the photon carries enough energy to excite a ∆(1232)
intermediate state whose large width and strong γN∆ coupling makes it dominate the ampli-
tudes. At lower energies, the Delta should play a less pronounced role. This will be confirmed
in the list of all contributions in the following sub-sections.
1. Regime I: ω m
π
∼
The Compton graphs that contribute up to the order to which we work can be separated into
the three classes of Figs. II.1 to II.3. References for the actual form of the resulting amplitudes
are given in Appendix A.
Wefirstdiscuss thegraphsseparately forphotonenergiesω m . Forcontributionswithout
π
∼
an explicit ∆(1232), the only expansion scale in this regime is P δ2, and the power counting
≡
is that of HBχPT, with only even powers of δ contributing [63].
The first class consists of the “Born” or “non-structure” amplitudes of Fig. II.1 and contains
no loops. All graphs are calculated in the gauge v A = 0, so direct γN couplings do not
·
occur at lowest order. Thus in regime I, the leading, (e2δ0 e2P0), contribution is the
O ∼
(2)
“seagull” diagram Fig. II.1(i), with a vertex from (II.4); this gives the Thomson amplitude.
LπN
Correctionsat (e2δ2 e2P)are: minimalcouplingtotheprotonchargeandmagneticmoment
O ∼
from (2) (II.4) in diagram (ii)(a); 1/M2 seagulls from (3) in diagram (ii)(b), and coupling
LπN N LπN
to the anomaly via (4) (II.8) in graph (ii)(c). At (e2δ4 e2P2), corrections enter from a
ππ
L O ∼
(3)
γNN vertex from in diagram (iii)(a), and finally also from the short-distance contributions
LπN
(4)
δα , δβ to the scalar polarisabilities in (II.9), diagram (iii)(b).
E1 M1 LπN
The second class of contributions comprises the pion-loop diagrams of Fig. II.2. At leading-
one-loop order in this regime, (e2δ2 e2P), the diagrams of Fig. II.2(i) contain only the
O ∼
9
FIG. II.1: (Colour online) Tree diagrams that contribute to Compton scattering in the ǫ v = 0 gauge,
·
ordered by the typical size of their contributions in the two regimes ω m δ2 and ω ∆ δ,
∼ π ∼ ∼ M ∼
respectively. The leading-order contribution in a particular regime is indicated by (LO). The vertices
(1) (2) (3) (4) (4),a
are from: (no symbol), (square), (triangle), (diamond), (disc). Permuted
LπN LπN LπN LπN Lππ
and crossed diagrams not shown.
FIG. II.2: (Colour online) Pion-nucleon loop diagrams , ordered by their typical size for ω m δ2
π
∼ ∼
and ω ∆ δ, respectively. Notation as in Fig. II.1. Permuted and crossed diagrams not shown.
∼ M ∼
vertices of Eqs. (II.1) and were first calculated by Bernard et al. [54]. At the next order,
(e2δ4 e2P2), oneinsertionof (2) leadstotwo types ofcontribution, depicted inFig. II.2(ii).
O ∼ LπN
Those of diagrams (a) to (k) involve 1/M terms which are required by relativity and therefore
N
would already be present in a leading-order covariant calculation of the πN loop amplitudes,
while those of graphs (l) to (r) involve in addition the LECs which are NLO in both the
10