Baryon PT for Two-Boson Exchange GraphBaryon ˜PT for Two-Boson Exchange Graph Vadim Lensky Johannes...
Transcript of Baryon PT for Two-Boson Exchange GraphBaryon ˜PT for Two-Boson Exchange Graph Vadim Lensky Johannes...
Baryon χPT for Two-Boson Exchange Graph
Vadim Lensky
Johannes Gutenberg Universität Mainz
September 30, 2017
Vadim Lensky (U. Mainz) Electroweak Box Workshop September 30, 2017 1 / 31
Motivation
Consider the γγ box at low energies (important corrections to epscattering, µ atoms)χPT — the low-energy EFT of QCD — is a suitable tool in this regimeRemove the lepton line =⇒ proton Compton scattering (CS)— wealth of exp. dataCalculate CS in χPT, confront data, make predictions for γγ box
Extend to γZ and γW at low energies?
Vadim Lensky (U. Mainz) Electroweak Box Workshop September 30, 2017 2 / 31
Outline
1 χPT framework
2 Results for nucleon Compton scattering and µHRCSVVCSVCSLamb shift and HFS
3 Some thoughts on extension to γZ and γW
Vadim Lensky (U. Mainz) Electroweak Box Workshop September 30, 2017 3 / 31
Outline
1 χPT framework
2 Results for nucleon Compton scattering and µHRCSVVCSVCSLamb shift and HFS
3 Some thoughts on extension to γZ and γW
Vadim Lensky (U. Mainz) Electroweak Box Workshop September 30, 2017 4 / 31
χPT framework
inlcude nucleons, photons, pions Weinberg, Gasser and Leutwyler, ...
count powers of small momenta p ∼ mπ
numerically e ∼ mπ/MN — count as p
also include the Delta isobar Hemmert, Holstein, ...
∆ = M∆ −MN is a new energy scale =⇒ δ-counting: Pascalutsa, Phillips (2002)
numerically δ = ∆/MN ∼√
mπ/MN , count ∆ ∼ p1/2 if p ∼ mπ
complications due to the spin-3/2 field (consistent couplings etc.)
two energy regimes:ω ∼ mπ:
n = 4L− 2Nπ − NN −12
N∆ +∑
kVk
ω ∼ ∆:n = 4L− 2Nπ − NN − N∆ − 2N1∆R +
∑kVk
Vadim Lensky (U. Mainz) Electroweak Box Workshop September 30, 2017 5 / 31
NLO: Born and πN Loops
Born graphs
responsible for low energy (Thomson) limit; point-like nucleonO(p2) and O(p3) (a.m.m. coupling)
π0 anomaly and πN loops
leading-order contribution to polarisabilities: O(p3)
Vadim Lensky (U. Mainz) Electroweak Box Workshop September 30, 2017 6 / 31
NNLO: Delta pole and π∆ loops
Delta pole and π∆ loops
different counting in different energy regimes
Delta pole: O(p4/∆) = O(p7/2) at ω ∼ mπ ; O(p) at ω ∼ ∆π∆ loops: O(p4/∆) = O(p7/2) at ω ∼ mπ ; O(p3) at ω ∼ ∆
at ω ∼ ∆ one needs to dress the 1∆R propagator
iΣ =
at ω ∼ ∆ corrections to γN∆ vertex are O(p2)
+= +
Vadim Lensky (U. Mainz) Electroweak Box Workshop September 30, 2017 7 / 31
Outline
1 χPT framework
2 Results for nucleon Compton scattering and µHRCSVVCSVCSLamb shift and HFS
3 Some thoughts on extension to γZ and γW
Vadim Lensky (U. Mainz) Electroweak Box Workshop September 30, 2017 8 / 31
Outline
1 χPT framework
2 Results for nucleon Compton scattering and µHRCSVVCSVCSLamb shift and HFS
3 Some thoughts on extension to γZ and γW
Vadim Lensky (U. Mainz) Electroweak Box Workshop September 30, 2017 9 / 31
Results: RCS observables
covariant baryon calculationVL, McGovern, Pascalutsa (2015)
VL, Pascalutsa (2009)
NNLO (O(p7/2)) at ω ∼ mπ
NLO (O(p2)) at ω ∼ ∆
prediction of ChPT
polarisabilities are seenstarting at ∼ 50 MeVpion loops are important at lowenergies and around pionproduction thresholdDelta pole dominates in theresonance region
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Vadim Lensky (U. Mainz) Electroweak Box Workshop September 30, 2017 10 / 31
Scalar polarisabilities: status
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for the proton, χPT calculations somewhat differ from other extractions (inparticular, TAPS value)
there are hints that the issue might be due to exp. dataKrupina, VL, Pascalutsa, in preparation
neutron polarisabilities are less well constrained — there is no freeneutron target
Vadim Lensky (U. Mainz) Electroweak Box Workshop September 30, 2017 11 / 31
Outline
1 χPT framework
2 Results for nucleon Compton scattering and µHRCSVVCSVCSLamb shift and HFS
3 Some thoughts on extension to γZ and γW
Vadim Lensky (U. Mainz) Electroweak Box Workshop September 30, 2017 12 / 31
VVCS
forward VVCS amplitude
T (ν,Q2) =fL(ν,Q2) + (~ε ′∗ · ~ε ) fT (ν,Q2)
+ i~σ · (~ε ′∗ × ~ε ) gTT (ν,Q2)− i~σ · [(~ε ′∗ − ~ε )× q̂] gLT (ν,Q2)
low-energy expansion of the amplitude is
fT (ν,Q2) = f BT (ν,Q2) + 4π
[Q2βM1 + (αE1 + βM1)ν2] + . . .
fL(ν,Q2) = f BL (ν,Q2) + 4π(αE1 + αLν
2)Q2 + . . .
gTT (ν,Q2) = gBTT (ν,Q2) + 4πγ0ν
3 + . . .
gLT (ν,Q2) = gBLT (ν,Q2) + 4πδLTν
2Q + . . .
ν-dependent terms can be treated as functions of Q2 and related tomoments of nucleon structure functions
Vadim Lensky (U. Mainz) Electroweak Box Workshop September 30, 2017 13 / 31
VVCS: results
NLO/LO [O(p4/∆)/O(p3)]VL, Alarcón, Pascalutsa (2014)
HB χPT O(p4)Kao, Spitzenberg, Vanderhaeghen (2003)
IR O(p4)Bernard, Hemmert, Meissner (2003)
covariant χPT O(ε3)Bernard, Epelbaum, Krebs, Meissner (2013)
MAID
HB and IR do not provide adequatedescription
covariant χEFT works much better,especially in γ0 (HB is off the scale there)
δLT puzzle: difference between the twocovariant calculations (the one of Bernardet al. contains π∆ loops subleading in ourcounting)data on δLT from JLab expected
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Vadim Lensky (U. Mainz) Electroweak Box Workshop September 30, 2017 14 / 31
Outline
1 χPT framework
2 Results for nucleon Compton scattering and µHRCSVVCSVCSLamb shift and HFS
3 Some thoughts on extension to γZ and γW
Vadim Lensky (U. Mainz) Electroweak Box Workshop September 30, 2017 15 / 31
VCS: response functions
++=
VCS Born VCS non-BornBethe-Heitler
e
γ
e′
p
e
p
e
e
p
p
pp
e
e
p′
Im ...Ν(p) (p’)Ν
γ∗(q) γ (q’)
Ν
γπ
N
Ν
γ∗
low-energy expansion (small ω′):Guichon, Liu, Thomas (1995)
d5σVCS =d5σBH+Born
+ ω′Φ Ψ0(Q2, ε, θ, φ) +O(ω′2);
Ψ0(Q2, ε, θ, φ) = V1
[PLL(Q2)−
PTT
ε
]+ V2
√ε(1 + ε)PLT (Q2)
at Q2 = 0:
PLL(0) =6Mαem
αE1
PLT (0) = −3M
4αemβM1
response functions ofVCS in covariant χETVL, Pascalutsa, Vanderhaeghen
(2016)
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Q2 @GeV2D0
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-20
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2 Dmore data from MAMI expected soonlow-Q2 expts. at MESA?
Vadim Lensky (U. Mainz) Electroweak Box Workshop September 30, 2017 16 / 31
Outline
1 χPT framework
2 Results for nucleon Compton scattering and µHRCSVVCSVCSLamb shift and HFS
3 Some thoughts on extension to γZ and γW
Vadim Lensky (U. Mainz) Electroweak Box Workshop September 30, 2017 17 / 31
Two-photon corrections
TPE — all TPE — elastic
polarisability = all − elastic
elastic is calculated from formfactors(empirical data)
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review by Hagelstein, Miskimen, Pascalutsa (2016)
muonic hydrogen Lamb shift in theory [energies in meV, Rp in fm]:
∆ELS = 206.0336(15)− 5.2275(10)R2p + ETPE Antognini et al (2013)
ETPE = 0.0332(20); E (pol) = 0.0085(11) Birse, McGovern (2012)
ETPE is an important source of th. uncertaintypolarisability corrections depend on the VVCS amplitude
Vadim Lensky (U. Mainz) Electroweak Box Workshop September 30, 2017 18 / 31
Lamb shift
πN loops give the leading contributionDelta pole strongly suppressedπ∆ loops not included
(b) (c)(a)
(d) (e) (f )
(g) (h) (j)
the O(p3) result is∆E (pol)
2S = −8.2(+1.2−2.5) µeV
Alarcon, VL, Pascalutsa (2014)
consistent with othercalculations
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∆ddddddddµddd
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Vadim Lensky (U. Mainz) Electroweak Box Workshop September 30, 2017 19 / 31
HFS (preliminary)
πN loops give the leading contributionDelta pole importantπ∆ loops not included
(b) (c)(a)
(d) (e) (f )
(g) (h) (j)
the O(p3) result isE (pol)
2S,HFS = −1.4(+1.6−1.2) µeV
Hagelstein, VL, Pascalutsa in preparation
large cancellationsdoes not agree with dispersivecalculations
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Vadim Lensky (U. Mainz) Electroweak Box Workshop September 30, 2017 20 / 31
Lamb shift: HB vs. covariant
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∆E (pol)nS =
αem
πφ2
n
∫ Qmax
0
dQQ2
w(τ`)[T (NB)
1 (0,Q2)− T (NB)2 (0,Q2)
]
one can expect larger error in HB sincethe integral converges more slowly thereneither HB nor covariant (nor otherresults), however, can explain the missing∼ 300 µeV
HBΧPT
BΧPT
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Vadim Lensky (U. Mainz) Electroweak Box Workshop September 30, 2017 21 / 31
Outline
1 χPT framework
2 Results for nucleon Compton scattering and µHRCSVVCSVCSLamb shift and HFS
3 Some thoughts on extension to γZ and γW
Vadim Lensky (U. Mainz) Electroweak Box Workshop September 30, 2017 22 / 31
Some thoughts (and summary)
covariant baryon χEFT works well in nucleon CS (RCS/VCS/VVCS) andgives a O(p3) prediction for µH Lamb shift and hyperfine splittingby connecting γγ box to nucleon CS, we cross-check our χPT calculationand better understand these processeshow can χPT be applied also to γZ and γW boxes?
naïvely: can work at low energies (P2@MESA, β-decays)could follow a route similar to γγ box:
calculate the analog of CS (“vector boson production”)calculate the boxes
could provide a systematic evaluation of γZ and γW boxesfeedback and suggestions welcome!
Vadim Lensky (U. Mainz) Electroweak Box Workshop September 30, 2017 23 / 31
Backup
Vadim Lensky (U. Mainz) Electroweak Box Workshop September 30, 2017 24 / 31
χPT for nucleon CS and µH
real CS in χPTprediction at O(p4/∆)compare with datapredict nucleon polarizabilities
virtual and doubly-virtual CS in χPTprediction at O(p4/∆)nucleon generalized polarizabilities (different in VCS and VVCS!)
we can calculate µH (Lamb shift and HFS) in χPTprediction at O(p3)
=⇒ RCS, VCS, VVCS and Lamb shift calculated in a single χPT frameworkcomplementary information about properties of the nucleon(verify sum rules in χPT etc.)
our choice is to use the covariant formulation of χPT
Vadim Lensky (U. Mainz) Electroweak Box Workshop September 30, 2017 25 / 31
Lamb shift and Compton scattering
Compton scattering amplitude (forward VVCS)
Tµν(q, p) =
(−gµν +
qµqν
q2
)T1(ν,Q2) +
1M2
(pµ −
p · qq2
qµ)(
pν −p · qq2
qν)
T2(ν,Q2)
−1Mγµναqα S1(ν,Q2)−
1M2
(γµνq2 + qµγναqα − qνγµαqα
)S2(ν,Q2)
spin-dependent terms contribute to hyperfine splitting
nth S-level shift is given by
∆E (pol)nS =
αem
πφ2
n
∫ ∞0
dQQ2
w(τ`)[T (NB)
1 (0,Q2)− T (NB)2 (0,Q2)
]
w(τ`): the lepton weighting function
w(τ`) =√
1 + τ` −√τ`, τ` =
Q2
4m2`
weighted at low virtualities
wHΤΜL
wHΤeL
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Vadim Lensky (U. Mainz) Electroweak Box Workshop September 30, 2017 26 / 31
Compton amplitudes
T1(ν,Q2) and T2(ν,Q2) can be related, via dispersive integrals, withnucleon structure functions (ν̃ = 2Mν/Q2):
T1(ν,Q2) =8παM
1∫0
dxx
f1(x ,Q2)
1− x2ν̃2 − i0, T2(ν,Q2) =
16παMQ2
1∫0
dxf2(x ,Q2)
1− x2ν̃2 − i0
the integral for T1 needs a subtraction: unknown function T1(0,Q2)
high-Q2 behaviour of T1(0,Q2) needs to be modelledformfactors Pachucki (1999), Martynenko (2006)
χPT-inspired formfactors Carlson, Vanderhaeghen (2011), Birse, McGovern (2012)
empirical fits Tomalak, Vanderhaeghen (2016)
something we know about T1(0,Q2): low-energy theorem
T NB1 (0,Q2) = 4πβM1Q2 + · · · , T NB
2 (0,Q2) = 4π(αE1 + βM1)Q2 + · · ·
=⇒ nucleon polarisabilities
Vadim Lensky (U. Mainz) Electroweak Box Workshop September 30, 2017 27 / 31
Polarisabilities
point particle (or low energies) =⇒ charge, mass, a.m.m.higher energies: response of the nucleon to external e.m. field
=⇒ static polarisabilities: low-energy constants of effective γN interaction
H(2)eff = −
12
4π(αE1~E2 + βM1
~H2),
H(3)eff = −
12
4π(γE1E1~σ · ~E × ~̇E + γM1M1~σ · ~H × ~̇H − 2γM1E2Eijσi Hj + 2γE1M2Hijσi Ej
),
H(4)eff = −
12
4π(αE1ν~̇E2 + βM1ν
~̇H2)−1
124π(αE2E2
ij + βM2H2ij ), . . .
Aij =12
(∇i Aj +∇j Ai ), A = ~E , ~H
this EFT breaks down around the pion production thresholdwe can calculate the polarisabilities from our more high-energy theory —
χPT... or find from fits to data (with some help of χPT)
Vadim Lensky (U. Mainz) Electroweak Box Workshop September 30, 2017 28 / 31
One more step: fit at O(p4) (partial)
add a dipole polarisabilities contact term; fit δαE1 and δβM1 to data
L(4)πN = πe2N
(δβM1FµρFµρ + 2
M2 (δαE1 + δβM1)∂µFµρF νρ∂ν
)N
results:VL, McGovern (2014)
αE1 = 10.6± 0.5
βM1 = 3.2± 0.5
Baldin constrainedχ2/d.o.f. = 112.5/136
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Vadim Lensky (U. Mainz) Electroweak Box Workshop September 30, 2017 29 / 31
Sum rules
sum rules connecting VCS, RCS, and VVCS:Pascalutsa, Vanderhaeghen (2015)
δLT = −γE1E1 + 3Mαem
[P′ (M1M1)1(0)− P′ (L1L1)1(0)
],
I′1(0) =κ2
N12〈r2
2 〉+M2
2
{1αem
γE1M2 − 3M[P′ (M1M1)1(0) + P′ (L1L1)1(0)
]}
verified in covariant and HB χPT VL, Pascalutsa, Vanderhaeghen, Kao (2017)
connect experimentally accessible quantitiesallow to obtain complementary information on, e.g., static spinpolarisabilitieshigher-order scalar sum rules VL, Pascalutsa, Vanderhaeghen, Hagelstein in preparation
Vadim Lensky (U. Mainz) Electroweak Box Workshop September 30, 2017 30 / 31
Sum rules and δLT puzzle
δLT = −γE1E1 + 3Mαem
[P′ (M1M1)1(0)− P′ (L1L1)1(0)
]
δLT puzzle shows here
the result of Bernard et al. for δLTseems to be in contradiction withMAID
new JLab data for the proton δLTare expected to shed light on thispuzzle
Vadim Lensky (U. Mainz) Electroweak Box Workshop September 30, 2017 31 / 31