some theory tools for neutrino interactions with nucleons RICHARD - - PowerPoint PPT Presentation
some theory tools for neutrino interactions with nucleons RICHARD HILL UKentucky and Fermilab WWONNI Fermilab 6 November, 2017 thanks to many collaborators and colleagues, including: J.Arrington, M. Betancourt, R. Gran, P. Kammel, A.
RICHARD HILL
UKentucky and Fermilab
1
thanks to many collaborators and colleagues, including: J.Arrington, M. Betancourt, R. Gran, P. Kammel, A. Kronfeld, G.Lee,
thanks Andreas and Pilar!
2
3
4
why bother with neutrino interactions? Isn’t this too hard/ too different/ somebody else’s problem?
4
why bother with neutrino interactions? Isn’t this too hard/ too different/ somebody else’s problem?
“The good news is that it’s not my problem”
HEP
10 20 30 200 400 600 800 1000 (GeV)
ν
E 1 10 CC evts/GeV/10kt/MW.yr
µ
ν 200 400 600 800 1000 Appearance Probability 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
CC spectrum
µ
ν =n/a
cp
δ = 0.0,
13
θ 2
2
sin /2 π =-
cp
δ = 0.1,
13
θ 2
2
sin =0
cp
δ = 0.1,
13
θ 2
2
sin /2 π =+
cp
δ = 0.1,
13
θ 2
2
sin
2
= 2.4e-03 eV
31 2
m ∆ CC spectrum at 1300 km,
µ
ν
5
long baseline neutrino oscillation experiment is simple in conception:
LBNE, 1307.7335
10 20 30 200 400 600 800 1000 (GeV)
ν
E 1 10 CC evts/GeV/10kt/MW.yr
µ
ν 200 400 600 800 1000 Appearance Probability 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
CC spectrum
µ
ν =n/a
cp
δ = 0.0,
13
θ 2
2
sin /2 π =-
cp
δ = 0.1,
13
θ 2
2
sin =0
cp
δ = 0.1,
13
θ 2
2
sin /2 π =+
cp
δ = 0.1,
13
θ 2
2
sin
2
= 2.4e-03 eV
31 2
m ∆ CC spectrum at 1300 km,
µ
ν
5
long baseline neutrino oscillation experiment is simple in conception:
LBNE, 1307.7335
10 20 30 200 400 600 800 1000 (GeV)
ν
E 1 10 CC evts/GeV/10kt/MW.yr
µ
ν 200 400 600 800 1000 Appearance Probability 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
CC spectrum
µ
ν =n/a
cp
δ = 0.0,
13
θ 2
2
sin /2 π =-
cp
δ = 0.1,
13
θ 2
2
sin =0
cp
δ = 0.1,
13
θ 2
2
sin /2 π =+
cp
δ = 0.1,
13
θ 2
2
sin
2
= 2.4e-03 eV
31 2
m ∆ CC spectrum at 1300 km,
µ
ν
5
long baseline neutrino oscillation experiment is simple in conception:
LBNE, 1307.7335
6
long baseline neutrino oscillation experiment is difficult in practice: simple picture is complicated by
neutrino interaction cross sections
need theory for σνe/σνμ, at ~% precision of measurement and also
aided by near detector but need theory for σνμ, at a precision depending on the experimental capabilities
7
current paradigm: constrain neutrino interactions by
modifications folk paradigms: constrain neutrino interactions by
constrain neutrino interactions by
8
in any paradigm: near detector has access to primarily νμ neutrinos νe appearance signal is directly impacted by νμ/νe cross section differences
having talked the talk, do some walking:
nuclear corrections: see talks of W. Van Order, S. Pastore, A. Ankowski, N. Jachowicz, A. Lovato. experiment: S. Bolognesi; lots of references: NUSTEC white paper 1706.03621
9
beyond neutrino oscillations related applications relying on quantitative nucleon structure:
entering a precision realm where percent level corrections to nucleon structure need to be calculated, not just estimated QED is “easy”. But QED + nucleon structure is “hard” Notes:
10
first, e-p elastic scattering second, ν-n CC scattering
11
first, e-p elastic scattering second, ν-n CC scattering
recall scattering from extended classical charge distribution:
e−
e− ρ(r)
dσ dΩ = ✓ dσ dΩ ◆
pointlike
|F(q2)|2
for the relativistic, QM, case, define radius as slope of form factor
F(q2) = Z d3r eiq·rρ(r) = Z d3r 1 + iq · r 1 2(q · r)2 + . . .
= 1 1 6hr2iq2 + . . .
hJµi = γµF1 + i 2mp σµνqνF2
GE = F1 + q2 4m2
p
F2
GM = F1 + F2
r2
E ≡ 6 d
dq2 GE(q2)
12
(up to radiative corrections)
Radius extraction requires data over a Q2 range where a simple Taylor expansion of the form factor is invalid
[sensitivity studies based on bounded z expansion fit]
13
data of Bernauer et al. (A1 collaboration), PRL 105, 242001 (2010)
Radius extraction requires data over a Q2 range where a simple Taylor expansion of the form factor is invalid
size of rE anomaly (hydrogen) [sensitivity studies based on bounded z expansion fit]
13
data of Bernauer et al. (A1 collaboration), PRL 105, 242001 (2010)
Radius extraction requires data over a Q2 range where a simple Taylor expansion of the form factor is invalid
size of rE anomaly (hydrogen) [sensitivity studies based on bounded z expansion fit]
13
Cut used for radius extraction
data of Bernauer et al. (A1 collaboration), PRL 105, 242001 (2010)
Radius extraction requires data over a Q2 range where a simple Taylor expansion of the form factor is invalid
size of rE anomaly (hydrogen) convergence radius for simple Taylor expansion [sensitivity studies based on bounded z expansion fit]
13
Cut used for radius extraction
data of Bernauer et al. (A1 collaboration), PRL 105, 242001 (2010)
14
coefficients in rapidly convergent expansion encode nonperturbative QCD
tcut F(q2) = X
k
ak[z(q2)]k
experimental kinematic region That’s ok: underlying QCD tells us that Taylor expansion
q2
particle thresholds
z
z(q2, tcut, t0) = p tcut − q2 − √tcut − t0 p tcut − q2 + √tcut − t0 ,
15
Reanalysis of scattering data reveals strong influence of shape assumptions Errors larger, but discrepancy remains
proton radius[fm]
electron combination
2S − 2P 1
22S − 2P 1
22S − 2P 3
22S − 4S 1
22S − 4D 5
22S − 4P 1
22S − 4P 3
22S − 6S 1
22S − 6D 5
22S − 8S 1
22S − 8D 3
22S − 8D 5
22S − 12D 3
22S − 12D 5
21S − 3S 1
2Mainz data
muonic H
15
Reanalysis of scattering data reveals strong influence of shape assumptions Errors larger, but discrepancy remains
proton radius[fm]
electron combination
2S − 2P 1
22S − 2P 1
22S − 2P 3
22S − 4S 1
22S − 4D 5
22S − 4P 1
22S − 4P 3
22S − 6S 1
22S − 6D 5
22S − 8S 1
22S − 8D 3
22S − 8D 5
22S − 12D 3
22S − 12D 5
21S − 3S 1
2Mainz data
reanalysis of Mainz data reanalysis of other world data Lee, Arrington, Hill (2015)
muonic H
16
0.8 0.9 1 1.1
2S − 2P1/2 2S − 2P1/2 2S − 2P3/2 2S − 4S 1/2 2S − 4D5/2 2S − 4P1/2 2S − 4P3/2 2S − 6S 1/2 2S − 6D5/2 2S − 8S 1/2 2S − 8D3/2 2S − 8D5/2 2S − 12D3/2 2S − 12D5/2 1S − 3S 1/2
e-p Mainz e-p world CODATA 2010 electron comb. e-p Mainz (z exp.) e-p world (z exp.) CODATA 2014 electron comb. µD + iso. H 2S-4P (sensitivity) low-Q2 e-p (sensitivity) µ-p (sensitivity)
rp
E (fm)
CREMA µH 2010→ CREMA µH 2014→
update: Beyer et al. (Science, 2017)
17
first, e-p elastic scattering second, ν-n CC scattering
18
Start with the basic process
n p μ- νμ
σ(νn → µp) = | · · · FA(q2) · · · |2
A common ansatz for FA has been employed for the last ~40 years:
F dipole
A
(q2) = FA(0) ✓ 1 − q2 m2
A
◆2
rA = 0.674(9) fm
1 FA(0) dFA dq2
⌘ 1 6r2
A
Typically quoted uncertainties are (too) small (e.g. compared to proton charge form factor!) Inconsistent with QCD.
19
n p μ- νμ p p
deuteron Deuterium bubble chamber data
Fermilab 15-foot deuterium bubble chamber, PRD 28, 436 (1983)
28
HIGH-ENERGY QUASIELASTIC v„n ~@ p SCATTERING IN. . . 439
80
60
OtP co.
h4, =1.05 GeV
tion, the
following assumptions
are made:
(1) time-
reversal invariance and charge symmetry, (2) partially
con-
served axial-vector
current
(PCAC} for the small pseudo-
scalar term,
and
(3) isotriplet-conserved-vector-current
(CVC) hypothesis. The first assumption,
which requires all form factors to
be real, yields Eq —
—
F~—
—
0, leading to the absence of second
class currents. With the second
assumption,
Fp(Q
) is
given by
20-
Fp(Q )=2M Fg(Q~)/(Q
+m ),
where
'0
2
Q' (Gev')
The
Q distribution
for the
selected quasielastic events.
The solid curve represents
the differential
cross section
scattering for the neutron
in deuteron.
Q'= (P —
P„)'—
(E„—
E„)' .
The contribution to the cross section from this term in the
energy region E„&5 GeV is less than 0.1%, and conse- quently
this term is neglected.
The third
assumption re- lates Fz and Fz to the isovector Sachs electric and mag- netic form factor, Gz and G~ determined from electron- scattering experiments as follows: near /=0 . The shaded area corresponds
to the addition-
al events found from the rescan. Using the average of the events with P between —
90
and
126
(dashed line), we calculated
the event bias to be
S%%uo. This does not neces-
sarily represent
the true
loss of events because
three-point plot per event.
We examined the true event
loss from the event bias in Fig. 4 by using a Monte Carlo simulation.
This
event loss amounts
to 8% and
is not recovered by rescanning (shaded
area).
Hence, a correc- tion of 1.08+0.05 has been made to the data independent
efficiency. Figure
5 shows
the Q distribution
for the quasielastic
events.
The curve in Fig. 5 is the best fit obtained
by us- ing the prediction of the differential
cross section for reac- tion
(2) with
M~ —
—
1.05 GeV which
was obtained
from this experiment
(see Sec. III). The X value from this ftt was found to be 15 for 20 data points for Q between 0.1 and 3 GeV . Comparing
the observed Q distribution
to
the fitted curve, the correction factor for Q &0.1 GeV2 is estimated
to be 1.10+0.02. The overall
correction factor
including scanning-measuring
efficiency
is
1.34+0.07.
We note that this correction factor influences the value of
the neutrino flux but not the Mz value, because we use a flux-independent method
to determine
Mq.
2 2
Fy(Q') = G~(Q')+
—
G (Q')
1+
4M 4M
2
' — 1
Ff(Q )=[6M(Q
)—
GE(Q )]g
' 1+ 4M
2
' —
2
GE(Q }=6M(Q }(1+/)
=A(Q
) 1+
My
where M~ is the vector mass, Mv —
—
0.84 GeV, g is the
difference
between
the proton
and neutron anomalous magnetic moment,
g'=}Mp —
p„=3.708,
and
A,(Q
) (Ref. 1S) is the correction factor for the small
deviation
data from a pure di-
pole
form
factor.
We further
assume
the axial-vector form factor in a dipole form,
+g(Q )=+g(0)/(I+Q
/Mg
)
where
the value of F~(0)= —
1.23+0.01 is taken from P-
decay experiments. '
From these
assumptions, the differential
cross section
for the quasielastic
reaction can be expressed
in terms of
Mz, as
In the context of the V—
A theory,
the matrix
element
for the quasielastic
reaction,
v&n ~p p, can be written
as
a product of the hadronic
weak current and the leptonic
The general
form of the hadronic
weak current is written in terms of six complex form factors which are
functions
and
characterize
the nucleon
structure. These are Fs (induced scalar), Fp (induced pseudoscalar),
F~ (isovector
Dirac), Ff (isovector
Pauli), F~ (axial vec-
tor}, and Fr (induced
tensor).
The quasielastic
cross sec- tion can be expressed
in terms of these six form factors.
In order to simplify
the analysis of the quasielastic reac-
GMcos8c
2
2 (s
u)
&( ')+&(
)
dQ
C(Q2) (s —
u)
(7)
where
s —
u =4ME„Q
m&,
and M =(M„+—
Mp)—
/2.
The values of the Fermi constant
and of the Cabibbo angle
are taken
to
be G = 1.166 32 & 10
GeV
and
cos8c —
—
0.9737, respectively
(see Ref. 16). The structure
Best source of almost-free neutrons: deuterium
ANL 12-foot deuterium bubble chamber, PRD 26, 537 (1982) BNL 7-foot deuterium bubble chamber, PRD23, 2499 (1981) also:
20
[a1, a2, a3, a4] = [2.30(13), −0.6(1.0), −3.8(2.5), 2.3(2.7)] (31)
Cij = B B B @ 1 0.350 −0.678 0.611 0.350 1 −0.898 0.367 −0.678 −0.898 1 −0.685 0.611 0.367 −0.685 1 1 C C C A
]
2
[GeV
2
Q
1 2 3
)
2
(-Q
A
F
0.5 1
=4 z expansion
a
N = 1.014(14) dipole
A
m
z
0.2 0.4
(z)
A
F
0.5 1
=4 z expansion
a
N = 1.014(14) dipole
A
m
21
Derived observables: 1) axial radius
1 FA(0) dFA dq2
⌘ 1 6r2
A
capture
r2
A = 0.46(22) fm2 .
22
Derived observables: 2) neutrino-nucleon quasi elastic cross sections
[GeV]
ν
E
10 1 10
]
2
)[cm
ν
(E σ
5 10 15
10 ×
=4 z expansion
a
N = 1.014(14) dipole
A
m
[GeV]
ν
E
10 1 10
]
2
)[cm
ν
(E σ
5 10 15
10 ×
=4 z expansion
a
N = 1.014(14) dipole
A
m
σνn→µp(Eν = 1 GeV) = 10.1(0.9) × 10−39 cm2
−39 2
→
× σνn→µp(Eν = 3 GeV) = 9.6(0.9) × 10−39 cm2
]
2
[GeV
2
Q
0.5 1 1.5 2
]
2
/GeV
2
[cm
2
/dQ σ d
5 10 15 20
10 ×
GENIE RFG z-expansion GENIE RFG dipole MINERvA Data
23
discriminating nuclear models
n p μ- νμ poorly known axial form factor
σ(νn → µp) = | · · · FA(q2) · · · |2
want to extract nuclear and flux effects from this comparison: but large nucleon level form factor uncertainty
24
25
W + p µ− n νµ
muon capture from ground state of muonic hydrogen:
, FA
from RJH, Kammel, Marciano, Sirlin 1708.08462
26
W + p µ− n νµ
L = GF Vud p 2 ¯ νµγµ(1 γ5)µ ¯ dγµ(1 γ5)u + H.c. + . . . ,
H = p2 2mr α r + δVVP iG2
F |Vud|2
2 ⇥ c0 + c1(sµ + sp)2⇤ δ3(r) ,
Λ = G2
F |Vud|2 ⇥ [c0 + c1F(F + 1)] ⇥ |ψ1S(0)|2 + . . .
perturbative matching nonperturbative matching
L = LSM
weak hadronic atomic
factorization:
c0 = E2
ν
2πM2 (M mn)2 2M mn M mn F1(q2
0) + 2M + mn
M mn FA(q2
0) mµ
2mN FP (q2
0)
+ (2M + 2mn 3mµ)F2(q2
0)
4mN 2 , ⇢
27
W + γ p µ− n νµ
expansion in small quantities:
✏ ∼ ↵ ∼ m2
µ
m2
ρ
∼ mu − md mρ . 10−2
ensure against numerical enhancements (need these to ~100%)
28
momentum expansion:
q2
0 ⌘ m2 µ 2mµEν = 0.8768 m2 µ.
1 + g1, gA
g2
r2
1, r2 A, gP
sensitivity to momentum dependence in the capture process in our power counting, rA2 competes with gP , and other well-determined quantities (g≡normalization, r2≡slope)
∼ ✏
FP (q2
0) = 2mNgπNNfπ
m2
π − q2
− 1 3 gA m2
N r2 A + . . .
gpiNN: pion-nucleon scattering, and NN scattering
gA: neutron lifetime
g1,g2,r12: e-p, e-n scattering + H, muH (see below)
29
α expansion: Sirlin g function (IR subtraction)
RC = RC(electroweak) + RC(finite size) + RC(electron VP) ,
matching, running in 4 Fermi theory
hadronic matrix element
computed within QM
RC(electroweak) = α 2π 4 log mZ mp 0.595 + 2C + g(mµ, βµ = 0)
RC(electron VP) = +0.0040(2),
RC(finite size) = 0.005(1)
finite terms (estimate with OPE) large log
(should be done better: computed in large nucleus ansatz rE>>rA)
30
isospin violation:
hn|(V µ Aµ)|pi = ¯ un F1(q2)γµ+ iF2(q2) 2mN σµνqν FA(q2)γµγ5 FP (q2) mN qµγ5 + FS(q2) mN qµ iFT (q2) 2mN σµνqνγ5
vector form factors: CC from isovector NC 2nd class currents: deviations in F1(0): second order in IV (definition of CVC) deviations in F1(q2): first order in IV plus first order in q2 deviations in F2(0): first order in IV plus 0.5 order in kinematic prefactor (numerical estimate: 3.2e-4 << %)
contribution of FS,FT: first order in IV plus 0.5 order in kinematic prefactor
31
results:
¯ gP
MuCap
A=0.46(22) fm2 = 8.19 (48)exp (69)¯
gA (6)RC = 8.19(84)
¯ gP
theory = 8.25(25)
gMuCap
πNN
= 13.04 (72)exp (8)gA (67)r2
A (10)RC = 13.04(99)
, gexternal
πNN
= 13.12(10)
r2
A(MuCap) = 0.43 (24)exp (3)gA (3)gπNN (3)RC = 0.43(24) fm2.
turning the tables, take QCD for granted and extract rA2: competitive with other methods with existing data, and potential for improvement
A
δr2
A(future exp.) = (0.08)exp (0.03)gA (0.03)gπNN (0.03)RC = 0.10 fm2.
factor 3 improvement
32
0.2 0.4 0.6 0.8 1
r2
A (fm2)
νd (dipole) [17] eN → eNπ (dipole) [17] νC (dipole) [20] νd (z exp.) [19] MuCap this work LHPC [21] ETMC [22] CLS [23] PNDME [24] lattice QCD
W + p µ− n νµ
experiment
muon capture constraints
lattice average: see also Yao, Alvarez-Ruso, Vicente-Vacas 1708.08776 [ rA2=0.26(4) ] RJH, Kammel, Marciano, Sirlin 1708.08462
33
2 4 6 8 10 0.7 0.8 0.9 1 1.1 1.2
Eν (GeV) σ (10−38cm2)
existing error (no external radius constraint) with radius constraint: ( hatched:
external radius error δrA2=20% )
implications for quasielastic neutrino cross sections
34
0.2 0.4 0.6 1.26 1.27 1.28 1.29
r2
A (fm2)
gA
δΛ = 1% δΛ = 0.33%
test of electron-muon universality
electron coupling (neutron lifetime) current uncertainty muon coupling (current uncertainty)
35
36
p p p p p p p p
⇥
k µ p
p p
= e pµ p · k
⇒ exponentiation of IR divergences, cancellation between real and virtual
+ . . .
Yennie, Frautschi, Suura (1961)
But exponentiation of IR divergences does not imply exponentiation of the entire first order correction
Large logarithms spoil QED perturbation theory when -q2=Q2~GeV2
|F(q2)|2 → |F(q2)|2 ✓ 1 − α π log Q2 m2
e
log E2 (∆E)2 + . . . ◆
≈ 0.5
Experimental ansatz sums exponentiates 1st order:
|F(q2)|2 ✓ 1 − α π log Q2 m2
e
log E2 (∆E)2 + . . . ◆ → |F(q2)|2 exp − α π log Q2 m2
e
log E2 (∆E)2
Q ∼ E , ∆E ∼ me
As consistency check, error budget should contain the difference from resumming:
log2 Q2 m2
e
log Q2 m2
e
log E2 (∆E)2
vs.
e− e− e− e−
37
∆E q e− e−
38
0.75 0.80 0.85 0.90 0.95 1.00 rE [fm] 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Q2
max [GeV2]
0.6
Q2 [GeV2] δ
1st order in α 2nd order in α
scattering data are 0.2-0.5 %
subleading logarithms, recoil, nuclear charge and structure
total radiative correction
∆E = 5 MeV
electron energy loss cut:
E = 1 GeV
electron energy:
38
0.75 0.80 0.85 0.90 0.95 1.00 rE [fm] 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Q2
max [GeV2]
0.6
> 5σ discrepancy
Q2 [GeV2] δ
1st order in α 2nd order in α
scattering data are 0.2-0.5 %
subleading logarithms, recoil, nuclear charge and structure
total radiative correction
∆E = 5 MeV
electron energy loss cut:
E = 1 GeV
electron energy:
38
0.75 0.80 0.85 0.90 0.95 1.00 rE [fm] 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Q2
max [GeV2]
0.6
potentially large uncertainty from radiative corrections
Q2 [GeV2] δ
1st order in α 2nd order in α
scattering data are 0.2-0.5 %
subleading logarithms, recoil, nuclear charge and structure
total radiative correction
∆E = 5 MeV
electron energy loss cut:
E = 1 GeV
electron energy:
39
p p ⇥ p p ⇥
treat the problem in stages:
proton electron
p p ⇥
p p p p
⇥⇥
static source,
static source,
with recoil corrections with nuclear charge corrections (two photon exchange)
40
dσ / H ✓Q2 µ2 ◆ J ✓m2 µ2 ◆ R ✓m2 µ2 , p · p0 m2 ◆ S ✓∆E µ , p · p0 m2 , E m, E0 m ◆
=
+ +
+
[ remainder function starting at 2-loop (collinear anomaly/rapidity logs) ]
Becher, Melnikov (2007) Chiu, Golf, Kelley, Manohar (2007) Becher, Neubert (2010) Chiu, Jain, Neill, Rothstein (2012) …
41
Hard
Large logarithms regardless of choice for μ FS: exponentiates (evaluate at any scale) FJ: evaluate at μ~m FH: evaluate at μ~M~Q
FH(µ) = 1 + α 4π − log2 Q2 µ2 + 3 log Q2 µ2 − 8 + π2 6
FS(µ) = 1 + α 4π 2 log λ2 µ2 ✓ log Q2 m2 − 1 ◆
Collinear FJ(µ) = 1 + α
4π log2 m2 µ2 − log m2 µ2 + 4 + π2 6
Sudakov form factor at one loop: (two-loop matching, real+virtual see 1605.02613)
42
Two photon exchange
(helicity counting: 3 amplitudes at leading power in me/Q)
FH(µ)γµ ⊗ γµ →
3
X
i=1
ci(µ) Γ(e)
i
⊗ Γ(p)
i
1-photon exchange and 2-photon exchange contributions to ci
photon and 2-photon contributions.
radiative corrections are universal
43
p S(µ, ∆E = 0) = Z(e)
h Z(p) h
+ + + + +
⇤ ⇥ ⇤
loop order)
dσ = H(M) × H(µ) H(M) × J(µ) × S(µ)
44
d log H d log µ = 2 γcusp(¯ α) log Q2 µ2 + γcusp(v · v0, ¯ α) + 2γcusp(¯ α) log v · p0 −v · p − i0 + γ(¯ α)
proton : Mvµ electron : pµ
universal functions
log H(µL) H(µH) = − α 2π log2 µ2
H
µ2
L
+ . . .
¯ hiv · Dh → ¯ h(0)S†
viv · DSvh(0) = ¯
h(0)iv · ∂h(0) , Sv(x) = P exp i Z 0
−∞
dsv · As(x + sv)
v0 governed by Wilson loops with cusps: renormalization of hard function of interest: solution, summing large logarithms:
45
)
2
(GeV
2
Q
0.2 0.4 0.6 0.8 1
δ
0.35 − 0.3 − 0.25 − 0.2 − 0.15 −
LL NLL NLO
dσ = H(M) × H(µ) H(M) × J(µ) × S(µ)
numerically: αL2 = α log2 Q2
m2 ∼ 1
αL ∼ α
1 2
, etc.
O(1) O(α
1 2 )
O(α)
correct through:
∆E = 5 MeV
electron energy loss cut:
E = 1 GeV
electron energy:
46
Comparison to previous implementations of radiative corrections, e.g. in A1 analysis of electron-proton scattering data
)
2
(GeV
2
Q
0.2 0.4 0.6 0.8 1
δ
0.25 − 0.24 − 0.23 − 0.22 − 0.21 − 0.2 −
resummed EFT result naive exponentiation of 1-loop, (μ2=Q2 in two-photon piece) naive exponentiation of 1-loop, (μ2=M2 in two-photon piece)
shape variations when fitting together with backgrounds
correction models (cf. 0.2-0.5% systematic error budget of A1 experiment)
total radiative correction
∆E = 5 MeV E = 1 GeV
47
EFT analysis clarifies several issues involving conflicting and/or implicit conventions and scheme choices
1) Scheme choice and definition of radius and “Born” form factors 2) Scheme dependence of two-photon exchange 3) Limitations of naive exponentiation
48
1) Scheme choice and definition of radius and “Born” form factors Fi(q2)Born ⌘ ˜ Fi(q2)F 1
S (w, µ = M)
hJµi = ¯ uv0 ˜ F1γµ + ˜ F2 i 2σµν(v0
ν vν)
hard coefficient soft function
˜ Fi = FHFS
Multiple conventions in the literature. Different “Born” form factors, different radii (differences typically below current precision)
FH(q2, µ = M) ≡
Massive particle form factor (e.g. for proton):
49
2) Scheme dependence of two-photon exchange
As for form factors, define hadronic functions in the general 2→2 scattering process as the hard component in the factorization formula at factorization scale μ=M Prevailing conventions have used conflicting μ=M for 1 photon exchange, μ=Q for 2 photon exchange
)
2
(GeV
2
Q
0.2 0.4 0.6 0.8 1
δ
0.25 − 0.24 − 0.23 − 0.22 − 0.21 − 0.2 −
A scale-variation estimate of uncertainty in the 2 photon exchange subtraction μ=M μ=Q
total radiative correction
50
3) Limitations of naive exponentiation
⇒ New terms at order α2 L3, α2 L2, α3 L4, …
S(2) = 1 2![S(1)]2 − 16π2 3 (L − 1)2 .
⇒ New terms at order α2 L2
log H(µL) H(µH) = − α 2π log2 µ2
H
µ2
L
+ . . .
complete analysis: account for floating normalizations, correlated shape variations when fitting together with backgrounds.
S = X
n
⇣ α 4π ⌘n S(n)
a difficult archeological problem. PRP from e-p appears to require something more (expt. syst.: ? / theory systematic: hard TPE)
51
52
better for elementary amplitudes
and world’s best (in a tie) rA determination
exclusive νe/νμ analysis and cautionary tale for % level
differences and ν amplitudes at the nucleon level