1

20 April 2016RICHARD HILL

some discussion points for precision muon physics

1) proton radius puzzle

2) muon capture and neutrino cross sections

3) some interesting η decays

2

• muons a great tool

- new physics sensitivity:

- nuclear size from spectroscopy:

| (0)µH |2

| (0)eH |2 ⇠m3

µ

m3e

⇠ 106

�aµ�ae

⇠(m2

µ/M2)

(m2e/M

2)⇠ 104

- nuclear structure from capture:

mµ +mp > mn (> me +mp)

EPJ Web of Conferences

0.8 0.9 1 1.1

PSfrag replacements

2S � 2P1/22S � 2P1/22S � 2P3/22S � 4S 1/22S � 4D5/22S � 4P1/22S � 4P3/22S � 6S 1/22S � 6D5/22S � 8S 1/22S � 8D3/22S � 8D5/22S � 12D3/22S � 12D5/21S � 3S 1/2

e-p Mainze-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)

rpE (fm)

CREMA µH 2010!

CREMA µH 2014!

Figure 13. Status of the proton radius puzzle circa 2016, with prospects for new data. The upper pane isreproduced from Fig. 1. The middle pane shows updated results. The cyan points give updated fits to electronscattering data using z expansion (final two points in Fig. 4, from Ref. [15]. The black point represents the 2014CODATA [1] combination of hydrogen and electron-proton scattering determinations. The red point is fromthe 2016 CREMA muonic deuterium Lamb shift measurement using the regular hydrogen-deuterium isotopeshift [73]. The bottom pane shows expected sensitivities of anticipated results in: regular hydrogen [78] (blue);low-Q2 electron-proton scattering [90] (cyan); and muon-proton scattering [92] (magenta). See text for details.

4.5 Summary of status and prospects

Figure 13 displays the current status of the proton radius puzzle. Compared to Fig. 1, the muonichydrogen error bar has been increased to reflect updates and a revised treatment of TPE in Ref. [71],and the new muonic deuterium data point has been included. The electron scattering results reflectthe treatment of form factor nonlinearities and more conservative systematic errors from Ref. [15]. In

1702.01189

1) the proton radius puzzle

3

4

Implications: 1) new window on fundamental constants

- 6.9 sigma shift in Rydberg

2) canary in the gold mine for lepton-nucleon interactions, critical for neutrino program

3) something else?

- form factor nonlinearities

- radiative corrections21

0.75

0.80

0.85

0.90

0.95

1.00

r E[fm

]

FIG. 9: Illustrative fit with modified radiative correctionsgiven by Eq. (41) using �E = 10MeV. Lower and upperdashed blue lines correspond to the plus sign and minus signin Eq. (41), respectively. Fits are for the 657 point rebinnedA1 MAMI dataset with 0.3–0.4% uncorrelated systematic un-certainties using the z expansion with t

0

= 0, Gaussian priorswith |ak|max

= |bk|max

/µp = 5, kmax

= 12. Black solid linesreproduce the curves in Fig. 6. For orientation, the dash-dotted red line indicates the muonic hydrogen value for rE .

are fixed by infrared divergences whose form is dictatedby soft photon theorems [75]. Equivalently, an e↵ectivetheory renormalization analysis between hard (⇠ Q) andsoft (⇠ me) scales determines the relevant Sudakov formfactor. However, in practice �E can be large comparedto me, introducing another scale into the problem, andassociated large logarithms not captured by the naive ex-ponentiation of one-loop corrections. A complete analy-sis is outside the scope of the present paper, but to illus-trate the potential impact, let us consider in place of theansatz that makes the replacement (31) in Eq. (29), thefollowing expressions:

(1 + �) !1 ±

✓� +

↵

⇡

log2

Q

2

m

2

e

◆�±1

⇥ exp

✓�↵

⇡

log2

Q

2

m

2

e

◆. (41)

These expressions agree with the known correctionsthrough one-loop order, and resum the leading loga-rithms to all orders in perturbation theory when thereis only one large ratio of scales.

Fig. 9 illustrates the impact of applying the correctionon the right hand side of Eq. (41) in place of the ansatz

(31). For definiteness, the plot takes �E = 10MeV.As indicated in the figure, the shifts in the radii underthis correction are a factor ⇠ 2–3 larger than those al-lowed in Table IX, which considered corrections vary-ing by 0.5% over beam-energy/spectrometer combina-tions. The variation of the correction (41) over beam-energy/spectrometer combinations (i.e., the magnitudeof a in Eq. (35)) ranges between 0.9% and 2.6%, with anaverage 1.5%.

D. Final radius extractions

Q

2

max

[GeV2]

�r[fm

]

FIG. 10: Statistical error on rE (bottom, red squares) and rM

(top, blue circles) as a function of Q2

max

. Solid symbols arefor the 1422 point A1 MAMI dataset, and open symbols arefor the world cross section and polarization dataset. Fits usethe z expansion with t

0

= 0, Gaussian priors with |ak|max

=|bk|max

/µp = 5, kmax

= 12.

A global analysis combining Mainz and other worlddata will artificially favor the Mainz data, as the un-certainties associated with each cross section measure-ment include only a small part of the total uncertainty.Thus, we provide best fit values separately for our anal-yses of Mainz and world data. To determine an opti-mal Q

2

max

, Fig. 10 illustrates the statistical uncertaintyon rE and rM found using our default fit both to the1422 point Mainz dataset and to the world dataset. Forthe Mainz data, the uncertainty is minimized by tak-ing Q

2

max

& 0.5 GeV2, with negligible improvement be-yond this point. In order to maximize the statisticalpower of the data, while minimizing potential system-atic e↵ects in higher Q

2 data, we take for definiteness

21

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0Q2

max [GeV2]

0.6

FIG. 9: Illustrative fit with modified radiative correctionsgiven by Eq. (41) using �E = 10MeV. Lower and upperdashed blue lines correspond to the plus sign and minus signin Eq. (41), respectively. Fits are for the 657 point rebinnedA1 MAMI dataset with 0.3–0.4% uncorrelated systematic un-certainties using the z expansion with t

0

= 0, Gaussian priorswith |ak|max

= |bk|max

/µp = 5, kmax

= 12. Black solid linesreproduce the curves in Fig. 6. For orientation, the dash-dotted red line indicates the muonic hydrogen value for rE .

are fixed by infrared divergences whose form is dictatedby soft photon theorems [75]. Equivalently, an e↵ectivetheory renormalization analysis between hard (⇠ Q) andsoft (⇠ me) scales determines the relevant Sudakov formfactor. However, in practice �E can be large comparedto me, introducing another scale into the problem, andassociated large logarithms not captured by the naive ex-ponentiation of one-loop corrections. A complete analy-sis is outside the scope of the present paper, but to illus-trate the potential impact, let us consider in place of theansatz that makes the replacement (31) in Eq. (29), thefollowing expressions:

(1 + �) !1 ±

✓� +

↵

⇡

log2

Q

2

m

2

e

◆�±1

⇥ exp

✓�↵

⇡

log2

Q

2

m

2

e

◆. (41)

These expressions agree with the known correctionsthrough one-loop order, and resum the leading loga-rithms to all orders in perturbation theory when thereis only one large ratio of scales.

Fig. 9 illustrates the impact of applying the correctionon the right hand side of Eq. (41) in place of the ansatz

(31). For definiteness, the plot takes �E = 10MeV.As indicated in the figure, the shifts in the radii underthis correction are a factor ⇠ 2–3 larger than those al-lowed in Table IX, which considered corrections vary-ing by 0.5% over beam-energy/spectrometer combina-tions. The variation of the correction (41) over beam-energy/spectrometer combinations (i.e., the magnitudeof a in Eq. (35)) ranges between 0.9% and 2.6%, with anaverage 1.5%.

D. Final radius extractions

Q

2

max

[GeV2]

�r[fm

]

FIG. 10: Statistical error on rE (bottom, red squares) and rM

(top, blue circles) as a function of Q2

max

. Solid symbols arefor the 1422 point A1 MAMI dataset, and open symbols arefor the world cross section and polarization dataset. Fits usethe z expansion with t

0

= 0, Gaussian priors with |ak|max

=|bk|max

/µp = 5, kmax

= 12.

A global analysis combining Mainz and other worlddata will artificially favor the Mainz data, as the un-certainties associated with each cross section measure-ment include only a small part of the total uncertainty.Thus, we provide best fit values separately for our anal-yses of Mainz and world data. To determine an opti-mal Q

2

max

, Fig. 10 illustrates the statistical uncertaintyon rE and rM found using our default fit both to the1422 point Mainz dataset and to the world dataset. Forthe Mainz data, the uncertainty is minimized by tak-ing Q

2

max

& 0.5 GeV2, with negligible improvement be-yond this point. In order to maximize the statisticalpower of the data, while minimizing potential system-atic e↵ects in higher Q

2 data, we take for definiteness

} discrepancy with muonic hydrogen

� e2

Q2F (Q2) ! � e2

Q2F (Q2)⌥ g2

Q2 +m2V

• depending on mass, consistent with reH ~ rμH < re-p

new force?

1505.01489

5

)2 (fm2Ar

0.3 0.4 0.5 0.6

Ag

1.25

1.26

1.27

1.28

1.29

1.3

(GeV)νE0 2 4 6 8 10

)2 c

m-3

8 (1

0σ

0.8

1

1.2

1603.03048

w/ P. Kammel, W. Marciano, A. Sirlin

- there is a surprisingly large uncertainty on the (CCQE) signal cross section for neutrino experiments

- cross section uncertainty traced to axial-vector nucleon form factor

- form factor uncertainty dominated by nucleon axial radius

2) muonic hydrogen and the nucleon axial radius

- new constraints from capture rate in μH (currently: competitive with neutrino-deuteron scattering. future: better)

6

This yields a branching fraction,

�(⌘ ! ⇡

0

µ

+

µ

�)BSM

�tot

⇡ 3⇥ 10�3 ⇥✓0.07

⇠

◆2

⇥✓�R

2

p ��R

2

n

�0.054fm2

◆2

. (37)

A comparison with the current experimental limit, shows that this contact interaction model

is already disfavored by the data by 3 orders of magnitude. Let us explore whether other

SM extensions provide viable phenomenological extensions of the atomic EFT.

IV. LIGHT MEDIATOR

The constraints of ⌘ ! ⇡

0

µ

+

µ

� decay can be alleviated if the contact interaction for

the atomic system is viewed as arising from a model with light scalar mediator coupling to

muons and light quarks,

L =1

2(@µS)

2 � 1

2m

2

SS2 + gµSµµ+ guSuu+ gdSdd . (38)

At energies below mS, we recover the contact interactions with

cu =gµgu

m

2

S

, cd =gµgd

m

2

S

. (39)

For atomic applications where mS is not integrated out we may also consider the hadronic

e↵ective lagrangian,

L =1

2(@µS)

2 � 1

2m

2

SS2 + gµSµµ+ gpSp

†p+ gnSn

†n . (40)

The couplings gp,n are related to gu,d as

gp =⌃⇡N

mu +md[(gu + gd) + (gu � gd)⇠] ,

gn =⌃⇡N

mu +md[(gu + gd)� (gu � gd)⇠] , (41)

A. Muonic spectroscopy

Let us examine the constraints on the scalar couplings gµ,p,n and scalar mass mS from

muonic spectroscopy. In the notation of Ref. [4],

gpgµ

m

2

S

f

✓mS

↵mr

◆= �e

2

6

�[R2

p](µ) � [R2

p](e)�,

10

This yields a branching fraction,

�(⌘ ! ⇡

0

µ

+

µ

�)BSM

�tot

⇡ 3⇥ 10�3 ⇥✓0.07

⇠

◆2

⇥✓�R

2

p ��R

2

n

�0.054fm2

◆2

. (37)

A comparison with the current experimental limit, shows that this contact interaction model

is already disfavored by the data by 3 orders of magnitude. Let us explore whether other

SM extensions provide viable phenomenological extensions of the atomic EFT.

IV. LIGHT MEDIATOR

The constraints of ⌘ ! ⇡

0

µ

+

µ

� decay can be alleviated if the contact interaction for

the atomic system is viewed as arising from a model with light scalar mediator coupling to

muons and light quarks,

L =1

2(@µS)

2 � 1

2m

2

SS2 + gµSµµ+ guSuu+ gdSdd . (38)

At energies below mS, we recover the contact interactions with

cu =gµgu

m

2

S

, cd =gµgd

m

2

S

. (39)

For atomic applications where mS is not integrated out we may also consider the hadronic

e↵ective lagrangian,

L =1

2(@µS)

2 � 1

2m

2

SS2 + gµSµµ+ gpSp

†p+ gnSn

†n . (40)

The couplings gp,n are related to gu,d as

gp =⌃⇡N

mu +md[(gu + gd) + (gu � gd)⇠] ,

gn =⌃⇡N

mu +md[(gu + gd)� (gu � gd)⇠] , (41)

A. Muonic spectroscopy

Let us examine the constraints on the scalar couplings gµ,p,n and scalar mass mS from

muonic spectroscopy. In the notation of Ref. [4],

gpgµ

m

2

S

f

✓mS

↵mr

◆= �e

2

6

�[R2

p](µ) � [R2

p](e)�,

10

• new physics models for proton radius puzzle have large isospin violation, enhanced muon couplings

• consider the rare process:

• η decays very constraining: present experimental limit, Br < 5 x 10-6, disfavors scalar contact interaction model

• light scalar mediator extensions a potential target for future experiments

electron-muon radius differences for proton, neutron (from μH, μD)

B. Model independent contact interaction: quark level

In order to make contact with UV physics, let us now consider a quark-level contact

interaction lagrangian,

L =⇥cuuu+ cddd

⇤µµ . (25)

Here cu,d are coe�cients of mass dimension minus two, scaling under QCD renormalization as

quark masses. Using the notation of Ref. [11], the matching onto the nucleon-level lagrangian

results in the equations,

cu

mumNfu,N +

cd

mdmNfd,N =

�d

2N

mµmN, (26)

for N = n, p. Consider the matrix elements

⌃⇡N ⌘ mu +md

2hN |(uu+ dd)|Ni = 44(13)MeV ,

⌃� ⌘ (mu �md)hN |(uu� dd)|Ni = ±2(2)MeV , (27)

with the upper (lower) sign referring to N = p (N = n), and introduce

⇠ =mu +md

md �mu

|⌃�|2⌃⇡N

= 0.07(7) , (28)

using mu/md = 0.49(13). Then Eqs. (26) are equivalent to

⌃⇡N

mu +md[(cu + cd) + (cu � cd)⇠] =

�d

2p

mµmN,

⌃⇡N

mu +md[(cu + cd)� (cu � cd)⇠] =

�d

2n

mµmN, (29)

or solving for cu,d,

cu + cd =mu +md

2⌃⇡N

✓�d

2p

mµmN+

�d

2n

mµmN

◆,

cu � cd =mu +md

2⌃⇡N

1

⇠

✓�d

2p

mµmN� �d

2n

mµmN

◆. (30)

Note that the smallness of the neutron coe�cient requires a cancellation between isovector

and isoscalar combinations, and an enhanced isovector combination to compensate for the

smallness of ⇠. These values provide canonical points that explain the muonic hydrogen

anomaly while remaining consistent with muonic deuterium.

9

• light vector models, η→γV (V→ee, …) another potential target

3) η decays and new light particles

e.g.