Physics Letters B 746 (2015) 315-317

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Physics Letters B

www.elsevier.com/locate/physletb

Hyperfine splitting in muonic hydrogen constrains new pseudoscalar interactions

W.-Y. Keunga'c, D. Marfatiab,c'*

a Department of Physics, University of Illinois at Chicago, Chicago, IL 60607, USA b Department of Physics and Astronomy, University of Hawaii, Honolulu, HI 96822, USA c Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA

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A R T I C L E I N F 0

Article history:

Received 28 March 2015

Received in revised form 4 May 2015

Accepted 11 May 2015

Available online 14 May 2015

Editor: M. Cvetic

A B S T R A C T

We constrain the possibility of a new pseudoscalar coupling between the muon and proton using a recent measurement of the 2S hyperfine splitting in muonic hydrogen.

© 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

Recent measurements of 2S-2 P transition frequencies in the exotic atom constituted by a proton orbited by a muon [1,2] find the proton charge radius to be la smaller than the 2010-CODATA [3] value obtained using ordinary hydrogen and e-p scattering. The 2S hyperfine splitting deduced from the same measurements shows excellent agreement with predictions [1]. The discrepancy in the proton radius has generated a lot of interest, including the invocation of new fundamental interactions as an explanation. Here, we focus on the implications of the hyperfine splitting for new interactions between the muon and proton. Specifically, we consider the possibility of a new pseudoscalar particle that couples to the muon and proton. Such an interaction is spin and velocity dependent and has a negligible effect on the Lamb shift (which is used to extract the proton radius) in the nonrelativistic limit [4], but has a significant effect on the hyperfine splitting.

The measured value of the 2S hyperfine splitting (HFS) [1]

AEhfs = 22.8089 ± 0.0051 meV

is to be compared with the theoretical prediction [5]

AEffFS = (22.9843 ± 0.0030) - (0.1621 ± 0.0010)rz + 5Ea (2)

in meV, where the Zemach radius [6] rZ = 1.045 ± 0.004 fm

is obtained from e-p scattering.1 S Ea is the contribution to HFS from the new pseudoscalar interaction. Taking the experimental and theoretical uncertainties in quadrature, the best-fit to the experimentally measured AEHpS and rZ occurs for rZ = 1.045 fm and SEa = -0.006 meV, and

-0.018 meV < 5Ea < 0.006 meV at 2a.

We now compute 8Ea, and subject it to the above 2a constraint. In the nonrelativistic (NR) limit, the pseudoscalar vertex becomes

J5 = u(p')iy5U(p) -X ix'T^X - ix'TX ,

where x and x' are 2-component Pauli spinors. The ¡-p interaction in terms of the muon line (given by x-¡) and the proton line (given by xp, -p) is then (see Fig. 1)

J5,iJ5,p = mx1 • (p - P')xi 2m;xpi-p • (P - P')xp,

and the NR scattering amplitude for p + P x p' + P' is iM = ifi J5,i -r—2 ifpJ5,p , with q = p - p' = P' - p .

The couplings of the light pseudoscalar a of mass ma to the muon and to the proton are fand fp, respectively. Then,

* Corresponding author at: Department of Physics and Astronomy, University of Hawaii, Honolulu, HI 96822, USA.

E-mail address: dmarf8@hawaii.edu (D. Marfatia).

1 The use of the value of rZ obtained from e-p scattering is appropriate here

because the correction to rZ from using the new ¡-p interaction arises at loop

order.

http://dx.doi.org/10.1016Zj.physletb.2015.05.025

0310-2693/© 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

Fig. 1. Pseudoscalar exchange in muonic hydrogen.

fy fp x ' t

xy * y • qxy xp * p • qxp

- fy fp 1 q2 x /1

"4mymp 3q xy *yxy ' xp *PxP q2+m2 '

with the relative angle averaged for the s wave. The effective Hamiltonian is

5 Ha = -

1 fy fp

3 4mymp

83 (r)

m2 e-mar

* y • * p

so that

8 Ea = [lf(0)|2 - mlf |f(r )|2 e4nr d3 r

where f is the wave function of the 2S state:

^(r ) =

(1 - 20!)e 2aB .

2J2na3B aB

Here, aB = am is the Bohr radius for muonic hydrogen with mr = mp/(my + mp), the reduced mass of the system. On convolving,

we obtain

fy fp a

8 Ea =

3mymp 8n

F (x) = 1 - x

2 a2 + 2x2 2 (a + x)4

It is important to distinguish between mr and my in the equations above. The mr dependence comes from the Bohr radius aB, and my from the NR reduction. The function F (x) interpolates between 1 and 0 for x = 0 and x ^ro which is consistent with decoupling behavior.

In Fig. 2, we show the 2a allowed values of fy fp as a function of ma. The region between the solid curves is allowed. We restrict ma < 100 MeV so that fy fp remains comfortably in the perturba-tive regime.

Note that in the potential model of the proton with nonrela-tivistic quarks, the proton pseudoscalar coupling fp arises from the pseudoscalar couplings fu, fd of the up and down quarks, which are of the same order of magnitude. In this simplified picture, we have fp = 4 fd - 3 fu (as for the magnetic moments). If fu = fd, we have the simple result, fp = fu = fd.

In principle, the anomalous magnetic moment of the muon places a stringent independent constraint on fy since for the ma of interest, pseudoscalar couplings yield a negative contribution to ay [7,8],2 while the measured value is higher than the standard model expectation: Aay = aeyXp - ay = (29 ± 9) x 10-10 [9]. However, the scalar sector may be more intricate than envisioned here,

2 In Eq. (11) of Ref. [7], Cp should be replaced by |Cp|2, since Cp, as defined in

Eq. (9) therein, is complex.

Fig. 2. The values of fy fp allowed at 2a lie between the solid curves.

and may offer a fine-tuned (and perhaps unnatural) cancellation of the pseudoscalar contribution.

For the sake of comparison, the QED contribution at leading order is

8 H qed =

8 (r ) * y

6 4mymp

Here, gy (^ 2), and gp (^ 5.5857) are the gyromagnetic ratios for the muon and proton. Correspondingly,

8 Eqed :

a 4m3 12mymp

gy gp .

The above QED result, though simple, represents the first three significant digits of the dedicated theoretical calculation, and is consistent with the recent measurement of Ref. [1].

The ratio of the pseudoscalar contribution to the leading QED contribution is

2 fy f

8 E qed 4na g y gp

2 fy fp

yJp F I ma

4na gy gp

mi a2 + 2(ma/mr)2 m2 2(a + ma /mr )4

In sum, the 2S hyperfine splitting in muonic hydrogen constrains the product of the pseudoscalar couplings of the muon and proton fy fp to lie in the 2a ranges [-0.00040,0.00013], [-0.00173, 0.00058] and [-0.015, 0.005] for ma = 0, 10 MeV and 100 MeV, respectively. As the pseudoscalar mass is further increased, the constraint is weakened. The couplings have no impact on the discrepant measurements of the proton radius.

For ma < 100 MeV, no direct limits on fy fp exist from colliders, although limits for higher ma were obtained by the CMS experiment using the dimuon channel in pp collisions [10]. The CMS upper limits on the cross section times branching fraction, a • B(pp ^ a ^ y+y-), directly constrain fy fp in the mass ranges, 5.5-8.8 GeV and 11.5-14 GeV. CLEO's nonobservation of the decay J/^ ^ ya with a invisible [11] gives the 90% C.L. constraint | fp | < 0.029 (assuming the J/^-a coupling to be fp) for ma < 100 MeV [4].

Acknowledgements

This work was supported by the DOE under Grant Nos. DE-SC0010504 and DE-FG-02-12ER41811, and by the Kavli Institute for Theoretical Physics under NSF Grant No. PHY11-25915.

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