fmf_W1

vardef cross_bar (expr p, len, ang) = ((-len/2,0)–(0,0)) rotated (ang + angle direction length(p) of p) shifted point length(p) of p enddef; style_def crossed expr p = cdraw p; ccutdraw cross_bar (p, 2mm, 30); ccutdraw cross_bar (p, 2mm, -30) enddef; \fmfcmdvardef port (expr t, p) = (direction t of p rotated 90) / abs (direction t of p) enddef; \fmfcmdvardef portpath (expr a, b, p) = save l; numeric l; l = length p; for t=0 step 0.1 until l+0.05: if t¿0: .. fi point t of p shifted ((a+b*sind(180t/l))*port(t,p)) endfor if cycle p: .. cycle fi enddef; \fmfcmdstyle_def brown_muck expr p = shadedraw(portpath(thick/2,2thick,p) ..reverse(portpath(-thick/2,-2thick,p)) ..cycle) enddef;

arrow_len3mm \fmfsetarrow_ang12 \fmfsetwiggly_len3mm \fmfsetwiggly_slope75 \fmfsetcurly_len2mm

FERMILAB-PUB-16-465-T

WSU-HEP-1607

November 29, 2016

Nucleon spin-averaged forward virtual Compton tensor at large

Richard J. Hill and Gil Paz

Perimeter Institute for Theoretical Physics, Waterloo, ON, N2L 2Y5 Canada,

TRIUMF, 4004 Wesbrook Mall, Vancouver, BC, V6T 2A3 Canada,

Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USA, and

The University of Chicago, Chicago, Illinois, 60637, USA

Department of Physics and Astronomy

Wayne State University, Detroit, Michigan 48201, USA

The nucleon spin-averaged forward virtual Compton tensor determines important physical quantities such as electromagnetically-induced mass differences of nucleons, and two-photon exchange contributions in hydrogen spectroscopy. It depends on two kinematic variables: , the virtual photon energy in the proton rest frame, and , the photon’s invariant four-momentum squared. Using the operator product expansion, we calculate the tensor’s large- behavior for , including for the first time the full spin-2 contribution and correcting a previous result in the literature for the spin-0 contribution. Implications for the proton radius puzzle are discussed.

## 1 Introduction

The forward Compton amplitude describes virtual photon scattering on a hadron, depicted in Fig. 1, and encodes important structure information. For example, the spin-averaged proton forward Compton amplitude determines two-photon exchange (TPE) contributions to hydrogen spectroscopic transitions. These contributions dominate the uncertainty of the proton charge radius measured in muonic hydrogen [1] and have been the subject of recent intense scrutiny and controversy [2, 3, 4, 5]. The difference between spin-averaged proton and neutron forward Compton amplitudes determines the electromagnetic contribution to the proton-neutron mass splitting, also the subject of recent controversy [6, 7, 8, 9, 10].

The tensor representation of the spin-averaged Compton amplitude
(cf. Eq. (1) below) involves two scalar functions,
and . The imaginary parts of these
functions are determined by (elastic and inelastic) electron
scattering data. can be reconstructed from its
imaginary part using a dispersion relation in . However,
requires a subtraction in the dispersion relation.
Uncertainties related to the subtraction function for the proton,
, dominate the
error budget for the muonic hydrogen Lamb shift. Similarly,
uncertainties in the difference of subtraction functions for proton and
neutron, , dominate the error budget for the
proton-neutron electromagnetic self energy. It is important to
constrain this subtraction function.^{1}^{1}1
The need for subtraction is readily demonstrated:
at low (see e.g. Ref. [12]), in contradiction with the prediction from
an unsubtracted dispersion relation,
,
which is manifestly positive using the optical theorem for (see e.g. Ref. [2]).
Reference [10] argues that after subtraction of a term
motivated by Reggeon dominance, the remaining structure functions
satisfy unsubtracted dispersion relations; however, a reliable evaluation
of such a subtraction has not been presented.

While not accessible directly from experimental data, we can constrain in the small- and large- limits. For small one can calculate [2] in terms of the Wilson coefficients of Non Relativistic QED (NRQED) [11, 12]. These coefficients are in turn determined by low-energy experimental measurements, or by nonperturbative QCD computations using lattice QCD or chiral perturbation theory. The Operator Product Expansion (OPE) may be used to show that for large [7, 2]. In this paper we perform the explicit computation of the coefficient in the OPE determination of .

There are four types of operator that contribute to the coefficient of in the OPE expression for : spin-0 quark, spin-0 gluon, spin-2 quark and spin-2 gluon. Previous analyses of are based on the work of Collins from 1979 [7], which was concerned with the renormalization of the Cottingham formula for electromagnetic mass splittings, and considered only the spin-0 operators. In this paper we calculate the Wilson coefficients for both types of operator. We correct an error in the expression for the spin-0 contribution in Ref. [7]. Applications to muonic hydrogen require inclusion of the spin-2 operators. Since these were not included before, existing applications are incomplete; in fact, we show that the spin-2 contribution is numerically dominant.

The remainder of the paper is structured as follows. In Sec. 2 we perform the analytic calculations of the Wilson coefficients. In particular we perform the one-loop calculations needed to extract the Wilson coefficients of gluon operators. In Sec. 3 we evaluate the relevant local nucleon matrix elements. Focusing on the proton case, we numerically evaluate at large . In Sec. 4 we discuss the implications of our results for TPE contributions to the muonic hydrogen Lamb shift. In Sec. 5 we present our conclusions.

## 2 Wilson coefficients

Here we introduce notation for the Compton amplitude and determine the relevant operators in the OPE. We then perform matching onto quark and gluon operators.

### 2.1 Preliminaries

The nucleon spin-averaged forward virtual Compton scattering tensor is defined as (cf. Fig. 1)

(1) |

where is the nucleon three-momentum and its spin,
is the electromagnetic current, and
the subscript denotes symmetrization in and .^{2}^{2}2Symmetrization in and can be shown equivalent to spin averaging:
if is expanded in a tensor basis, terms that are odd under
are linear in the nucleon spin vector, , and
vanish under spin averaging;
terms symmetric under are independent of as displayed in
Eq. (2).
Here or denotes
a proton or neutron. Also,
and . Notice that some authors refer to
this quantity as [7, 8].

Using current conservation, and invariance of electromagnetic interactions under parity and time-reversal, can be expressed as

(2) |

We will use relativistic normalization of states, , for nucleon spinors throughout. The imaginary parts are related to physical cross sections. Inserting a complete set of states into (1),

(3) |

where is the momentum of the state . We can now perform a separation between nucleon and non-nucleon (i.e., excited) states. The nucleon contributions to can be expressed in terms of nucleon form factors. Contributions from other final states can be expressed in terms of inelastic structure functions. Using dispersion relations, can be reconstructed from its imaginary part, but requires a subtraction in order to have a convergent dispersion relation. Thus

(4) |

Since is not directly related to measured quantities it is a major source of uncertainty for quantities like the TPE contribution to the Lamb shift in muonic hydrogen [2] and the isovector nucleon electromagnetic self-energy [8]. As discussed in the introduction, we can constrain in the small- and large- limits. The small- behavior was discussed in Ref. [2]. Here we will calculate the large behavior.

We will determine the OPE for the operator,

(5) |

where curly braces around indices denote symmetrization,
i.e., for four-vectors and .
Note that the Compton tensor is obtained by taking the matrix element,
.
For the OPE evaluation we match onto the lowest
dimension QCD operators, which begin at dimension four. There are
four^{3}^{3}3As we will see below, there is another possible
structure .
The matrix element of the corresponding contribution to
between proton states vanishes, so we do not include it in Eq. (2.1).
See Appendix A.
relevant operator types

(6) |

where is the spacetime dimension, and runs over active quark flavors. The superscript label denotes operator spin. The spin-2 operators are traceless, i.e. they satisfy .

Current conservation implies up to operators whose physical
matrix elements vanish.
This implies that the general form^{4}^{4}4A fourth possible term
is omitted since its matrix elements between spin-averaged proton states is zero.
of may be written

(7) | |||||

Using the matrix elements (20) and (3.2) below, we see that and will contribute to and to and . Let us proceed to match onto quark and gluon operators at leading order in QCD perturbation theory.

### 2.2 Quark operators

In order to match onto the quark operators, we compute forward scattering amplitudes using on-shell quarks with electromagnetic charge (in units of the proton charge), mass , and four-momentum as external states; cf. Fig. 2. The matrix element for between such states is

(8) |

After simplification this gives the results

(9) |

The expression for agrees with equation 2.14 of Ref. [7]. The coefficients and were not considered in Ref. [7].

### 2.3 Gluon operators

The calculation of is simplified by using a background field analysis in Fock-Schwinger gauge [13]. We then have the following expressions for the gluon contributions to ,

(10) | |||||

where we separate into three pieces, , , and , as illustrated in Fig. LABEL:fig:gluon. The contributions and are equal and IR divergent, while is IR finite. In the following we will work in dimensions to regularize these IR divergences and set the quark masses to zero. The effective theory loop diagrams pictured in Fig. 3 then vanish. We note also that none of , or obeys current conservation by itself, only their sum.

Since is IR finite we can set and get

(11) |

where is an operator structure whose nucleon matrix element vanishes, cf. Appendix A. is IR divergent and we calculate it in dimensions. We find

(12) | |||||

Expanding in and adding and , we have the total gluon contribution

(13) | |||||

from which we may read off the coefficients . Here we have introduced the renormalized strong coupling in the scheme,

(14) |

As an alternative approach, inspection of Eq. (7) shows that the coefficients , and may be extracted from the quantity , for arbitrary timelike unit vector ():

(15) |

Following the notation of Ref. [14], we write^{5}^{5}5
For our case, we need the “ZZ”
contribution from Ref. [14], with axial coefficient and vector coefficient .

(16) |

Neglecting quark masses, we have^{6}^{6}6
A non-propagating typo appears in Eq. (83) of Ref. [14]:
in , in the term, the opposite sign should appear in front of .

(17) |

with . Thus, either from Eq. (13) or from Eqs. (2.3) and (16), we read off the bare matching coefficients,

(18) |

Renormalizing operators in the scheme [14], we find the renormalized coefficients,

(19) |

## 3 Nucleon matrix elements

Here we evaluate the relevant nucleon matrix elements. In Secs. 3.1 and 3.2, we discuss the spin-0 and spin-2 matrix elements for both proton and neutron. In Sec. 3.3 we specialize to the proton case and provide explicit numerical results for the leading OPE evaluation of .

### 3.1 Spin zero

Following Ref. [15] let us define^{7}^{7}7
Compared to the quantity in Ref. [15],
we have . The conventional
prefactor is designed to simplify the expression for the sum rule (21) when .

(20) |

where we are using relativistic normalization for nucleon spinors.^{8}^{8}8
Recall that the nonrelativistic normalization was
used in Ref. [15].
The gluon matrix element is determined by the mass sum rule,

(21) |

Here and denote the QCD beta function and mass anomalous dimension, whose leading expansions are

(22) |

where is the number of active quark flavors.^{9}^{9}9Higher order terms are listed in Ref. [15]. In numerical applications
of Eq. (21), we
evaluate and through .
For the quark matrix elements we use

(23) |

In terms of the quark mass ratio we have

(24) |

Numerical values for spin-zero matrix elements are summarized in Table 1. As detailed in Ref. [15], the matrix elements are identical for and , up to power corrections.

quantity | value | reference |
---|---|---|

[16, 17, 18, 19, 20, 21] | ||

[22, 23] | ||

[24] | ||

[25] | ||

(lattice) | [26, 27, 19] | |

(pQCD) | [15] |

### 3.2 Spin two

quantity | value |
---|---|

0.346(6) | |

0.192(5) | |

0.034(2) | |

0.0088(3) |

Let us define

(25) |

The gluon matrix element is determined by the momentum sum rule,

(26) |

Up to corrections proportional to and , the proton and neutron matrix elements are related as,

(27) |

For the numerical evaluation, we use the inputs of Table 2 for at , with gluon matrix element given by Eq. (26). For different in the limited range considered, we employ the leading log renormalization (cf. Table 5 of Ref. [15]).

### 3.3 Large behavior of : numerical results

Let us restrict attention to the subtraction function for the proton, . For notational simplicity we suppress the superscript, , in the following. In terms of the matrix elements of the previous section, we have the leading power result

(28) |

where the first nonvanishing order for coefficients of each operator are given by Eqs. (9) and (2.3). Let us consider separately the spin-0 and spin-2 contributions.

#### 3.3.1 Spin-0: numerical evaluation for proton

Substituting the Wilson coefficients from Eqs. (9) and (2.3), the spin-0 contribution is

(29) |

This contribution is displayed in Fig. 4 using and . Having neglected quark masses, our result is formally valid in the regime. Alternatively, an evaluation using is formally valid in the regime . We show below that is numerically dominated by the spin-2 contribution, and do not pursue a more elaborate analysis of charm quark mass dependence in the spin-0 contribution.

We take in our final result. Fig. 4 compares results using different values of the charm scalar matrix element, . As default, we use the perturbative QCD value obtained from expanding in [15]. For comparison, Fig. 4 displays the range of from currently available lattice QCD evaluations [26, 27, 19]. The gluon matrix element is given by Eq. (21), with and evaluated through . We choose default renormalization scale , and perturbative scale uncertainty is estimated by varying . We remark that a partial cancellation occurs between quark and gluon matrix elements, as illustrated in the figure.

The spin-0 part disagrees with Collins’s calculation [