Sum rules across the unpolarized Compton processes involving generalized polarizabilities and moments of nucleon structure functions

We derive two new sum rules for the unpolarized doubly virtual Compton scattering process on a nucleon, which establish novel low-$Q^2$ relations involving the nucleon's generalized polarizabilities and moments of the nucleon's unpolarized structure functions $F_1(x,Q^2)$ and $F_2(x,Q^2)$. These relations facilitate the determination of some structure constants which can only be accessed in off-forward doubly virtual Compton scattering, not experimentally accessible at present. We perform an empirical determination for the proton and compare our results with a next-to-leading-order chiral perturbation theory prediction. We also show how these relations may be useful for a model-independent determination of the low-$Q^2$ subtraction function in the Compton amplitude, which enters the two-photon-exchange contribution to the Lamb shift of (muonic) hydrogen. An explicit calculation of the $\Delta(1232)$-resonance contribution to the muonic-hydrogen $2P-2S$ Lamb shift yields $-1 \pm 1$ $\mathsf\mu$eV, confirming the previously conjectured smallness of this effect.


I. INTRODUCTION
formalism, we take the nucleon as the target particle and hence limit ourselves to the spin-1/2 case. We therefore consider the doubly virtual Compton scattering process on the nucleon, where λ and λ denote the photon helicities (0, ±1) and s and s are the nucleon helicities (±1/2).

A. Tensor decomposition
The VVCS tensor T µν can be Lorentz decomposed into 18 invariant amplitudes as (in the notation of Ref. [24]) with P = 1 2 (p + p ). The 18 independent tensors T µν i in Eq. (2) are constructed to be gauge invariant [25]. Note that in the most general case one has to use the basis consisting of all 21 tensor amplitudes introduced in Ref. [24] in order to avoid kinematic constraints; however, as long as only the non-Born part of the VVCS amplitude is important (which is the case in the present work, as detailed below), one can use the minimal decomposition of Eq. (2); see Refs. [24,25].
The invariant amplitudes B i depend in general on four kinematic invariants. The incoming (outgoing) photon virtualities are denoted by q 2 (q 2 ), respectively. We also define the usual virtualities Q 2 = −q 2 and Q 2 = −q 2 that are positive for spacelike virtual photons. These two definitions of a photon's virtuality can be used interchangeably, multiplying the appropriate sign factors where needed. Furthermore, the variable q · q = (q 2 + q 2 − t)/2 is related with the momentum transfer to the nucleon, t ≡ (p − p) 2 . The crossing symmetric variable q · P ≡ M ν, with M the nucleon mass, can be expressed in terms of the Mandelstam variables s and u: M ν ≡ (s − u)/4.

B. Born contribution
An important contribution to the nucleon Compton amplitude at low energies corresponds with a nucleon intermediate state in the blob of Fig. 1, referred to as the Born term. This contribution is, by definition, not affected by structure-dependent constants, such as polarizabilities. The Born term is defined by using the electromagnetic vertex for the transition γ * (q) + N (p) → N (p + q) given as with F D and F P the Dirac and Pauli form factors of nucleon N , normalized as F D (0) = e N and F P (0) = κ N , where e N is the charge in units of the proton charge e (e N = 0 for the neutron) and κ N is the anomalous magnetic moment in units of the nuclear magneton e/2M ; σ µν = (i/2)[γ µ , γ ν ]. With this choice, the Born contribution to the spin-independent VVCS amplitudes is given by where ν B ≡ −q · q /(2M ), and we introduced the Sachs magnetic form factor, G M = F D + F P . The Born contribution of Eq. (8) can be split into pole and nonpole contributions. The pole contributions (also called elastic contributions) are singular at ν = ν B . The only nonpole piece in Eq. (8) is obviously the first term, i.e., B np 1 = F P (q 2 )F P (q 2 )/4M 3 . The rest of the Born terms are the pole contributions.

C. Low-energy expansions
The non-Born part of the VVCS amplitudes (denoted asB i ) can be expanded for small values of q 2 , q 2 , q · q , and ν, with the expansion coefficients given by polarizabilities. We use the low-energy expansions (LEXs) in k = {q, q } established in Ref. [24], where the parameters b i,x are structure constants. We notice that in order to fully specify the low-energy structure of the spin-independent doubly virtual Compton amplitude one requires two constants at the lowest order (b 1,0 and b 2,0 ) and nine additional constants when going to the next order: six coefficients arising from higher-order terms inB 1 andB 2 and the three lowest-order coefficients in the amplitudes B 3 ,B 4 , andB 19 , which are the amplitudes which are accompanied by tensor structures of higher order in k = {q, q }. The RCS process, corresponding with q 2 = q 2 = 0, allows one to constrain the two lowest-order coefficients inB 1 andB 2 as well as four of the next-order coefficients inB 1 andB 2 . We detail the connection between these coefficients and the polarizabilities, accessible through RCS, in Appendix A. These relations are given by (with the fine-structure constant α em ≡ e 2 /4π 1/137): Besides the electric (magnetic) dipole polarizabilities α E1 (β M 1 ), the above relations involve the corresponding electric (magnetic) dispersive polarizabilities α E1,ν (β M 1,ν ) and the electric (magnetic) quadrupole polarizabilities α E2 (β M 2 ). Furthermore, there are recoil terms (proportional to 1/M relative to the quadrupole polarizability terms), which involve the lowest-order nucleon spin polarizabilities γ M 1M 1 , γ E1M 2 , and γ M 1E2 , as well as recoil terms (proportional to 1/M 2 ), which involve the scalar polarizabilities α E1 and β M 1 .
In Appendix B, we show that the nonforward VCS process, corresponding with an outgoing real photon, i.e., q 2 = 0, and an initial spacelike virtual photon with virtuality q 2 , provides a second limit for the doubly virtual Compton scattering. Its measurement allows us to constrain two more of the next-order coefficients inB 1 andB 2 as which involve the slopes at Q 2 = 0 of the magnetic (β M 1 ) and electric (α E1 ) GPs, defined through Eqs. (B7a) and (B7b). Furthermore, the recoil terms (proportional to 1/M and 1/M 2 relative to β M 1 or α E1 ) involve, besides α E1 and β M 1 , also the RCS spin polarizabilities γ E1E1 , γ E1M 2 , as well as the longitudinal-transverse spin polarizability δ LT at Q 2 = 0, which is accessed from a moment of the nucleon spin-dependent structure functions g 1 and g 2 . We note that all quantities entering the rhs of Eqs. (10a)-(10f) and Eqs. (11a)-(11b) are observables which are accessed either through the RCS process, VCS process, or forward structure functions.

D. Forward limit
Besides the low-energy RCS and VCS processes, we can consider as another limit of the doubly virtual Compton process of Eq. (1) the forward VVCS limit, which corresponds with q = q and p = p. Notice that for this process q 2 = q 2 = q · q = −Q 2 < 0. The helicity averaged forward VVCS process is described by two invariant amplitudes, denoted by T 1 and T 2 , which are functions of two kinematic invariants: Q 2 and ν. Its covariant tensor structure can be written as where α em is conventionally introduced in defining the forward amplitudes T 1 and T 2 . The optical theorem relates the imaginary parts of T 1 and T 2 to the two unpolarized structure functions of inclusive electronnucleon scattering as where x ≡ Q 2 /2M ν and where F 1 and F 2 are the conventionally defined structure functions parametrizing inclusive electron-nucleon scattering. The imaginary parts of the forward scattering amplitudes, Eqs. (13), get contributions from both elastic scattering at ν = ν B ≡ Q 2 /(2M ), or equivalently x = 1, as well as from inelastic processes above the pion production threshold, corresponding with ν > ν 0 ≡ m π +(Q 2 +m 2 π )/(2M ) with m π the pion mass, or equivalently x < x 0 ≡ Q 2 /(2M ν 0 ).
Expressing the doubly virtual Compton tensors of Eq. (4) in the forward limit, the VVCS amplitudes T 1 and T 2 can be readily expressed in terms of the B i amplitudes of Eqs. (3) as where the amplitudes B i also depend on ν and Q 2 for forward kinematics. Using Eq. (8), we can express the Born contributions in the forward limit as and the corresponding pole parts as Using the LEXs of the non-Born amplitudesB i , given in Eqs. (9a) and (9b), we can obtain from Eqs. (14) and (15) LEXs for the non-Born partsT 1,2 of the amplitudes T 1,2 . Up to fourth order in k = {ν, Q}, these LEXs are given bȳ Besides the low-energy coefficients constrained from RCS and VCS, as given in Eqs. (10a) and (10f) and Eqs. (11a) and (11b), the knowledge of the amplitudesT 1 andT 2 to fourth order requires in addition the knowledge of the constants b 3,0 , b 4,1 , and b 19,0 , which we will discuss in the next section.
In the following, we will also be interested in the amplitudeT 1 at zero energy (ν = 0), which plays the role of a subtraction function in a dispersive framework for the VVCS amplitude. From Eq. (18), we see that its LEX can be expressed as

E. Sum rules
Using the RCS constraints of Eqs. (10a) and (10f) and the VCS constraints of Eqs. (11a) and (11b) on the low-energy coefficients, we can express the spin-independent VVCS amplitudes of Eqs. (18) and (19) including all terms up to fourth order in either Q or ν as We notice that the quadratic terms are fully determined by the proton electric (α E1 ) and magnetic (β M 1 ) dipole polarizabilities. The terms of order ν 4 inT 1 and of order Q 2 ν 2 inT 2 are also fully determined by the electric and magnetic dispersive and quadrupole polarizabilities which are observables in RCS. The term of order Q 2 ν 2 inT 1 involves in addition the slopes at Q 2 = 0 of the electric and magnetic GPs, as well as the RCS spin polarizabilities and the longitudinal-transverse spin polarizability δ LT , all of which are also observable quantities either through RCS, VCS, or using moments of spin structure functions. The only unknown in this Q 2 ν 2 term arises from the low-energy coefficient b 4,1 . This term could in principle also be accessed from the VCS process through the LEX of the amplitude f 3 , as given by Eq. (B1c), using the LEX of Eq. (9b) as by using, e.g., a BChPT calculation for the VCS process [26]. However, it will be difficult to extract this constant empirically as it would involve higher-order GPs which have not been quantified so far. In the following, we will show, however, that a forward sum rule will allow us to fix this term. Finally, we notice that the quartic terms of order Q 4 involve the unknown low-energy coefficients b 3,0 forT 1 and b 19,0 forT 2 . These coefficients cannot be obtained from RCS or VCS because the corresponding tensors vanish when one or both photons are real. In this section, we will show that b 19,0 can also be determined from a forward sum rule, involving the longitudinal electroabsorption cross section on a proton. The only unknown parameter which remains then is b 3,0 . Its determination will require an observable from the doubly virtual Compton process.
Having established the LEXs of the non-Born parts of the forward VVCS amplitudes T 1 and T 2 , we are ready to use the analyticity in ν, for fixed spacelike momentum transfer q 2 = −Q 2 ≤ 0. Both amplitudes are even functions of ν. We will present the relations for the nonpole parts of the amplitudes, T np i.e., the well-known pole amplitudes given by Eq. (17) are subtracted from the full amplitudes.

Spin-independent amplitude T 1
The dispersion relation (DR) for T 1 requires one subtraction, which we take at ν = 0, in order to ensure high-energy convergence, Because the nonpole amplitudes are analytic functions of ν, they can be expanded in a Taylor series around ν = 0 with a convergence radius determined by the lowest singularity, the threshold of pion production at ν = ν 0 . Analogous to the low-energy expansion of RCS, the series in ν, at fixed value of Q 2 , for forward VVCS takes the following form [2], where M can, respectively, be expressed through the second and fourth moments of the unpolarized nucleon structure function F 1 as Furthermore, in the second equalities of Eqs. (26) and (27), we introduced the transverse electroabsorption cross section (σ T ) on a nucleon through where K is a conveniently defined virtual photon flux factor; e.g., in the definition by Hand [27], it is given To obtain the low-energy expansion of the nonpole part T np 1 entering Eq. (24), we also need to account for the difference between the Born and pole parts, which can be easily read off Eq. (16) as where r 2 1 is the squared Dirac radius of the proton and where the Q 4 term involves the curvature of the Dirac form factor at Q 2 = 0, defined as As the difference between the Born and pole term contributions to T 1 is independent of ν, it can be fully absorbed in the subtraction function. The non-Born part of the subtraction functionT 1 (0, Q 2 ) can be read off Eq. (21) asT Apart from the well-known fact that the expansion ofT 1 (0, Q 2 ) in powers of Q 2 starts from the term β M 1 Q 2 , this relation constrains the next term in the expansion, proportional to Q 4 : Combining Eqs. (29) and (31), the subtraction function T np 1 (0, Q 2 ) entering the DRs of Eq. (24), including terms up to order O(Q 4 ), is given by In order to completely fix the term of O(Q 4 ) in the subtraction function, one needs to determine the low- Its determination requires a measurement of the doubly virtual Compton process with a spacelike initial and timelike final photon. The ν-dependent terms in the expansion of Eq. (25) can all be determined from sum rules in terms of electroabsorption cross sections on a nucleon, as given, e.g., by Eqs. (26) and (27). For Q 2 = 0, one can use the LEX of Eq. (21) to obtain the Baldin sum rule [7] and a higher-order generalization thereof as where σ T (ν ) is the total photoabsorption cross section on a proton. We can next write down a new generalized Baldin sum rule for the term proportional to Q 2 ν 2 in the LEX of Eq. (21): The structure function moment M is an observable and has been measured at the Jefferson Laboratory (JLab) [28]. If one could determine the low-energy coefficient b 4,1 from the VCS process using Eq. (23), the sum rule of Eq. (37) provides an exact nonperturbative relation which relates observables in RCS, VCS, and VVCS. A direct determination of b 4,1 , however, involves higher-order GPs, which may be quite complicated to extract from experiment. In practice, one can use the measured value on the lhs of the sum rule of Eq. (37) in order to determine the low-energy coefficient b 4,1 as For the amplitude T 2 , which is even in ν, one can write down an unsubtracted DR in ν: For the amplitude T 2 , there is no difference between the Born and pole contributions, as seen from Eq. (16). The expansion of the amplitude T np 2 at small k = {ν, Q} can therefore be directly read off Eq. (22). By evaluating Eq. (39) at ν = 0, taking its derivative with respect to Q 2 at Q 2 = 0, and using the relation one recovers from the Q 2 term inT 2 the Baldin sum rule of Eq. (35) and from the Q 2 ν 2 term inT 2 the higher Baldin sum rule of Eq. (36). Furthermore, Eq. (39) allows us to expressT 2 (0, Q 2 ) for general Q 2 as with M where the second identity in Eq. (42) has been obtained by expressing F 2 through the sum of transverse (σ T ) and longitudinal (σ L ) electroabsorption cross sections on a proton as We can then express the low-energy expansion ofT 2 including all terms up to fourth order in k = {ν, Q} as 1 where M 1 (0) are given by Eqs. (35) and (36) respectively. The term of order O(Q 4 ) involves the first derivative at Q 2 = 0 of Eq. (42) and can be obtained through a sum-rule relation from Eq. (22) as As the slope M 2 (0) is also an observable, the knowledge of it therefore allows us to determine the low- Let us note that it is also of interest to use the following combination of structure functions, as its absorptive part can be related to the longitudinal electroabsorption cross section on a nucleon as Its low-energy expansion, obtained by substitutingT 1 andT 2 from Eqs. (21) and (22) into the above definition, goes asT with where the last line has been obtained by using Eqs. (36), (37), and (45). On the other hand, we recognize that α L is the value at Q 2 = 0 of the usual α L (Q 2 ). It satisfies the sum rule in Eq. (5.36) of Hagelstein et al. [5], which at Q 2 = 0 corresponds with the first line in Eq. (50). Furthermore, the term of order O(Q 4 ) in Eq. (49) is given by 1 Note that M To conclude this section, we would like to note that in Refs. [13,14] a different choice of basis was used for the purpose of evaluating the Cottingham formula for the proton-neutron mass difference. The basis (denoted here by the superscript GL) used in that work is related to ours as The LEX of the corresponding non-Born amplitudesT GL To obtain the total amplitudes T GL 1 and T GL 2 , one needs to add the Born terms, which read The use of amplitudes T GL 1 and T GL 2 is equivalent to that of T 1 and T 2 as far as the quartic constraints derived in this work are concerned. Indeed, the ν 2 -dependent term in T GL 1 and the Q 2 -dependent term in T GL 2 lead to the two new sum rules of Eqs. (37) and (45) respectively.

III. SUM-RULE VERIFICATIONS IN BARYON CHIRAL PERTURBATION THEORY AND EMPIRICAL ESTIMATES FOR THE LOW-ENERGY COEFFICIENTS
In this section, we verify the sum rules derived in Eqs. (37) and (45). For this purpose, we will use a covariant next-to-leading-order BChPT calculation of the non-Born part of the CS process. Furthermore, we will provide empirical estimates for the low-energy coefficients entering the sum rules.
In several previous works, we have provided next-to-leading-order BChPT results for moments of nucleon structure functions [29], nucleon polarizabilities entering the RCS process [30,31], and generalized polarizabilities entering the VCS process [26].   The latter two coefficients can be obtained directly from a calculation of the VVCS process in the offforward regime. We performed such a calculation for the Delta-exchange graph, extending the CS calculation of that graph to the most general VVCS kinematics, obtaining the respective contributions to b 3,0 and b 19,0 . This allows us to verify Eqs. (32) and (45), too, albeit only for the Delta-exchange graph contribution at O(p 4 /∆).
An additional remark is in order regarding our calculation of the Delta-exchange graph. As explained in Ref. [33], the magnetic γN ∆ coupling g M is complemented by the dipole form factor, inferred from vector meson dominance considerations, and needed phenomenologically for a satisfactory description of electromagnetic nucleon-Delta transitions, with the dipole mass Λ 2 = 0.71 GeV 2 . The form factor changes the slopes of the VCS GPs and the VVCS structure function moments which enter the sum rules and the analyticity constraint, specifically, the values of β M 1 , M 2 (0), andT 1 (0). 2 However, the sum rules and the analyticity constraint are not affected. This can be seen explicitly from the expressions for the respective Delta-exchange contributions. In general, we checked that it is possible to add an arbitrary Q 2 dependence to the couplings, e.g., by including form factors, without violating the spin-independent sum rules considered herein, or the constraint on the derivative of the subtraction functionT 1 (0), as discussed in Section IV.
Source M      We show the BChPT estimates for all terms entering the sum rule (37) for M   Table II. The values in both tables include the contribution of the dipole form factor in the Delta-exchange graph; values without those contributions can be obtained by adding 4β ∆ M 1 /Λ 2 = 1.57 × 10 −4 fm 5 where appropriate. Remember that b 4,1 was known from BChPT before [26], while the coefficient b 19,0 was previously unknown. For the Delta-exchange contribution, b 19,0 has been calculated directly from the off-forward VVCS process, whereas the πN -and π∆-loop contributions were deduced from the sum rule and the BChPT predictions for the remaining quantities in Eq. (45).
Having verified the sum rules, we can provide empirical estimates of the low-energy coefficients. The lefthand sides of both Eq. (37) and Eq. (45) can be estimated from the measured moments of proton structure functions. We show the empirical Bosted-Christy (BC) fit [34] for the moments M  Tables I and II. Furthermore, we use the dispersive estimates of Ref. [35] for the higher-order real Compton polarizabilities and of Refs. [2,36] for the GPs. We use the phenomenological MAID2007 fit [37] as input for the πN -channel contribution in the DRs. The recoil terms on the right-hand sides of Eqs. (37) and (45), which are proportional to 1/M and 1/M 2 , depend on the lowest-order spin and scalar polarizabilities, respectively. To estimate these terms, we use the empirical values listed in Table III. Using the sum rules for M  Tables I and II. One can see from Tables I and II that there is a reasonable agreement between the BChPT values and the empirical ones for most terms entering Eqs. (37) and (45). We see differences for M (2) 1 (0), which is close to zero in BChPT but is negative in the empirical fit, as well as for b 19,0 which is very small in the empirical extraction. As both of these quantities yield relatively small contributions to the respective sum rules shown in Tables I and II, the differences can partly be attributed to cancellations between different terms in these relations. Figures 2 and 3 also demonstrate that there is qualitative agreement between the BChPT and the BC fit results for M according to the empirical Bosted-Christy (BC) fit (black solid curve) [34], in comparison with the πN +∆+π∆ BChPT calculation. For the latter, we also show the result when an additional form factor dependence is included in the ∆-exchange contribution as given by Eq. (56); blue dashed (magenta dashed-dotted) curves show the results with (without) the form factor. The blue band shows the uncertainty of the BChPT result with the form factor, estimated as in Ref. [26]. At the real photon point, the observable yields the Baldin sum-rule value for α E1 + β M 1 [5]. The data point at Q 2 = 0.3 GeV 2 is from JLab/HallC [28].
The uncertainty bands on the BChPT curves are calculated as detailed in Ref. [26] and represent a conservative estimate of corrections due to higher orders in the chiral expansion. On the other hand, one can see that the use of the form factor in the γN ∆ vertex is an important part of the presented result. To estimate the uncertainty due to the form factor, one notes that empirical data on electromagnetic nucleon-Delta transitions at low Q 2 allow one to extract the form factor with a precision of the order of 2%; see,

BChPT
BChPT with Δ according to the empirical Bosted-Christy (BC) fit (black solid curve) [34], in comparison with the πN +∆+π∆ BChPT calculation. For the latter, we also show the result when an additional form factor dependence is included in the ∆-exchange contribution as given by Eq. (56); blue dashed (magenta dashed-dotted) curves show the results with (without) the form factor. The blue band shows the uncertainty of the BChPT result with the form factor, estimated as in Ref. [26]. At the real photon point, the observable yields the Baldin sum-rule value for α E1 + β M 1 [5].

Value
Source  e.g., Ref. [38]. Varying the form factor within this range would result in changes of M 2 (Q 2 ) at least an order of magnitude smaller than the shown uncertainty bands. We thus neglect the uncertainty due to this source, expecting that any form factor that describes electromagnetic nucleon-Delta transitions reasonably well should give results close to those presented here.
The arguments concerning the uncertainty estimate also apply to the subtraction functionT 1 (0, Q 2 ); see a more detailed discussion thereof in Section IV.
Finally, we can also extract an empirical estimate for the longitudinal polarizability in Eq. (50). For the term M     Tables I and II, we then obtain an empirical estimate for α L : This polarizability has been calculated in BChPT at NLO [29]: α L 2.3 · 10 −4 fm 5 . We have checked that the same value is obtained by evaluating the separate BChPT contributions in Eq. (50).

IV. LOW-Q BEHAVIOR OF THE SUBTRACTION FUNCTION
In this section, we study the Q 2 dependence of the subtraction function,T 1 (0, Q 2 ), which is of interest for the (muonic) hydrogen Lamb shift calculations. It is the part of the TPE correction in the lepton-proton system noncalculable through the sum rules. In what follows, we will verify the analyticity constraint derived in Eq. (32) and give estimates for the low-energy coefficient b 3,0 . As a result, one constrains the subtraction contribution to the Lamb shift.
The LEX given in Eq. (32) relates the second derivative of the subtraction function,T 1 (0), to scalar and spin polarizabilites known from RCS, the GP slope β M 1 known from VCS, and the low-energy coefficient b 3,0 . Analogously to Section III, we verify Eq. (32) with the Delta-exchange graph contribution at O(p 4 /∆) in BChPT. As explained earlier, the validity of the constraint is not affected by adding a dipole form factor dependence to the magnetic coupling g M or, in general, by the inclusion of an arbitrary Q 2 dependence of the γN ∆ couplings. Once the constraint is verified, it can be used to make a prediction for b 3,0 at NLO in BChPT. As before, we rely on the results previously derived in Refs. [26,[29][30][31]. The corresponding BChPT values [again, with the use of the form factor in the Delta pole, as given by Eq. (56)], as well as empirical and dispersive estimates of all quantities entering Eq. (32), are given in Table IV It is interesting to note that the value of b 3,0 obtained in BChPT turns out to be rather small compared to other quantities entering Eq. (32) and is driven by the Delta-exchange graph, with πN and π∆ loops giving negligible contributions. The smallness of the πN -and π∆-loop terms in b 3,0 could be considered accidental, given that it results from very efficient cancellations between the different terms in Eq. (32).
Let us now compare the behavior of the subtraction function in different approaches. In Fig. 4, we show T 1 (0, Q 2 )/Q 2 as obtained in BChPT and heavy-baryon chiral perturbation theory (HBChPT) [40] (note that the latter calculation uses a dipole form factor [with the slope matched to the HBChPT expansion at low Q 2 ] to model the large-Q 2 behavior of the subtraction function) and an estimate from the superconvergence relation [41]. At the real photon point,T 1 (0, Q 2 )/Q 2 is given by the magnetic dipole polarizability β M 1 , cf. Eq. (31). The figure shows that the BChPT curve with no γN ∆ form factor is close to the HBChPT one; note that the static value in the latter curve was fixed to the PDG value of β M 1 = (2.5 ± 0.4) × 10 −4 fm 3 [42]    rather than the larger value β M 1 = (3.15 ± 0.50) × 10 −4 fm 3 (which is typical of modern HBChPT [43] and BChPT [44] fits), used in Ref. [40]. The form factor on the magnetic γN ∆ coupling increases the (negative) slope of the subtraction function at Q 2 = 0, as can be seen from Table IV by  Let us now turn to the contribution of the subtraction term in the TPE correction to the Lamb shift in µH and in particular to the effect of the ∆(1232) excitation, shown in Fig. 5. As seen in Fig. 4, the subtraction function changes a lot depending on the treatment of the Delta-exchange contribution. However, as argued in Ref. [45], the total contribution of the Delta exchange to the Lamb shift in µH turns out to be rather small due to cancellations between the subtraction and inelastic terms. This picture as well as the value of the total Delta-exchange contribution only very weakly depend on the parametrization of the γN ∆ transition. We will demonstrate it in detail below; for this purpose, we briefly recall the TPE formalism (see, e.g., Ref. [46]). The nth S-level shift in the (muonic) hydrogen spectrum due to forward TPE is related to the spin-independent forward VVCS amplitudes, where m is the lepton mass, φ 2 n = 1/(πn 3 a 3 ) is the wave function at the origin, a −1 = α em m r is the inverse Bohr radius and m r is the reduced mass of the lepton-proton system. Recall also that the Lamb shift is the difference between the shifts of the 2P and 2S levels; the TPE contribution to the former is negligible, and the TPE contribution to the Lamb shift is thus just −∆E TPE (2S). Obviously, the polarizability effect on the hydrogen spectrum is described by the non-Born amplitudesT 1 andT 2 . This effect can be split into the contribution of the subtraction functionT 1 (0, Q 2 ), with v l = 1 + 4m 2 /Q 2 , and contributions of the inelastic structure functions (Ref. [5], Sec. 6): where τ = Q 2 /(4M 2 ). The ∆(1232)-exchange contribution to theT 1 (0, Q 2 ) subtraction function reads [47] T 1 (0, Here, the second row contains terms proportional to the subleading Coulomb coupling g C . In Table V, we show the effect of TPE with intermediate ∆(1232) excitation on the 2S level in µH. 3 As mentioned above, the magnetic coupling can be multiplied by a dipole form factor in order to model a vector-meson type of dependence; the use of the form factor is specified in the table. For the prediction in the last row, the γN ∆ couplings were replaced by the Jones-Scadron nucleon-to-Delta transition form factors (see Ref. [47] for the details of the calculation). These transition form factors were related to nucleon form factors by the finite-momentum transfer extension of their large-N c limit [48]. The nucleon form factors were in turn described by an empirical parametrization [49]. As one can see from the table, the relatively large contribution of the subtraction function,T 1 (0, Q 2 ) (second column), is largely cancelled by the contributions of the inelastic structure functions, F 1 and F 2 (third and fifth columns). The total effect of the ∆(1232) resonance on the shift of the 2S state in µH is small [47] (quoting the calculation with the Jones-Scadron form factors), compared to the leading effect of chiral dynamics [45], At the same time, a calculation of the TPE with ∆(1232) excitation, employing again Jones-Scadron form factors, allows for a meaningful prediction of the contribution of the subtraction term (i.e., a prediction independent from its combination with the inelastic contribution into the polarizability contribution, cf. the discussion in Ref. [45], Sec. 3) to the shift of the 2S state at LO plus ∆ in BChPT, which is in good agreement with dispersive predictions [40,46]. Table IV shows a comparison of separate contributions to ∆E TPE (2S) in different frameworks. 4 To conclude this section, we note that ChPT here is an example which satisfies the sum rules. However, the hope is that the sum rules will provide a data-driven evaluation, independent of ChPT. For that, one would need to have an experimental determination of the constant b 3,0 , which can become possible in future doubly virtual Compton scattering measurements. 3 Note that the structure functions not only contain the ∆ production, i.e., terms proportional to δ(x − x∆) with x∆ = Q 2 M 2 ∆ −M 2 +Q 2 , but also contain terms proportional to δ(x). 4 A different HBChPT prediction of the subtraction term that does not use form factors to model the high-Q 2 dependence and includes the leading and subleading πN and π∆ loops, respectively, can be found in Ref. [51].

V. CONCLUSIONS
The main result of this work is given by the VVCS sum rules in Eqs. (37) and (45), and the LEX constraint in Eq. (32). For the derivation, the known CS formalism, reviewed in the beginning of Section II, was used. At second order in energy (ν 2 ) or momentum transfer (Q 2 ), the unpolarized nucleon response in the CS process is fully described in terms of electric and magnetic dipole polarizabilities. In this work, we have fully quantified the response of the double virtual CS to fourth order, including terms in ν 4 , ν 2 Q 2 , and Q 4 . The new forward sum rules we have derived establish relations between RCS, VCS, and VVCS observables at this order. In particular, they give access to the VVCS low-energy coefficients b 4,1 and b 19,0 through moments of the nucleon structure functions, VCS GPs, and static scalar and spin polarizabilities; see Eqs. (38) and (46) Tables I and II, were found to be in reasonable good agreement.
The remaining unknown in the doubly virtual CS process at order Q 4 results from the low-energy coefficient b 3,0 , which enters the VVCS subtraction functionT 1 (0, Q 2 ). The latter is also the main hadronic uncertainty in the estimate of the TPE correction to the muonic-hydrogen Lamb shift. Our NLO BChPT calculation yields a very small value for b 3,0 . We have shown that this result originates predominantly from the ∆-pole contribution. The corresponding NLO BChPT prediction of the subtraction function displays a sign change induced by the form factor dependence of the ∆-exchange graph. The LO plus ∆ BChPT prediction for the polarizability contribution (subtraction term and inelastic term) to the µH Lamb shift is found to be in good agreement with dispersive calculations. Studying in particular the TPE with intermediate ∆ excitation, we have shown that the sizeable contribution of the subtraction term is largely cancelled by the inelastic contribution, leading to a small polarizability effect of the ∆(1232) in the µH Lamb shift.
To check the smallness of the low-energy coefficient b 3,0 , as predicted by our NLO BChPT calculation, we noted that there is at present no direct experimental access to the slope of the VVCS subtraction function. In order to have some empirical guidance, we compared our BChPT result with the estimate based on a superconvergence relation [41]. The latter yields a much smaller value (in absolute size) for the Q 4 term in the subtraction functionT 1 (0, Q 2 ), which then yields a significantly larger value for b 3,0 . The superconvergence estimate of Ref. [41] at lower values of Q 2 1 GeV 2 is constrained by existing nucleon structure function data in the resonance region (W < 3 GeV) as well as by HERA data at high energies (W > 10 GeV). However, in the intermediate W region (3 W 10 GeV) at finite Q 2 , the empirical estimate is quite uncertain because of the scarce data situation in that region. Forthcoming structure function data from the JLab 12 GeV facility will allow us to further improve such superconvergence relation estimates for b 3,0 . It may also be very worthwhile to directly access b 3,0 through a low-energy doubly virtual CS experiment. The formalism laid out in the present work provides the unpolarized hadronic tensor entering the description of such a process. We leave the study of the doubly virtual CS observables necessary to measure the low-energy coefficient b 3,0 as a topic for future work.
Note that the remaining two amplitudes B 3 and B 19 which are needed to fully specify the spin-independent doubly virtual Compton amplitude of Eq. (3) cannot be accessed in the VCS process, as the corresponding tensors decouple when the outgoing photon is real (q 2 = 0).
The VCS experiments at low outgoing photon energies can also be analyzed in terms of LEXs, as proposed in Ref. [55]. For this purpose, the VCS tensor has been split in Ref. [55] into a Born part, which is defined as the nucleon intermediate state contribution using the γ * N N vertex of Eq. (7), and a non-Born part. The latter describes the response of the nucleon to the quasistatic electromagnetic field, due to the nucleon's internal structure. To obtain the lowest-order nucleon structure terms, one considers the response linear in the energy of the produced real photon. This linear response of the non-Born VCS tensor, i.e., the limit q → 0 at arbitrary virtuality Q 2 of the initial photon, can be parametrized by six independent GPs [55,56]. The GPs can be accessed in experiment through the eN → eN γ process; see the reviews [1,2] for more details. At lowest order in the outgoing photon energy, there are two spin-independent GPs, denoted by P (L1,L1)0 and P (M 1,M 1)0 , and four spin GPs, denoted by P (L1,M 2)1 , P (M 1,L2)1 , P (L1,L1)1 , and P (M 1,M 1)1 , which are all functions of Q 2 . 5 In this notation, L stands for the longitudinal (or electric) and M stands for the magnetic nature of the transition respectively. One usually defines the electric and magnetic GPs as which are related to the RCS static polarizabilities as α E1 (0) = α E1 , β M 1 (0) = β M 1 .