Effects of light-by-light scattering in the Lamb shift and
hy-perfine structure of muonic hydrogen
A. E.Dorokhov1,2,∗,A. P.Martynenko2,∗∗,F. A.Martynenko2,∗∗∗, andA. E.Radzhabov2,3,∗∗∗∗
1Joint Institute of Nuclear Research, BLTP, 141980, Moscow region, Dubna, Russia
2Samara University, 443086, Samara, Russia
3Matrosov Institute for System Dynamics and Control Theory SB RAS, 664033, Irkutsk, Russia
Abstract.We calculate the meson exchange contribution to the interaction op-erator of muon and proton, which is determined by the meson coupling with two photon state. For the construction of transition form factorMeson→γγ we use the existing parametrizations based on experimental data including the monopole parametrization over photon four-momenta. For an estimate of the form factor value at zero photon four-momenta squared we use experimental data on the decay widthΓMeson→γγ. It is shown that scalar, pseudoscalar, axial
vector and tensor mesons exchanges give significant contribution to the Lamb shift (LS) and hyperfine splitting (HFS) in muonic hydrogen which should be taken into account for a comparison with precise experimental data.
1 Introduction
The discrepancy between results for the proton charge radius obtained by different methods got the name "proton radius puzzle". In particular, measurements using electronic hydrogen lead to a different proton charge radius compared with those using muonic hydrogen. This problem arose after the CREMA experiments on Lamb shift measurement in muonic hydro-gen [1, 2]. Increasingly accurate experiments require corresponding theoretical calculations. To achieve high accuracy it is necessary to take into account the problem of a more accurate construction of the particle interaction operator in muonic atoms and the inclusion of new contributions to this operator. Emerging new experimental data on the Lamb shift in electron hydrogen, as well as a new analysis of experiments already performed on the scattering of leptons by nucleons, show that the values of the proton charge radius obtained from electron and muon systems are approaching [3, 4]. The problem of the proton charge radius is grad-ually beginning to be solved. However, the analysis of new interactions between the proton and the muon is important for future more accurate experiments. Among the interactions of the proton and the muon there are those in which two virtual photons turn into a meson, which leads to an effective one-meson potential. The calculation of the transition form factor (TFF) of two photons into a meson can be performed within the framework of nonperturba-tive quantum chromodynamics. In this work we continue our investigations [5–9] (see also [10–14]) of the role of one-meson exchange interactions in muonic hydrogen (µp).
∗e-mail: [email protected]
∗∗e-mail: [email protected]
∗∗∗e-mail: [email protected]
2 One meson exchange contribution
2.1 Scalar meson exchange
Our approach to the calculation of one-meson exchange contributions to the energy levels of (µp) is based on quasipotential approach in quantum field theory [15–18]. The general parametrization of scalar meson→γ∗+γ∗vertex function takes the form [14]:
TSµν(t,k1,k2)=4πα
A(t2,k12,k22)(gµν(k1·k2)−kν1k µ
2)+ (1)
B(t2,k21,k22)(kµ2k21−kµ1(k1·k2))(kν1k 2 2−k
ν
2(k1·k2))
,
whereA(t2,k2 1,k
2 2),B(t
2,k2 1,k
2
2) are two scalar functions,k1,2are four momenta of virtual
pho-tons,tis the four momentum of scalar meson. Then the muon-proton interaction amplitude via the scalar meson exchange can be written as follows:
iMS= α 2g
s
π2
d4kA(t2,k2,k2)(gµν(k
1·k2)−kν1k µ
2)[ ¯u(q1)γµ( ˆp1−kˆ+m1)γνu(p1)][¯v(p2)v(q2)] k4(k2−2m
1k0)(t2+M2s)
,
(2) wherep1,p2are four momenta of particles in initial state,q1,q2are four momenta of particles
in final state. We sett = q1−p1 = 0 because this momentum is small and we consider
further the leading order in fine structure constantαcontribution to the interaction operator (|t| ∼ µα). This leads to the cancelation of the term with the functionB(p2,k2
1,k 2
2). Using
projection operator on muon-proton states with spin S=0, S=1 [15] we can construct the interaction operator for these states. In the case of triplet state we find:
iMS(3S1)=
α2gs
16m2 1m
2 2π2
d4
k4(k2−2k 0m1)
A(t2,k2,k2)(gµνk2−kµkν) (3)
T r1+γ0
2√2 ε( ˆˆ
p2−m2)( ˆq2−m2) ˆε∗
1+γ0
2√2 ( ˆ
q1+m1)γµ( ˆp1−kˆ+m1)γν( ˆp1+m1)
1
t2+M2 s
,
wherem1,m2are the masses of a muon and proton,MS is the mass of scalar meson. After
the trace calculation using the package Form [19] we obtain in the leading order:
T1S0 =T3S1=k2(3k0+2m1)−2m1k20. (4)
So in the leading order the scalar meson exchange doesn’t contribute to hyperfine structure. The typical momentum integral contributing to the shift has the following form:
I1=
d4(k2+2k2
0) k2(k4+a2
1k 2 0)
1 (k2+1)2 =
π2 12
−9+36 ln 2+2a21(−7+12 ln 2)−12(3+2a21) lna1
,
(5) wherea1 =2m1/Λ. An analytical valueI1is presented after an expansion over 2m1/Λup
to terms of the second order. Such integral appears if we suppose that the parametrization of functionA(t2,k2,k2) for scalar mesons has monopole form for variablesk21andk22:
A(t2,k2,k2)=AS
Λ4
(Λ2−k2 1)(Λ
2−k2 2)
,k1=k, k2 =−k, (6)
whereAS =A(0,0,0). Then the interaction potential takes the form :
∆V2SLs,S(t)= α 2g
sm1AS
6
−9+36 ln 2+2a21(−7+12 ln 2)−12(3+2a21) lna1
1
t2+M2 s
2 One meson exchange contribution
2.1 Scalar meson exchange
Our approach to the calculation of one-meson exchange contributions to the energy levels of (µp) is based on quasipotential approach in quantum field theory [15–18]. The general parametrization of scalar meson→γ∗+γ∗vertex function takes the form [14]:
TSµν(t,k1,k2)=4πα
A(t2,k21,k22)(gµν(k1·k2)−kν1k µ
2)+ (1)
B(t2,k21,k22)(kµ2k21−kµ1(k1·k2))(kν1k 2 2−k
ν
2(k1·k2))
,
whereA(t2,k2 1,k
2 2),B(t
2,k2 1,k
2
2) are two scalar functions,k1,2are four momenta of virtual
pho-tons,tis the four momentum of scalar meson. Then the muon-proton interaction amplitude via the scalar meson exchange can be written as follows:
iMS =α 2g
s
π2
d4kA(t2,k2,k2)(gµν(k
1·k2)−kν1k µ
2)[ ¯u(q1)γµ( ˆp1−kˆ+m1)γνu(p1)][¯v(p2)v(q2)] k4(k2−2m
1k0)(t2+M2s)
,
(2) wherep1,p2are four momenta of particles in initial state,q1,q2are four momenta of particles
in final state. We sett = q1−p1 = 0 because this momentum is small and we consider
further the leading order in fine structure constantαcontribution to the interaction operator (|t| ∼ µα). This leads to the cancelation of the term with the functionB(p2,k2
1,k 2
2). Using
projection operator on muon-proton states with spin S=0, S=1 [15] we can construct the interaction operator for these states. In the case of triplet state we find:
iMS(3S1)=
α2gs
16m2 1m
2 2π2
d4
k4(k2−2k 0m1)
A(t2,k2,k2)(gµνk2−kµkν) (3)
T r1+γ0
2√2 ε( ˆˆ
p2−m2)( ˆq2−m2) ˆε∗
1+γ0
2√2 ( ˆ
q1+m1)γµ( ˆp1−kˆ+m1)γν( ˆp1+m1)
1
t2+M2 s
,
wherem1,m2are the masses of a muon and proton,MS is the mass of scalar meson. After
the trace calculation using the package Form [19] we obtain in the leading order:
T1S0 =T3S1=k2(3k0+2m1)−2m1k20. (4)
So in the leading order the scalar meson exchange doesn’t contribute to hyperfine structure. The typical momentum integral contributing to the shift has the following form:
I1=
d4(k2+2k2
0) k2(k4+a2
1k 2 0)
1 (k2+1)2 =
π2 12
−9+36 ln 2+2a21(−7+12 ln 2)−12(3+2a21) lna1
,
(5) wherea1 =2m1/Λ. An analytical valueI1 is presented after an expansion over 2m1/Λup
to terms of the second order. Such integral appears if we suppose that the parametrization of functionA(t2,k2,k2) for scalar mesons has monopole form for variablesk21andk22:
A(t2,k2,k2)=AS
Λ4
(Λ2−k2 1)(Λ
2−k2 2)
, k1 =k, k2=−k, (6)
whereAS =A(0,0,0). Then the interaction potential takes the form :
∆V2SLs,S(t)= α 2g
sm1AS
6
−9+36 ln 2+2a21(−7+12 ln 2)−12(3+2a21) lna1
1
t2+M2 s
. (7)
Averaging (7) over wave functions of 2S-state we obtain the shift of 2S-level in the form:
∆ELsσ(550)(2S)=α 5µ3g
sm1AS
96πM2 s
(2+WM22
S
)
(1+MSW)4
−9+36 ln 2+2a21(−7+12 ln 2)−12(3+2a21) lna1
(8)
=−0.011meV.
The contribution to 2P-state is suppressed by additional degrees ofα:
∆ELsσ(550)(2P)= α 7µ5g
sAS
288πm1M2s(1+MsW) 4
3
4+
W MS
+3
8
W2 M2 S
−9+a21(−5+6 ln 2)−6a21lna1− (9)
3m2 1 M2
S
−9+36 ln 2+2a21(−7+12 ln 2)−12(3+2a21) lna1
=0.000014µeV.
2.2 Pseudoscalar meson exchange
The general parametrization of pseudoscalar meson (π0, η, η′
)→two virtual photon vertex function is expressed through transition form factor in the form [20]:
Vµν(k1,k2)=iεµναβk1αk2β
α πFπ
Fπ0γ∗γ∗(t2,k21,k22), (10)
Transition form factor has normalizationFπ0γ∗γ∗(0,0,0)=1. At first we consider construction of the hyperfine part of interaction potential in the case of S-states. Using projection operators technique we present interaction amplitude via pseudoscalar meson exchange as follows:
iMP =4πZα 1 16m21m22
d4k
(2π)4
1 (p1−k)2−m21
Fπ0γ∗γ∗(t2,k21,k22) 1
t2+M2 (11)
kαtβεµναβT r[( ˆq1+m1)γν( ˆp1−kˆ+m1)γµ( ˆp1+m1) ˆΠ( ˆp2−m2)γ5( ˆq2−m2) ˆΠ+],
Introduction total and relative momenta of particles in the initial and final states p =
(0,p), q =(0,q) insteadp1,2, q1,2and taking into account that relative momenta are small
(|p| ∼ µα, |q| ∼ µα) we obtain for the numerator of the amplitude the result in the leading order (proportional tot2):
Nh f s=512
3 m
2
1m2[t2k2−(tk)2] (12)
As a result the hyperfine part of muon proton interaction potential takes the form:
∆Vh f s P (p,q)=
α2 6π2
gp m2Fπ
(p−q)2 (p−q)2−m2
πA
(t2), (13)
where
A(t2)= 2i
π2t2
d4k t 2k2
−(tk)2
k2(k−t)2(k2−2kp 1)
Fπγ∗γ∗(k2,(k−t)2). (14)
ForA(t2) there is a dispersion relation with one subtraction, which has the form:
A(t2)=A(0)−t
2
π
∞
0
dsImA(s)
Imaginary part ofA(t2) doesn’t depend on specific form of transition form factor parametriza-tion and is well known [21]:
ImA(t2)= π
2β(t2)ln
1−β(t2) 1+β(t2), β(t
2
)=
1−4m
2 1
t2 . (16)
Numerical value of the A(0) for an electron is equal Ae
(0) = −21.9±0.3 [22] and for a muonAµ
(0) = −6.1±0.3. Going in (13) to coordinate representation by means of the Fourier transform we obtain:
∆VP(r)= α 2g
p
6Fπm2π2{A
(0)
δ(r)− m
2 π
4πre −mπr
−
−1π
∞
0 ds
s ImA(s)
δ(r)+ 1
4πr(s−m2 π)
(m4πe−mπr−s2e−√sr)
} (17)
For numerical calculations we also use the Goldberg-Treiman relation for the pion-nucleon interaction constantgp = gπNN =
mpgA
Fπ withgA = 1.27,Fπ =0.0924 GeV. Averaging (17)
over wave functions of 1S and 2S-state we obtain the hyperfine splitting of 1S and 2S states:
∆Eh f s
P (1S)=−0.0017meV, ∆E h f s
(2S)=−0.0002meV (18)
In a similar way we obtain the potential of 2P1/2-state hyperfine splitting:
∆V2Ph f s,P
1/2(p,q)=− α2g
A
24π3F2 π
(pq)pq +qp−2pq
(p−q)2+m2 π
A(0) (19)
Averaging (19) over wave function of 2P-state we obtain the numerical value of the hyperfine splitting of the level 2P1/2:
∆E2Ph f s,P
1/2 =0.0004µeV. (20)
2.3 Axial vector meson exchange
The general parametrization of axial vector meson→γ∗γ∗vertex function has the form [23]:
Tµνα=4πiαεµνατ(kτ1k 2 2−k
τ 2k
2
1)FAVγ∗γ∗(k2,k2). (21)
Interaction amplitude via axial vector meson exchange has the following general structure:
iMAV=4πZα d4k
(2π)4
Tµνα
(p1−k)2−m21
[ ¯u1(q1)γµ( ˆp1−kˆ+m1)γνu1(p1)]Dαβ(t)[¯v2(p2)Γ (p) β v2(q2)],
(22) where the vertex function
Γ(p)β =γµγ5GA(t2)−iγ5 tµ
2m2
GP(t2), (23)
whereGA(t2) ,GP(t2) are axial and induced form factors respectively. Axial vector meson propagator isDαβ(t) = gαβ−tαtβ/M2A
t2+M2
A
. Using the Lorentz transformation for wave function of muon and proton and formalism of projection operators we obtain for numerator of the am-plitude in the case of state withF=1:
NAV =πiαερσταgµρgνσ(2k−t)τgαβF(0)(k21,k 2 2)G
A
Imaginary part ofA(t2) doesn’t depend on specific form of transition form factor parametriza-tion and is well known [21]:
ImA(t2)= π
2β(t2)ln
1−β(t2) 1+β(t2), β(t
2
)=
1−4m
2 1
t2 . (16)
Numerical value of the A(0) for an electron is equal Ae
(0) = −21.9 ±0.3 [22] and for a muonAµ
(0) = −6.1±0.3. Going in (13) to coordinate representation by means of the Fourier transform we obtain:
∆VP(r)= α 2g
p
6Fπm2π2{A
(0)
δ(r)− m
2 π
4πre −mπr
−
−π1
∞
0 ds
s ImA(s)
δ(r)+ 1
4πr(s−m2 π)
(m4πe−mπr−s2e−√sr)
} (17)
For numerical calculations we also use the Goldberg-Treiman relation for the pion-nucleon interaction constantgp = gπNN =
mpgA
Fπ withgA = 1.27,Fπ =0.0924 GeV. Averaging (17)
over wave functions of 1S and 2S-state we obtain the hyperfine splitting of 1S and 2S states:
∆Eh f s
P (1S)=−0.0017meV, ∆E h f s
(2S)=−0.0002meV (18)
In a similar way we obtain the potential of 2P1/2-state hyperfine splitting:
∆V2Ph f s,P
1/2(p,q)=− α2g
A
24π3F2 π
(pq)qp+qp−2pq
(p−q)2+m2 π
A(0) (19)
Averaging (19) over wave function of 2P-state we obtain the numerical value of the hyperfine splitting of the level 2P1/2:
∆E2Ph f s,P
1/2 =0.0004µeV. (20)
2.3 Axial vector meson exchange
The general parametrization of axial vector meson→γ∗γ∗vertex function has the form [23]:
Tµνα=4πiαεµνατ(k1τk 2 2−k
τ 2k
2
1)FAVγ∗γ∗(k2,k2). (21)
Interaction amplitude via axial vector meson exchange has the following general structure:
iMAV=4πZα d4k
(2π)4
Tµνα
(p1−k)2−m21
[ ¯u1(q1)γµ( ˆp1−kˆ+m1)γνu1(p1)]Dαβ(t)[¯v2(p2)Γ (p) β v2(q2)],
(22) where the vertex function
Γ(p)β =γµγ5GA(t2)−iγ5 tµ
2m2
GP(t2), (23)
whereGA(t2) ,GP(t2) are axial and induced form factors respectively. Axial vector meson propagator isDαβ(t) = gαβ−tαtβ/M2A
t2+M2
A
. Using the Lorentz transformation for wave function of muon and proton and formalism of projection operators we obtain for numerator of the am-plitude in the case of state withF=1:
NAV =πiαερσταgµρgνσ(2k−t)τgαβF(0)(k21,k 2 2)G
A
(0)× (24)
T r
( ˆq1+m1)γν( ˆp1−kˆ+m1)γµpˆ1+m1)
1+γ0
2√2 ε( ˆˆ p2−m2)γβγ5( ˆq2−m2) ˆε
∗1+γ0
2√2
,
For transition form factorFAVγ∗γ∗(k2,k2) we use the following representation [24]:
FAVγ∗γ∗(k2,k2)=FAVγ∗γ∗(0,0)
Λ8
(Λ2−k2)4,FAVγ∗γ∗(0,0)=24<e 2 q>R
′ (0)
√
2
√πM9/2 A
, (25)
whereR′(0) is the value of derivative of radial wave function at zero, < e2
q >is squared
effective charge of a light quark in a bound state in units of electron charge. As a result the potential contributing to hyperfine structure has the form of a Yukawa potential:
∆Vh f s
AV(p−q)==
32α2GA(0)FAVγγ(0,0)
3π2e(t2+M2 A)
id4k 2k 2+k2
0 k2(k2−2m
1k0) Λ8
(k2−Λ2)4 (26)
Equation (26) contains a characteristic loop momentum integral, that can be calculated ana-lytically:
I(m1
Λ)=
(−id4k) 2k
2+k2 0 k2(k2−2m
1k0) Λ6
(k2−Λ2)4 = (27)
− π
2
4(a2 1−1)
5/2
3
a2 1−1+a
2 1(2a
2
1−5)arcsec(a1)
=−6.78645,a1=
2m1 Λ
After the Fourier transform of the potential (26) and averaging it over the wave functions, we obtain the contribution to the HFS spectrum:
∆Eh f sf
1 (1S)=
32α5µ3Λ2gAV NNFAVγ∗γ∗(0,0)
2+WM22
A
3MA2π2e1+ W MA
2 I(
m1
Λ)=−0.0093meV, (28)
∆Eh f sf
1 (2S)=
2α5µ3Λ2g
AV NNFAVγ∗γ∗(0,0)
3M2 Aπ2e
1+MW
A
4 I(
m1
Λ)=−0.0012meV. (29)
2.4 Tensor meson exchange
Following [25] we will assume further that hadronic light-by-light amplitude for tensor mesons is dominated be helicityΛ =2 exchange. The amplitude of the processγ∗+γ∗→T
can be parameterized as follows [26, 27]:
TµναβT (k1,k2)=e2 k1k2
M Mµναβ(k1,k2)FTγ∗γ∗(k 2 1,k
2
2), (30)
whereFTγ∗γ∗(k12,k22) is a transition form factor (TFF) of tensor meson,
Mµναβ(k1,k2)=
Rµα(k1,k2)Rνβ(k1,k2)+
1
8(k1+k2)2
(k1k2)2−k21k22
Rµν(k1,k2)× (31)
(k1+k2)2(k1−k2)α−(k21−k 2
2)(k1+k2)α
×(k1+k2)2(k1−k2)β−(k21−k 2
2)(k1+k2)β
,
Rµν(k1,k2)=−gµν+
1
X
(k1k2)(k
µ 1k
ν 2+k
µ 2k
n 1u)−k
2 1k
µ 2k
ν 2−k
2 2k µ 1k ν 1
Then the muon-proton interaction amplitude via tensor meson exchange can be presented as
iMT = 4πZα
16m2 1m
2 2
� d4k
(2π)4
1
(p1−k)2−m21D
µµ′(t−k)Dνν′(k)Dα ′β′αβ
T (t)Mµ′ν′α′β′(k1,k2) (32)
[ ¯u(0)( ˆq1+m1)γµ( ˆp1−kˆ+m1)γν( ˆp1+m1)u(0)][¯v(0)( ˆp2−m2)Γ αβ
T NN( ˆq2−m2)v(0)],
where the vertex function of tensor meson nucleon interaction is [28]
ΓαβT NN(p2,q2)= GT NN
m2
�
(q2+p2)αγβ+(q2+p2)βγα
�
+FT NN
m22 (q2+p2)α(q2+p2)β.
We will use only first term of this vertex function, because different estimations [28] show, that there take place an inequalityGT NN ≫FT NN. The tensor meson propagator has the form:
DµναβT (t)=
1
t2−M2 T+iε
�1
2(gµαgνβ+gµβgνα−gµνgαβ)+ (33)
1 2 gµα
tνtβ M2 T
+gνβ tµtα M2 T
+gµβ tνtα M2 T
+gνα tµtβ M2 T
+
2 3 1 2gµν+
tµtν M2 T
1 2gαβ+
tαtβ M2 T
�
.
To obtain the interaction potential via tensor meson exchange we use the formalism of projec-tion operators. Projecting the interacprojec-tion amplitude on the muon-proton S-state withF =1 we obtain in the numerator of the amplitude (v=(1,0,0,0) is auxiliary time-like 4-vector):
T(3S1)=T r
� γε1
1+γ0
2√2 ( ˆq1+m1)γµ( ˆp1− ˆ
k+m1)γν( ˆp1+m1) (34)
1+γ0
2√2 γε2( ˆp2−m2)Γ
αβ
T NN( ˆq2−m2)
�1
3(−gε1ε2+vε1vε2).
Using the package Form for the trace calculation we obtain that in the leading order intthe contributions to the interaction amplitudes in3S
1and1S0states are the following:
T(3S1)=T(1S0)= GT NN
MT
4m1k4
1+
k4 0t
4
�
(kt)2−k2t2�2 +2
k2 0t
2
�
(kt)2−k2t2�
. (35)
The contribution to the ground state hyperfine splitting contains additional degreest:
Th f s= 4
3
GT NN MTm2
t2k4
1−
t4k4 0
�
(kt)2−k2t2�2
. (36)
The Belle Collaboration investigated the reactionγ∗γ∗→π0π0in a wide range of photon
virtualities and invariant massW of theπ0π0 system in the range 0.5GeV <W <2.1GeV
[29]. From analysis [29] it was obtained first empirical results for the tensor mesonf2TFF in
the dipole form on one photon four-momentum squared when other four-momentum squared is equal to 0. Because in the interaction amplitude both photons are virtual we use the dipole parametrization for the TFFγ∗+γ∗→T:
FTγ∗γ∗(k21,k22)=FTγ∗γ∗(0,0)
Λ4
Then the muon-proton interaction amplitude via tensor meson exchange can be presented as
iMT = 4πZα
16m2 1m
2 2
� d4k
(2π)4
1
(p1−k)2−m21D
µµ′(t−k)Dνν′(k)Dα ′β′αβ
T (t)Mµ′ν′α′β′(k1,k2) (32)
[ ¯u(0)( ˆq1+m1)γµ( ˆp1−kˆ+m1)γν( ˆp1+m1)u(0)][¯v(0)( ˆp2−m2)Γ αβ
T NN( ˆq2−m2)v(0)],
where the vertex function of tensor meson nucleon interaction is [28]
ΓαβT NN(p2,q2)= GT NN
m2
�
(q2+p2)αγβ+(q2+p2)βγα
�
+FT NN
m22 (q2+p2)α(q2+p2)β.
We will use only first term of this vertex function, because different estimations [28] show, that there take place an inequalityGT NN ≫FT NN. The tensor meson propagator has the form:
DµναβT (t)=
1
t2−M2 T+iε
�1
2(gµαgνβ+gµβgνα−gµνgαβ)+ (33)
1 2 gµα
tνtβ M2 T
+gνβ tµtα M2 T
+gµβ tνtα M2 T
+gνα tµtβ M2 T + 2 3 1 2gµν+
tµtν M2 T 1 2gαβ+
tαtβ M2 T � .
To obtain the interaction potential via tensor meson exchange we use the formalism of projec-tion operators. Projecting the interacprojec-tion amplitude on the muon-proton S-state withF =1 we obtain in the numerator of the amplitude (v=(1,0,0,0) is auxiliary time-like 4-vector):
T(3S1)=T r
� γε1
1+γ0
2√2 ( ˆq1+m1)γµ( ˆp1− ˆ
k+m1)γν( ˆp1+m1) (34)
1+γ0
2√2 γε2( ˆp2−m2)Γ
αβ
T NN( ˆq2−m2)
�1
3(−gε1ε2+vε1vε2).
Using the package Form for the trace calculation we obtain that in the leading order intthe contributions to the interaction amplitudes in3S
1and1S0states are the following:
T(3S1)=T(1S0)= GT NN
MT
4m1k4
1+
k4 0t
4
�
(kt)2−k2t2�2 +2
k2 0t
2
�
(kt)2−k2t2�
. (35)
The contribution to the ground state hyperfine splitting contains additional degreest:
Th f s= 4
3
GT NN MTm2
t2k4
1−
t4k4 0
�
(kt)2−k2t2�2
. (36)
The Belle Collaboration investigated the reactionγ∗γ∗→π0π0in a wide range of photon
virtualities and invariant massW of theπ0π0system in the range 0.5GeV <W <2.1GeV
[29]. From analysis [29] it was obtained first empirical results for the tensor meson f2TFF in
the dipole form on one photon four-momentum squared when other four-momentum squared is equal to 0. Because in the interaction amplitude both photons are virtual we use the dipole parametrization for the TFFγ∗+γ∗→T:
FTγ∗γ∗(k12,k22)=FTγ∗γ∗(0,0)
Λ4
(k2−Λ2)2, k1=k,k2=−k. (37)
The parametrization (37) is widely used for the description of experimental data for all ex-changed mesons (see previous sections). Making the transition to the Euclidean space we present the loop momentum integral in the integral form:
I=
� ∞
0
d4k k4−4k2
0m 2 1
FTγ∗γ∗(0,0)
Λ4
(k2−Λ2)2
1+
k40t4
�
(kt)2−k2t2�2 +2
k20t2
�
(kt)2−k2t2�
= (38)
=−FTγ∗γ∗(0,0) � π
0
sinψdψ
4�a2
1cosψ+a 2 1−2
�2 �
a21cos 2ψ−2 ln�a21cos2ψ�+a21−2�×
�
sin3ψ−3 sinψ�cos2ψ+3�+cos2ψ(cos 2ψ−7) ln �
2 sinψ+1 −1
��
wherea1= 2mΛ1,FTγ∗γ∗(0,0)=
2√5Γγγ
α√πMT. The integral (38) is calculated numerically. Then the
particle interaction operators for the Lamb shift and hyperfine splitting are:
∆VTLs(r)=−16α 2m
1GT NN MT
2� 5Γγγ
α√πMTI
1 4πre
−MTr, (39)
∆VTh f s(r)=8α 2G
T NN
3πm2MT
2� 5Γγγ
α√πMTJ
δ(r)−
M2 T
4πre −MTr
, (40)
J=
� π
0
sinψdψ (−2+a2
1+a 2
1cos 2ψ)2
�
a21cos 2ψ−2 ln�a21cos2ψ�+a21−2�×
�
sin3ψ+7 sinψ−3 sinψcos2ψ+2 cos4ψln �
2 sinψ+1 −1
�� .
There are several tensor mesons which can contribute to LS and HFS [30], but in our calcula-tion we take into account only the contribucalcula-tion off2(1270) meson, since we know for it more
or less reliable values of various parameters, including the coupling constant with the nucleon [28]. Using obtained potentials (39), (40) we can calculate contributions to the energy levels of muonic hydrogen:
∆ELsf
2(1S)=−
16α5µ3m 1GT NN
πMT
2�5Γγγ √
πMT
1 (MT +2W)2I
=−0.0528meV, (41)
∆ELsf
2(2S)=− 2α5µ3m
1GT NN
πMT
2� 5Γγγ √πM
T
(2MT2+W2) 2(MT +2W)4I
=−0.0066meV, (42)
∆Eh f sf
2 (1S)=− 8α5µ3G
T NN
3π2M Tm2
2�5Γγγ √πM
T 1−
M2T
(MT +2W)2
J=−0.0551µeV, (43)
∆Eh f sf
2 (2S)=−
α5µ3GT NN
3π2M Tm2
2�5Γγγ √
πMT
1−
M2 T(2M
2 T+W
2)
2(MT+W)4
J=−0.0069µeV, (44)
whereW =µZα. Another tensor mesona2(1320) can make an order of magnitude smaller
contribution to (41)-(44). Such an approximate estimate is due to the fact that we do not know fora2(1320) exactly the coupling constant with the nucleon, and the estimates of the
Reggeon coupling constants with the nucleon differ by an order of magnitude: f2Rpp =11.04, a2Rpp=1.68 [31]. If we use the dipole parameterization for each variablek21,k
2
2in TFF, then
3 Conclusion
The obtained numerical values of the meson contributions to the Lamb shift and the hyperfine structure show that they are significant and must be taken into account in a more accurate comparison with experimental data. The contribution of the tensor meson to the lamb shift is comparable to the contribution of the scalarσ-meson. Other tensor mesons apparently make a significantly smaller contribution, since their constant of interaction with the nucleon is much smaller. Experimental data [30] show that all tensor mesons have a significant decay width into a pair of pions, which interact well with the nucleon, therefore, such processes need to be investigated additionally. Work in this direction is in progress.
The work is supported by Russian Science Foundation (grant No. RSF 18-12-00128) and Russian Foundation for Basic Research (grant No. 18-32-00023) (F.A.M.).
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