Polarized Lepton-Nucleon Elastic Scattering and a Search for a Light Scalar Boson

# Polarized Lepton-Nucleon Elastic Scattering and a Search for a Light Scalar Boson

Yu-Sheng Liu1 and Gerald A. Miller2
11ysliu@uw.edu
22miller@uw.edu
###### Abstract

Lepton-nucleon elastic scattering, using the one-photon and one-scalar-boson exchange mechanisms considering all possible polarizations, is used to study searches for a new scalar boson and suggest new measurements of the nucleon form factors. A new light scalar boson, which feebly couples to leptons and nucleons, may account for the proton radius and muon puzzles. We show that the scalar boson produces relatively large effects in certain kinematic region when using sufficient control of lepton and nucleon spin polarization. We generalize current techniques to measure the ratio and present a new method to separately measure and using polarized incoming and outgoing muons.

affiliationtext: Department of Physics, University of Washington, Seattle, WA 98195-1560, USA

## 1 Introduction

Lepton-nucleon elastic scattering is important both in theory and experiment. The use of polarization techniques has yielded much new information. In this paper, we study lepton-nucleon elastic scattering using the one-photon and one-scalar-boson exchange mechanism by considering all possible polarizations. The differential cross sections are calculated in a general reference frame. There are two applications of our results in this paper. The first one is searching for a new light scalar boson. The second one, using only the one-photon exchange contribution, is to provide new ways to measure the nucleon form factor: generalizing current techniques to measure the ratio and presenting a new method to separately measure and using polarized incoming and outgoing muons.

The interest in a new light scalar boson arises from recent studies of the proton radius using the Lamb shift in the transition in muonic hydrogen [1, 2], the value of proton radius was reported to be 0.84087(39) fm, whereas the CODATA value [3] 0.8775(51) fm is 7 standard deviation away. The major difference between these two reported data is that the former extracts the proton radius from muonic hydrogen and the latter from electronic hydrogen and electron-proton scattering experiments. Although the different proton radius may arise due to subtle lepton-nucleon non-perturbative effects within the standard model [4], it could also be a signal of new physics caused by a violation of lepton universality.

Another candidate of new physics signal is the muon anomalous magnetic moment, which is defined as . The measurement at BNL [5] differs from the standard model prediction by 3 to 4 standard deviations,333The experimental and the standard model uncertainty are and , respectively.

 Δaμ=aexpμ−athμ= (287±80)×10−11\@@cite[cite][\@@bibrefDavier:2010nc] (1) = (261±78)×10−11\@@cite[cite][\@@bibrefHagiwara:2011af] (2)

the different values depending on the choice of lowest order hadronic contribution. This discrepancy is also possibly explained by new physics involving violation of lepton universality.

It is known that a light scalar boson with mass around 1 MeV is a candidate to explain both proton radius and muon anomalous magnetic moment puzzles simultaneously [4, 8, 9]. The non-relativistic potential between lepton and nucleon caused by exchange of a scalar boson is written as

 Vϕ(r)=−glgN4πe−mϕrr=glgNeleNVEM(r)e−mϕr, (3)

where is the mass of the scalar boson; and are scalar bosons coupling to lepton and nucleon, respectively. We adopt the constraints in [8] that for MeV

 gee≲2.3×10−4,gpe=gμe≲1.3×10−3,gn≲2×10−5. (4)

So the potential of the scalar boson is suppressed by and an exponentially decay factor comparing with the Coulomb potential. The fact that the potential of the scalar boson is intrinsically much smaller than the Coulomb potential makes it hard to find the scalar boson. There could be other possible particles, however, most of them are ruled out, e.g. pseudoscalar and axial vector are ruled out by hyperfine splittings [4, 10].

There are proposals to search for the light scalar boson, such as a direct detection method [9]. In this paper, we study the cross section of elastic lepton-nucleon scattering caused by one-photon and one-scalar-boson exchange. Since the muon is much heavier than the electron, the lepton mass can not be neglected, as is done for electron-proton scattering. The two-photon exchange contribution is expected (from perturbation expansion) and measured to be at a few percent level compared with one-photon exchange (OPE) [11, 12, 13, 14]. Therefore, the effect of one-scalar-boson exchange must be greater than few percent of OPE to be observed. We consider unpolarized and polarized elastic scattering cross sections. We find the following two cases that for certain kinematic regions where the effects of the scalar boson are dominant and potentially observable: electron-neutron cross section with incoming and outgoing electrons polarized, and muon-neutron cross section with incoming and outgoing neutrons polarized.

There are several facilities that can measure the lepton-nucleon elastic scattering, for electron, such as JLab, and for muon, MUSE at PSI and the J-PARC muon facility. Based on our results and the current experimental setup and capability [15, 16], although the polar angle resolution is sufficient, the kinematic region where scalar bosons may have significant effects is beyond the current polar angle measured range; the electron-neutron scattering is more promising than muon-neutron scattering due to the high intensity of the electron flux. Further estimates and discussions are in section 4.2 and 4.3.

The polarized cross section with one-photon exchange is used to measure the Sachs form factors, and [17, 18, 19]. The standard technique of measurement is using elastic electron-nucleon scattering using the following experiments: unpolarized Rosenbluth separation [20], polarized lepton beam and nucleon target, and polarized lepton beam and recoil nucleon. The latter two experiments, using the ratio technique (also known as polarization transfer method), were first developed in [21] and later discussed in more detail in [22]. We explore other possible ways to determine form factors by including lepton masses and other lepton polarization configurations different from conventional ones. In section 5.1, we generalize the current method to measure the ratio of form factors, , by including lepton mass, non-longitudinal lepton polarization, and more polarization configurations (polarized one lepton and one nucleon, either they are incoming or outgoing). In section 5.2, we present a new method to measure or directly for certain kinematic conditions in elastic muon-nucleon scattering cross section with polarized incoming and outgoing muons.

The outline of this paper is as follows. In section 2, we set up the formalism needed to compute cross sections. In section 3, the cross sections are calculated in a general reference frame with all possible polarizations, and the massless lepton limit is slso discussed. In section 4, we show that there are two cases in which the scalar boson is potentially observable: electron-neutron cross section with incoming and outgoing electrons polarized, and muon-neutron cross section with incoming and outgoing neutrons polarized. In section 5, comparing with the current method, we give a more general result for the ratio of form factors, , by including lepton mass, non-longitudinal lepton polarization, and more polarization configurations. We also discuss new measurements to selectively obtain the contribution from or in the cross section with polarized incoming and outgoing muons. A conclusion is presented in section 6.

## 2 Setup

The elastic lepton-nucleon scattering process is denoted as

 l(p1,s1)+N(p2,s2)→l(p3,s3)+N(p4,s4), (5)

where and stand for lepton and nucleon; and are momentum and spin polarization; the number 1, 2, 3, and 4 label the incoming lepton, incoming nucleon, outgoing lepton, and outgoing nucleon. This notation is used throughout the entire paper. In the lowest order, we consider the interaction by exchanging a scalar boson or a photon between the lepton and the nucleon. There are three contributions to the cross section: one-photon exchange, one-scalar-boson exchange, and the interference terms.

### 2.1 Kinematics

In the nucleon rest frame (lab frame), we choose the the coordinate such that is along axis and is in - plane to exploit the symmetry. With these choices, all the kinematics quantities which we need can be expressed in terms of only two variables, and the scattering angle . The scattering angle is an angle between outgoing and incoming lepton, and defined as . In the lab frame, the external momenta and space-like momentum transfer can be expressed as follows

 p1 =|p1|^z,E1=√|p1|2+m2l,p2=0,E2=mN, (6) p3 =(m2l+E1mN)cosθ+(E1+mN)√m2N−m2lsin2θ(E1+mN)2−cos2θ|p1|2|p1|^p3, (7) ^p3 =sinθ^x+cosθ^z, (8) E3 =(E1+mN)(m2l+E1mN)+cosθ|p1|2√m2N−m2lsin2θ(E1+mN)2−cos2θ|p1|2 (9) p4 =p1−p3,E4=E1+mN−E3, (10) q =p1−p3=p4−p2, (11) q2 =2|p1|2mN(mN+E1sin2θ−cosθ√m2N−m2lsin2θ)(E1+mN)2−cos2θ|p1|2. (12)

We use the mostly-plus metric, so that if is space-like. If we take spin polarization into account, each polarized particle needs two angular variables to specify the spin direction, see section 2.3 below.

### 2.2 Dynamics

The scalar boson interacts with lepton and nucleon through Yukawa coupling

 Lϕ⊃−12(∂ϕ)2−12m2ϕϕ2+glϕ¯ψlψl+gNϕ¯ψNψN, (13)

where and are Yukawa couplings between lepton-scalar and nucleon-scalar. There are two lowest order diagrams: one-photon and one-scalar-boson exchange, see figure 1.

The amplitude squared is given by

 |T|2=e2le2N(Iγγ+λIγϕ+λ2Iϕϕ), (14)

where and are couplings of leptons and nucleons, respectively; and using the mostly-plus metric are given by

 λ(q2)= glgNeleNq2q2+m2ϕ (15) Iγγ= 1q4Tr(u3¯u3γμu1¯u1γν)Tr(u4¯u4Vμu2¯u2Vν) (16) Iγϕ= 1q4[Tr(u3¯u3γμu1¯u1)Tr(u4¯u4Vμu2¯u2) +Tr(u3¯u3u1¯u1γμ)Tr(u4¯u4u2¯u2Vμ)] (17) Iϕϕ= 1q4Tr(u3¯u3u1¯u1)Tr(u4¯u4u2¯u2) (18) Vμ(q)= F1(q2)γμ−i2mNF2(q2)σμνqν (19)

where is mass of the scalar boson; ; and are the form factors of nucleon coupling. In general, there is a form factor, , of scalar-nucleon coupling, however, it always appears with , therefore we can include it into the constraint of .

### 2.3 Polarization

The product of spinors in mostly-plus metric is given by

 u¯u=12(1−γ5⧸s)(−⧸p+m) (20)

where is the spin polarization which satisfies and . We can solve for the spin polarization using the two constraints

 s=zE√m2+|p|2sin2α(|p|Ecosα,^s), (21)

where and are the polar and azimuthal angle of with respect to ; ; and becomes helicity if the particle is longitudinally polarized.

The polarization of leptons and nucleons in the lab frame in the coordinate we chose can be expressed as

 ^si= sinαicosβi^x+sinαisinβi^y+cosαi^zfor i=1 % or 2 (22) ^s3= sinα3cosβ3(^y×^p3)+sinα3sinβ3^y+cosα3^p3 = (sinα3cosβ3cosθ+cosα3sinθ)^x+sinα3sinβ3^y +(−sinα3cosβ3sinθ+cosα3cosθ)^z (23) ^s4= (sinα4cosβ4cosθ′−cosα4sinθ′)^x+sinα4sinβ4^y +(sinα4cosβ4sinθ′+cosα4cosθ′)^z (24)

where is angle between and ; .

There are two important special cases. For transverse polarization , . For longitudinal polarization ,

 sL=zEm(|p|E,^p). (25)

This is useful for a light lepton because if its mass is small compared with the beam energy, the light lepton is naturally longitudinally polarized.

#### 2.3.1 Massless Particle

The massless lepton limit is important for electron scattering or the high energy limit. In general, naively taking in physical quantities does not give us the correct result (see section 3.4 below), because the longitudinal polarization (25) blows up. The correct way to take the limit is to set the polarization to be longitudinal, , then take the massless limit. Then the product of spinors (20) becomes

 u¯u=−12(1+zγ5)⧸p(massless), (26)

where is the helicity of the lepton.

## 3 Cross Section

Combining with the amplitude squared (14), the differential cross section in a general reference frame is

 dσdΩ=A(Iγγ+λIγϕ+λ2Iϕϕ)=(dσdΩ)γγ+(dσdΩ)γϕ+(dσdΩ)ϕϕ, (27)

where

 A=α24|p3||E2p1−E1p2|[E1+E2−E3(p1+p2)⋅^p3|p3|]−1. (28)

In the lab frame, the kinematic factor becomes

 Alab=α24mN|p3||p1|[E1+mN−E3|p1||p3|cosθ]−1. (29)

In (27), it is worth mentioning that the pure one-photon or one-scalar-boson exchange contribution, and , is always greater or equal to zero, whereas the interference term can be positive or negative depending on the sign of Yukawa coupling and electric charge.

### 3.1 Unpolarized

The cross section (27) is proportional to , , and , so it is sufficient to show these three ’s instead of full cross section. The unpolarized ’s are defined as

 Iu=(12∑z1)(12∑z2)∑z3∑z4I1,2,3,4 (30)

where the superscripts refer to which particle is polarized and u for unpolarized. In a general reference frame,

 Iuγγ=G2E+τG2M1+τ(R2−1+ττ)+G2M(2−m2lτm2N) (31) Iuγϕ=−2GERmlτmN (32) Iuϕϕ=1+ττ(1+m2lτm2N) (33)

where and are Sachs form factors

 GM(q2)=F1(q2)+F2(q2),GE(q2)=F1(q2)−τF2(q2); (34)

is defined in the usual way444Note that there is no minus sign in mostly-plus metric, and if is space-like.; depends on Mandelstam variables555, , and .

 R=u−st=(p1+p3)⋅p2(p1−p3)⋅p2=−4p1⋅p2q2−1. (35)

As an example, in the lab frame, , with massless lepton limit (), the Rosenbluth cross section is

 (dσdΩ)Rosenbluth=(AIuγγ)lab,ml→0. (36)

The interference term (32) is proportional to lepton mass . The interference term for muon is about two order of magnitude bigger than for electron, see figure 2.

### 3.2 Partially Polarized

There is no contribution to cross section if only one particle is polarized in exchanging one-photon and one-scalar-boson. The reason is the following. The parity flips momentum and time reversal flips momentum and angular momentum. One can only flip spin polarization by combining parity and time reversal, and it is equivalent to change the overall sign of the polarization. Therefore, if the theory conserves , the spin asymmetry part can depend only on product of even number of spin polarizations to remain invariant under . For example, if we polarize three particles, there is no term depending on the product of all three polarizations, but there are terms depending on product of two polarizations, see section 3.3. On the other hand, one can consider an interaction which breaks time reversal to have an additional dependence on odd number of spin polarizations, such as exchanging a Z boson. In conclusion, in exchanging a photon and a scalar boson, one needs to polarize at least two particles to have a spin dependent part.

#### 3.2.1 One Lepton and One Nucleon Polarized

First we study the case that one lepton and one nucleon are polarized (each lepton and nucleon can be incoming or outgoing). In order to combine all four cases into one expression , we require that the first superscript, , to be lepton (1 or 3); the second superscript, , to be nucleon (2 or 4).

 (37)

where is the unpolarized part; , are particle labels and not summed; takes care of the factors due to the difference between average or sum over initial or final states

 ai,j=⎧⎪ ⎪⎨⎪ ⎪⎩1if (i,j)=(1,2)12if (i,j)=(1,4) or (3,2)14if (i,j)=(3,4) (38)

or more compactly . The results are

 (Ji,jγγ)μν= 2mlmNGM[bjF2(p2+p4)μqνq2−GEτ(gμν−qμqνq2)] (39) (Ji,jγϕ)μν= 2GM[R(gμν−qμqνq2)+(p2+p4)μ(p1+p3)νq2] (40) (Ji,jϕϕ)μν= 0 (41)

where depends only on nucleon label

 bj={1if j=2−1if j=4 (42)

or more compactly . The expression of (39) is the most compact form we found. However, the following expression for

 (Ji,jγγ)μν=2mlmNGM1+τ[ bjGM(p2+p4)μqνq2−GE(2bjpμjqνq2+1+ττgμν−qμqντq2)] (43)

is more useful for measuring the Sachs form factors, and , by polarized method, which is discussed in section 5.

It is worth noticing that it seems that in the massless lepton limit vanishes, however it is not true because there is another in the denominator when the lepton spin polarization becomes longitudinal (25). Therefore, this expression is well-behaved in the massless limit. We discuss the massless limit in detail in section 3.4.

#### 3.2.2 Incoming and Outgoing Leptons Polarized

For polarized incoming and outgoing leptons, the observable quantity is

 I1,3=(12∑z2)∑z4I1,2,3,4=12(1+s1⋅s3)Iu+s1μs3ν(J1,3)μν. (44)

The results are

 (J1,3γγ)μν= −G2Mgμν+2G2E+τG2M1+τ(qμqν2τq2−p{μ2pν}4+Rq[μpν]2q2) (45) (J1,3γϕ)μν= 2mlτmNGEq[μpν]2q2 (46) (J1,3ϕϕ)μν= −1+ττqμqνq2 (47)

where the curly and square brackets are symmetrization and anti-symmetrization notation

 p{μqν}=pμqν+pνqμ,p[μqν]=pμqν−pνqμ. (48)

The expression in terms of is the most compact form we found. The following expression of

 I1,3γγ=12Iuγγ+s1μs3ν{ −12G2Mm2lτm2Ngμν+2G2E+τG2M1+τ[ gμν4(R2−1+ττ)+qμqν2τq2−p{μ2pν}4+Rq[μpν]2q2]} (49)

is useful to show that and are separated, as unpolarized case, in spin asymmetry part. By choosing kinematics conditions, we can separately measure or . Further discussions are in section 5.2.

#### 3.2.3 Incoming and Outgoing Nucleons Polarized

For polarized incoming and outgoing nucleons,

 I2,4=(12∑z1)∑z3I1,2,3,4=12(1+s2⋅s4)Iu+s2μs4ν(J2,4)μν. (50)

The results are

 (J2,4γγ)μν= −G2M(gμν−m2lτm2Nqμqνq2+2pμ1pν1+pμ3pν3q2) −F22τ(R2−1+ττ)qμqνq2+2GMF1qμqν+Rq[μpν]1q2 (51) (J2,4γϕ)μν= −2mlmN(F2Rqμqνq2+GMτq[μpν]1q2) (52) (J2,4ϕϕ)μν= −(1+m2lτm2N)qμqνq2. (53)

The asymmetry part of contains the cross term, therefore it is harder to use it to measure the form factors separately.

### 3.3 Fully Polarized

From the argument at the beginning of section 3.2, the fully polarized cross section depends on the product of even number of spin polarizations. One can separate the unpolarized and partially polarized contributions, then leave the part which depends on all four polarizations,

 I1,2,3,4γγ= −14[5+s1⋅s3+s2⋅s4+(s1⋅s3)(s2⋅s4)]Iuγγ +12[(1+s2⋅s4)I1,3γγ+(1+s1⋅s3)I2,4γγ]+14I1,2γγ+12(I1,4γγ+I3,2γγ)+I3,4γγ −F1GM[(p2+p4){μgρ}[νqσ]+Rq[μgρ][νqσ]4q2+qμqνqρqσ+q[μpρ]2q[νpσ]1q4] +gμνgρσ+gμσgνρ4+q[μgσ][νqρ]+gν[μqρ]pσ1+gσ[μqρ]pν32q2} (54) I1,2,3,4γϕ= −Iuγϕ+I1,3γϕI2,4γϕIuγϕ+14I1,2γϕ+12(I1,4γϕ+I3,2γϕ)+I3,4γϕ −s1μs2νs3ρs4σGMmlτmN(q[ρgμ][νqσ]2q2+2Rq[μpρ]2q[νpσ]1q4) (55) I1,2,3,4ϕϕ= I1,3ϕϕI2,4ϕϕIuϕϕ. (56)

### 3.4 Massless Lepton Limit

As we discussed in section 2.3.1, naively taking the limit does not yield correct results. Only some results are still safe when we take the limit , such as , , , etc. We discuss the general massless lepton limit in this section.

For a lepton of negligible mass, from either direct calculation or conservation of angular momentum, the interference term vanishes for all cases. Therefore it is sufficient to display and .

1. The unpolarized is safe in the limit ,

 Iuγγ=G2E+τG2M1+τ(R2−1+ττ)+2G2M (57) Iuϕϕ=1+ττ. (58)
2. For one lepton and one nucleon polarized,

 (59)

the result can also be obtained by setting the lepton polarization to be longitudinal then taking limit,

 (Jjγγ)μ= GM[bjF2RqμmN−GEτ(p1+p3)μmN] (60) (Jjϕϕ)μ= 0. (61)
3. For incoming and outgoing leptons polarized, one can not directly take the limit , but the results are

 I1,3γγ= 12(1+z1z3)Iuγγ (62) I1,3ϕϕ= 12(1−z1z3)Iuϕϕ. (63)
4. For incoming and outgoing nucleons polarized, it is safe to take the limit ,

 I2,4=(12∑z1)∑z3I1,2,3,4=12(1+s2⋅s4)Iu+s2μs4ν(J2,4)μν. (64)

The results are

 (J2,4γγ)μν= −G2M(gμν+2pμ1pν1+pμ3pν3q2) −F22τ(R2−1+ττ)qμqνq2+2GMF1qμqν+Rq[μpν]1q2 (65) (J2,4ϕϕ)μν= −qμqνq2. (66)
5. Finally, for fully polarized case, the result is much simpler than (54)

 I1,2,3,4γγ= −Iuγγ+12(1+z1z3)I2,4γγ+14I1,2γγ+12(I1,4γγ+I3,2γγ)+I3,4γγ (67) I1,2,3,4ϕϕ= I1,3ϕϕI2,4ϕϕIuϕϕ. (68)

## 4 Searching for a Scalar Boson

If we want to see a new physics signal in elastic lepton-nucleon scattering using one photon and one-scalar-boson exchange, it needs to be greater than the next leading order contribution of the standard model, which is the two-photon exchange contribution (generically suppressed by one more fine structure constant than one-photon exchange). Therefore, the contribution involving scalar boson, and , must be greater than at least few percent of to be detected.

We adopt the constraints in [8] that for MeV

 gee≲2.3×10−4,gpe=gμe≲1.3×10−3,gn≲2×10−5. (69)

Therefore, recall (15), . In general, and are suppressed by and with respect to the leading standard model contribution , respectively. Although it seems to be hard to observe scalar boson in elastic scattering, we still find some kinematic regions where the scalar boson may be found. This is because for such kinematic region, goes or approaches to zero, whereas and