Cosmic Strings in Hidden Sectors: 1. Radiation of Standard Model Particles

In hidden sector models with an extra U(1) gauge group, new fields can interact with the Standard Model only through gauge kinetic mixing and the Higgs portal. After the U(1) is spontaneously broken, these interactions couple the resultant cosmic strings to Standard Model particles. We calculate the spectrum of radiation emitted by these"dark strings"in the form of Higgs bosons, Z bosons, and Standard Model fermions assuming that string tension is above the TeV scale. We also calculate the scattering cross sections of Standard Model fermions on dark strings due to the Aharonov-Bohm interaction. These radiation and scattering calculations will be applied in a subsequent paper to study the cosmological evolution and observational signatures of dark strings.


I. INTRODUCTION
In the Standard Model of particle physics, all of the matter fields are charged under the gauge group of the theory, and consequently all of the particles participate in the gauge interactions. It is natural to ask whether there can be new particles that do not participate in any of the Standard Model gauge interactions, and whose fields are singlets under the Standard Model gauge group. Such fields would be sequestered in a "hidden sector" where they participate in their own gauge interactions under which the SM fields are singlets. Despite their minimal nature, hidden sector models admit a rich phenomenology; they have been well-studied in the context of collider physics [1][2][3][4][5] as well as dark matter [6][7][8][9][10][11]. In Refs. [12,13] we have pointed out that these models may also contain cosmic string solutions, called "dark strings", that have novel interactions with Standard Model fields. The aim of the present paper is to derive the radiative and scattering properties of these strings. In a subsequent paper we will use these properties to study potential astrophysical and cosmological signatures of dark strings.
The Lagrangian for the hidden sector model under consideration is of the form L = L SM + L HS + L int . (I.1) The first term, L SM , is the Standard Model (SM) Lagrangian; the second term, is the hidden sector (HS) Lagrangian with S a complex scalar field charged under a U(1) X gauge group that hasX µ as its gauge potential, D µ = ∂ µ − ig XXµ ; and the third term, is the interaction Lagrangian with Φ the Higgs doublet and Y µ the hypercharge gauge field. The mass scale of the hidden sector fields is set by the parameter σ, and η = 174 GeV is the vacuum expectation value (VEV) of the Higgs field. The two terms in L int are called the Higgs portal (HP) term [14] and the gauge-kinetic mixing (GKM) term [15,16], respectively. For σ TeV, the HP and GKM couplings are well-constrained, |α|, | sin | 1 [17,18], but if σ is above the TeV scale, making HS particles inaccessible at laboratory energies, the hidden sector model is (as yet) unconstrained. In principle the hidden sector can be extended to include additional fields and interactions; we retain only the minimal degrees of freedom necessary to study radiation of SM particles from the cosmic string.
The VEV S = σ spontaneously breaks the U(1) X completely. Consequently the model admits topological (cosmic) string solutions [19]. The string tension is set by the symmetry breaking mass scale µ ≈ σ 2 , and we will use M ≡ √ µ ≈ σ through the text. In Ref. [13] we studied these "dark string" solutions, which were found to contain a non-trivial structure in the dark sector fields, S andX µ , as well as in the SM fields, Φ and Y µ . (See also [20] for the case when sin = 0.) In the decoupling limit, σ η, the dark fields form a thin core of thickness on the order of σ −1 , and the SM fields form a wide dressing with thickness η −1 . The dressing arises because the string core sources the SM Higgs and Z boson fields, φ H and Z µ . In the limit σ η we can integrate out the heavy HS fields leaving only the zero thickness string core. In Ref. [13] we found the effective interaction of the string core with the light SM fields to be where X µ denotes the location of the zero thickness string core, and the rest of the notation is defined in Appendix A. The coupling constants g H str and g Z str have been derived in Ref. [13] in terms of α, sin , and other Lagrangian coupling constants. We shall treat them as free parameters in the present paper. Note that the interaction in Eq. (I.4) is valid for σ η, when the string core is much thinner than the SM dressing. If the core and dressing widths are comparable, the effective interaction formalism breaks down and the full field theory equations must be solved to evaluate string-particle interactions.
The linear interactions given above arise because the Higgs gets a VEV, and the string acts as a source that modifies the VEV. In addition, the string also couples to the SM fields through the more generic quadratic interactions. Upon integrating out the heavy hidden sector fields, the effective quadratic interactions for the Higgs and Z boson are The quadratic Higgs interaction derives directly from the HP term in Eq. (I. 3), and we can estimate g HH str ≈ α up to order one factors related to integrals of the profile functions. The quadratic Z boson interaction results from the mixing of the Z boson with the heavyX µ field. The mixing angle goes like (sin )(η/σ) 2 [13], and therefore we obtain the quadratic interaction in Eq. (I.5) with g ZZ str ≈ sin 2 . The W bosons will have a coupling similar to the Z boson coupling in Eq. (I.5), and our results for the Z bosons carry over to the other weak bosons as well. The remaining bosonic SM fields, the gluons and the photons, do not couple to the string worldsheet at leading order [13].
In addition to interactions with φ H and Z µ , the string also couples to the SM fermions due to an Aharonov-Bohm (AB) interaction [21]. Upon circumnavigating the string on a length scale larger than the width of the SM dressing fields, the fermion wavefunction picks up a phase that is 2π times [13] θ q = − 2 cos θ W sin g X q.
(I. 6) where q is the electromagnetic charge on the fermion, and θ W is the weak mixing angle. The AB interaction is topological, insensitive to the details of the structure of the string, and in particular, does not assume σ η.
By virtue of the interactions in Eqs. (I. 4) and (I.5), dynamical dark strings will emit Higgs and Z bosons, and it will emit SM fermions through the AB interaction. In the following sections, we calculate the spectrum of radiation in the form of Higgs and Z bosons that is emitted from cusps, kinks, and kink collisions on cosmic string loops (see Fig. 1). The scalar boson radiation channels have been derived previously [22][23][24][25]. We refine these calculations by carefully estimating all dimensionless coefficients, and in some cases also correcting errors. Most importantly, we find that the calculation of Ref. [22] underestimates the scalar radiation by a factor of √ M L 1, which arises because the radiation from the cusp is highly boosted. The vector boson channels have not been worked out previously, and we present them here for the first time. We also estimate radiation from the Aharonov-Bohm interaction by drawing on results from the literature. Our results, it should be emphasized, are not unique to the dark string model; instead, the spectra derived here apply to any model with effective interactions of the form in Eqs. (I.4) and (I.5).
Particle radiation is expected to play an important role in the evolution of light cosmic string for which gravitational radiation is suppressed. Specifically, we find that Higgs boson emission is the dominant energy loss mechanism for light dark strings. The emission of SM particles may also lead to observational signatures of dark strings through astrophysics or cosmology, and we will explore this possibility in a companion paper [26].

II. RADIATION OF STANDARD MODEL PARTICLES
The interactions in Eqs. (I.4) and (I.5) allow a dark string to emit Higgs and Z bosons, and SM fermions are radiated by virtue of the non-local Aharanov-Bohm interaction. In the subsections below we first present the spectrum of Higgs and Z boson radiation from a general string configuration, and we then specify to the cases of cusps, kinks, and kink-kink collisions as these are expected to the be the three most copious sources of particle radiation. We leave the details of these calculations to the Appendices.

A. Higgs Boson Emission via Linear Coupling
The physical Higgs field, φ H (x), couples to the dark string through the effective interaction Since this term is linear in φ H it acts as a classical source term for the Higgs field and leads to radiation from the string. Note that the dimensional prefactor, η ≈ 174 GeV, is the vacuum expectation value of the where the integral is a functional of the string worldsheet, X µ (τ, σ), that describes the motion of the string loop. The kinematical variables are defined by k µ = ω , k with ω = (m 2 H + |k| 2 ) 1/2 and m H the Higgs boson mass. In the following subsections we specify X µ so as to evaluate the spectrum and total power of Higgs boson emission from cusps, kinks, and kink-kink collisions.

Higgs Emission from a Cusp
A cusp occurs when there is a point on the worldsheet where ∂ σ X = 0. At this point the velocity of the string segment approaches the speed of light, and the radiation is highly boosted. In the rest frame of the loop, the momentum of the emitted radiation cannot exceed the inverse string thickness, i.e. |k| < M where M = √ µ, else the point-like interaction in Eq. (II.1) is inapplicable, and the radiation is suppressed.
However, due to the large boost factor, γ boost ∼ √ M L, the radiation does not cut off until |k| ≈ M √ M L Appendix E.
Inserting the scalar integral from Eq. (D.12) into the spectrum in Eq. (II.2) we find where ψ ≈ 0.01 (see Eq. (C.18)), Θ ≈ 0.1 (below Eq. (C.13)), and 0.2 S (cusp) 10 (see below Eq. (D.12)). As explained above, the spectrum is cutoff in the UV by the (boosted) string thickness, and it cuts off in the IR due to a destructive interference that is manifest in the breakdown of the saddle point approximation. Since typically m H L 1, the radiation is ultra-relativistic and we can approximate |k| ≈ ω and d|k| ≈ dω.
The radiation is emitted into a cone that has an opening angle Θ(|k|L) −1/3 . Integrating over the solid angle, we find the spectrum to be The total energy emitted from a cusp is Since we are interested in heavy strings, M m H , we can neglect the second term in the parenthesis. If cusps appear on a loop with frequency f c /T where T = L/2 is the loop oscillation period, then the average power emitted per oscillation is

Higgs Emission from a Kink
A kink occurs where there is a discontinuity in the derivative of the string worldsheet ∂ σ X. We obtain the spectrum of Higgs radiation emitted from a single kink over the course of one loop oscillation period by evaluating Eq. (II.2) with Eq. (D.14), and we find where the dimensionless coefficient is typically in the range 0.05 S (kink) 10.
Here the upper bound on k is M , rather than M √ M L as for the cusp, since the string velocity at the kink is not highly boosted in the loop's rest frame. The lower bound on k is the same as in the case of the cusp as it arises from our use of the saddle point approximation in one of the worldsheet integrals I ± (see Appendix C). Unless the loop is very small, L < M 2 /m 3 H , the lower cutoff will exceed the upper cutoff; in this case, there is no Higgs radiation from the kink within our approximations. This argument is in contrast with the calculation of Ref. [27], where scalar radiation from the kink was also studied.
Radiation is emitted into a band that has an angular width Θ(|k|L) −1/3 and angular length ∼ 2π.
Integrating over the sold angle ∆Ω ≈ 2πΘ(|k|L) −1/3 gives the spectrum The total energy emitted by the kink into this channel during one loop oscillation is Note that the energy is logarithmically sensitive to both the upper and lower cutoffs of the spectrum. If the loop carries N k kinks, then the average power radiated during one loop oscillation period, T = L/2, is given by kink) . Taking N k ≈ 1 the dimensionless prefactor is estimated to be 10 −3 Γ (kink) H 1. This result disagrees with a previous calculation [27] of Higgs radiation from a kink, as explained in Appendix D 2.

Higgs Emission from a Kink-Kink Collision
A kink-kink collision occurs when two kinks momentary overlap at the same point on the string worldsheet. We find the spectrum of Higgs radiation at the collision using Eq. (II.2) along with the scalar integral in Eq. (D.16): The radiation is emitted approximately isotropically (no beaming), and the angular integration The total energy emitted by a kink-kink collision is found to be Defining N kk as the number of kink-kink collisions during one loop oscillation period, T = L/2, we can express the average power radiated by . We can estimate the number of collisions per loop oscillation period as N kk ≈ N 2 k , where N k is the number of kinks on the loop. Estimating N kk ≈ 1 we obtain a range 10 −2 < Γ (k−k) H < 10 for the dimensionless prefactor.

B. Higgs Boson Emission via Quadratic Coupling
The radial component of the Higgs field also couples to the dark string through the quadratic interaction Unlike in the linear type coupling discussed above, this interaction is not proportional to the Higgs field VEV, and it would exist even if the electroweak symmetry were unbroken. This quadratic interaction with the string produces two Higgs bosons, and thus the final state contains two different momenta, k andk. The spectrum of radiation is given by Eq. (B.12) with C = g HH str : where k µ = ω , k with ω = (m 2 H + |k| 2 ) 1/2 and m H the Higgs boson mass. The barred quantities are defined similarly.
From the arguments above, we obtain the cusp integral to be (II.23) The upper bound on θ kk implies that k andk are approximately parallel to one another, and the upper bound on θ k+k implies that their sum points along the direction of the cusp. The geometry is such that the radiation is emitted into a pair of overlapping cones, and the angular integrations yield 24) and the spectrum becomes The total energy emitted from a cusp is given by (II. 26) If the frequency of cusp appearance is f cusp = f c /T with T = L/2 is the loop oscillation period, then the average power emitted is Scalar boson pair radiation from a cusp has been calculated previously by Ref. [22]. Our calculation matches the UV-sensitive spectrum, Eq. (II.25), of the earlier reference. In calculating the total power, we integrate up to an energy of M √ M L where 1/M is the string thickness and √ M L is the boost factor that translates between the cusp and loop rest frames (see Sec. II A 1). This boost factor was overlooked in the previous calculations, and the power was found to be O(M/L), typically a significant underestimate compared to Eq. (II.27).

Higgs-Higgs Emission from a Kink
We calculate the spectrum of Higgs boson radiation from the kink by evaluating the spectrum in Eq. (II.17) using the scalar integral in Eq. (D.14). After also generalizing the saddle point criterion, as discussed in Sec. II B 1, we obtain where 0.05 < S (kink) < 10. The momenta k andk are separated by an angle θ kk , and their sum is oriented in a band of angular with Θ(|k| +| k|) −1/3 L −1/3 . Performing the angular integrations we obtain The spectrum is UV-sensitive, which allows us to neglect the lower limit, and upon integrating we find the total energy output to be where we have used 4/[3(2 1/3 − 1)] ≈ 5 in the second term. If the loop contains N k kinks, then the average power output during one loop oscillation period (T = L/2) is given by ) . Estimating N k ≈ 1 and using the range for S (kink) given above, the dimensionless prefactor can be estimated as 10 −4 < Γ (kink) HH < 10 −1 .

Higgs-Higgs Emission from a Kink-Kink Collision
To calculate the spectrum of Higgs boson radiation from a kink-kink collision we use the scalar integral from Eq. (D.16) in the spectrum from Eq. (II.17) to obtain The radiation can be emitted isotropically; performing the angular integration gives a factor of (4π) 2 and leaves The total energy output of a kink-kink collision is calculated as If there are N kk kink-kink collisions during a loop oscillation period T = L/2 then the average power is found to be HH < 10 −1 using the range for S (k−k) given above.

C. Z-Boson Emission via Linear Coupling
The interaction allows Z bosons to be radiated from the string. The radiation calculation is carried out in Appendix B. The spectrum is given by Eq. (B.21) after replacing C = g Z str (η/σ) 2 : In this expression ω = (|k| 2 + m 2 Z ) 1/2 with m Z the Z boson mass and Π(k) is a functional of the stringworldsheet, given by Eq. (B.22). In the following subsections we calculate the spectrum and total power in Z boson emission from cusps, kinks, and kink-kink collisions.

Z Emission from a Cusp
The spectrum of Z boson emission from a cusp is calculated using Eq. (II.37) with the integral in Eq. (D.18). Combining these formulae we obtain where the dimensionless coefficient takes values 0.5 T (cusp) 50. The direction of the outgoing Z boson lies within a cone centered at the cusp and has an opening angle Θ(|k|L) −1/3 . We integrate over the solid angle to obtain the spectrum we integrate over the momentum to obtain the energy output from a single cusp and if cusps arise with a frequency f c /T where T = L/2 is the loop oscillation period, then the average power per loop oscillation is found to be

Z Emission from a Kink
To calculate the spectrum of Z boson emission from a single kink, we use the expression Eq. (II.37) along with the expression Eq. (D.20) for Π(k) for a kink to find where 0.5 < T (kink) < 100 Radiation is emitted in a band with angular width Θ(|k|L) −1/3 , and we integrate over the solid angle to find The total energy emitted by a kink during one loop oscillation is and if there are N k kinks on the loop then the average radiated power during one loop oscillation period

Z Emission from a Kink-Kink Collision
Inserting Eq. (D.23) into Eq. (II.37) we obtain the spectrum of Z boson emission from a collision of kinks to be with the constant 0.1 < T (k−k) < 50. The emission is isotropic, and after performing the angular integra- The total energy emitted by a kink-kink collision is found to be If N kk such collisions occur during one loop oscillation period, T = L/2, then the average power is

D. Z Boson Emission via Quadratic Coupling
An interaction of the form also allows Z bosons to be radiated from the string. For heavy strings, the coefficient (η/σ) 4 is very small, and this radiation channel is negligible. However, we present the calculation of the radiation spectra for completeness. The spectrum is given by Eq. (B.28) after replacing C = g ZZ str (η/σ) 4 , where k µ = ω , k and ω = (m 2 Z + |k| 2 ) 1/2 with similar definitions for the barred quantities. Note the similarity between Eq. (II.51) and the spectrum of Higgs boson pair radiation given by Eq. (II.17). Since both spectra depend on the same scalar integral, I(k +k), we can simply carry over all the results from Sec. II B. We need only to make the replacement (g HH str ) 2 → 4(g ZZ str ) 2 (η/σ) 8 .

Z-Z Emission from a Cusp
We calculate the spectrum of Z boson radiation from a cusp following Sec. II B 1. We find the the energy radiated per cusp event and the average power output if cusps arise with frequency 2f c /L The dimensionless coefficient is defined as Γ and it may be estimated as

Z-Z Emission from a Kink
We calculate the spectrum of Z boson radiation from a kink following Sec. II B 2. We find the the energy radiated per kink during one loop oscillation and the average power emitted from a loop containing N k kinks and it can be estimated as

Z-Z Emission from a Kink-Kink Collision
We calculate the spectrum of Z boson radiation from a collision of two kinks following Sec. II B 2.
We find the spectrum the energy radiated during the collision (II.59) and the average power radiated from a loop that experiences N kk collisions during one loop oscillation

E. Fermion Emission via Aharonov-Bohm Coupling
The cosmic string can radiate fermions through a direct coupling, such as the ones we have been studying for the Higgs and Z bosons, or through a non-local AB interaction. SM fermions couple directly to the string worldsheet through interactions of the form where g ψψ str is a dimensionless coupling constant, and the factor of (η/σ) 2 arises from the mixing between the Higgs field and the HS scalar field [13]. Note that dimensional analysis requires the string mass scale to appear in the denominator. The radiation calculation with S (ψ) int is very similar to the case of Higgs radiation via the quadratic interaction, see Sec. B 5. We find the spectrum of ψ radiation to be where dN HH is the spectrum of Higgs radiation, given by Eq. (II.17). Because of the mixing angle factor, (η/σ) 4 1, this radiation channel is inefficient.
The non-local AB interaction provides an additional channel for particle production from the cosmic string [21]. Refs. [28][29][30] studied the AB radiation of scalars, fermions, and vectors from a string. In these calculations, the authors assumed that the string carries only one kind of magnetic flux, which is usually the case. The structure of the dark string, however, is more complex. As we saw in Ref. [13], the string core contains flux of the HS X µ field and the dressing contains flux of the SM Z µ field. When a fermion travels around the perimeter of the string, outside of both the core and the dressing, its wavefunction picks up an AB phase due to both fluxes, and the overall phase is given by 2πθ q , where θ q is defined in Eq. (I.6).
On the other hand, when the fermion makes a loop around the core by passing through the region of space containing the dressing fields, it will acquire a different AB phase.
In order to setup the radiation calculation we must know the effective AB interaction of the fermions with the string. The discussion above is intended to illustrate that this interaction will be scale dependent. At energies below the inverse dressing width, ∼ 1/η, the core plus dressing can be treated together as a zero width string. In this limit the structure of the string is unimportant, and the AB interaction can be derived following Refs. [28][29][30] with the AB phase given by θ q . At higher energies the Compton wavelength of the radiation drops below the dressing thickness. Here the effective coupling will presumably decrease as the particle "sees" less and less of the flux carried by the dressing. This behavior is in contrast with the Higgs and Z boson radiation channels that we considered previously. In those cases, the light SM fields coupled directly to the string core itself, and the dressing was neglected.
In light of the discussion above, we will proceed as follows. We calculate the spectrum of radiation due to the AB interaction where the coupling is set by the AB phase θ q . If the thickness of the string dressing is ∼ 1/η, then this spectrum is valid up to energies |k| ≈ η √ ηL for the cusp or |k| ≈ η for the kink and kink collision. At higher energies, we suppose that the effective coupling begins to decrease as the fermion radiation begins to penetrate inside of the dressing, and consequently the spectrum drops sharply.
The AB interaction can be treated perturbatively as follows. Let V µ (x) be the appropriate linear combination of the X µ and Z µ gauge fields that couples to the fermions, and let g ψ be the coupling constant.
Then the interaction is given by We treat V µ as a classical background field induced by the flux that the string carries: Φ = (2π/g ψ )θ q . This lets us write (Lorentz gauge, where the integration contour extends above the poles at p 0 = ± |p|, as in the calculation of a retarded where Π is given by Eq. (D.5).

Fermion AB Emission from a Cusp
We find the spectrum of radiation from a cusp by inserting Eq. (D.18) into Eq. (II.65): We calculate the total energy output as and the average power output per loop oscillation as . Using the range for T (cusp) given above, we can estimate 10 −5 Γ (cusp) AB 10 −2 .

Fermion AB Emission from a Kink
We find the spectrum of radiation from a kink by inserting Eq. (D.20) into Eq. (II.65): Recall that k +k is oriented in a ribbon with angular width θ k+k , and the opening angle between k andk does not exceed θ kk . After performing the angular integrations we obtain d|k|d| k| (|k| +| k|) 7/3 L 1/3 , ψ m ψ m ψ L < |k|,| k| < η . (II.71) We calculate the total energy output as and the average power output from N k kinks during one loop oscillation period (T = L/2) as Using the range for T (kink) given above along with N k ≈ 1, we can estimate 10 −2 Γ (kink) AB 1.

Fermion AB Emission from a Kink-Kink Collision
We find the spectrum radiation from a kink collision by inserting Eq. (D.23) into Eq. (II.65): with 0.1 < T (k−k) < 50. In this case, the emission is isotropic, and we can estimate (k +k) 2 ≈ 2ωω up to an O(1) factor associated with the angle between k andk. The angular integration is trivial, and we find (II.75) We calculate the total energy radiated as and the average power emitted from a loop which experiences N kk collisions during a loop oscillation . Using the parameter ranges given above along with N kk ≈ 1, we can estimate 10 −6 < Γ

III. SCATTERING CROSS SECTIONS
The interactions discussed in Sec. I allow SM particles to scatter off of the dark string. Interactions of the Higgs and Z bosons with the string, given by Eqs. (I.4) and (I.5), will lead to a "hard core" scattering, and the AB phases of the SM fermions, given by Eq. (I.6), will lead to a non-local AB scattering. If the couplings are comparable for the direct and the AB interactions, then the latter generally dominates [19], and therefore we focus on AB scattering here. Moreover, in the cosmological context the dark string will scatter from the SM plasma, which consists mostly of electrons and nuclei at late times.
The AB interaction allows fermions to scatter from a cosmic string. The scattering cross section (per length of string) was found to be [21] dσ AB dθ = sin 2 (πθ q ) 2πk ⊥ sin 2 (θ/2) (III.1) where the AB phase for SM fermions, θ q , is given in Eq. (I.6), and k ⊥ is the magnitude of the momentum transverse to the string. Inserting the expression for θ q and expanding in the θ q 1 limit gives where q is the electromagnetic charge of the fermion.
To study the motion of strings through the cosmological medium, we are interested in the drag (momentum transfer) experienced by the string. This is calculated in terms of a "transport cross section" (see [19]) given by To obtain the total drag due to the entire medium, we must sum over the various species with their respective charges q.
The derivation of the AB phase, given by Eq. (I.6), assumed that the particle circumnavigates the string on a length scale larger than the width of the SM dressing. In this way, the particle trajectory encloses both the flux carried by the thin HS string core and the thick SM dressing. This length scale is microscopic, ∆x ∼ η −1 ≈ 10 −16 cm, and therefore this assumption is well-justified for the cosmological medium at late times, where the inter-particle spacing is much larger than ∆x.
for Higgs emission via a quadratic coupling for Z boson emission via a quadratic coupling and for fermion emission via the AB interaction  3. The string loop also radiates gravitational waves from cusps, kinks, and kink collisions. The power output into this channel is well-known: P grav = Γ g GM 4 where µ = M 2 is the string tension, Γ g ≈ 100, and G is Newton's constant [19]. For comparison, P (cusp) HH ∼ M 3/2 /L 1/2 . If the string mass scale is large, then string loops will primarily radiate in the form of gravitational waves, as originally observed by Ref. [22]. However, it is important to emphasize that particle emission will dominate if the scale of symmetry breaking is low, e.g., for a TeV scale string. For instance, taking L ≈ 40 Gly to be the size of the horizon today we find P cusp HH /P grav ≈ 10 4 (M/ TeV) −5/2 . Moreover, in general Higgs emission dominates over gravitational emission for small loops: L < Throughout this analysis we have assumed that the light SM fields are coupled to the zero thickness dark string core, which is composed of the heavy HS fields. As we found in Ref. [13], the dark string has a much richer structure: the thin core is surrounded by a wide dressing made up of the SM Higgs and Z boson fields. The presence of this dressing could lead to a backreaction that was neglected in our particle production calculations, and this deserves further investigation. Additionally, as with most calculations of radiation from cosmic strings, we neglect the more familiar backreaction effect: a reduction in radiation power as cusps and kinks are gradually smoothed as a result of energy loss in the form of particle and gravitational radiation [31,32].

Comparing Higgs emission from a cuspy loop via the linear and quadratic interactions, we find
The particle production calculations that we have presented here play a central role in the study of astrophysical and cosmological signatures of cosmic strings. For instance, Higgs bosons emitted from the string at late times will decay and produce cosmic rays that are potentially observable on Earth [24]. In our followup paper [26], we will study the cosmological evolution of the network of dark strings and assess the prospects for their detection.

ACKNOWLEDGMENTS
We are very grateful to Yang Bai, Daniel Chung, Danièle Steer, and especially Eray Sabancilar for discussions. This work was supported by the Department of Energy at ASU.

Appendix A: Worldsheet Formalism
In this appendix we review the string worldsheet formalism (see, e.g., [19]). Let τ and σ be the time-like and space-like worldsheet coordinates, and let X µ (τ, σ) be the string worldsheet. Then d 2 σ = dτ dσ is the worldsheet volume element and dσ µν = dτ dσ µναβ ab ∂ a X α ∂ b X β is the worldsheet area element. Repeated Greek indices are summed from 0 to 4 and Latin indices from 0 to 1 with ∂ 0 X µ = ∂ τ X µ =Ẋ µ and ∂ 1 X µ = ∂ σ X µ = X µ . We define the pullback of the metric as We now specify to the conformal gauge by imposinġ Then we have where in the last equality we have used the fact thatẊ µ is not spacelike. We are also free to choose τ = t.
Solutions of the equation of motion for a free string,Ẍ = X , can be written as where we have introduced the right-and left-movers, a µ (σ − ) and b µ (σ + ), which are functions of σ ± ≡ (σ ± t). For regularly oscillating string loops, these functions obey the periodicity conditions in the center of mass frame of the loop. The derivatives arė We can use the residual gauge freedom to choose along with the condition that a and b should be null, which implies This parametrization lets us write Note that a · b = −1 − a · b ≤ 0 and therefore √ −γ ≥ 0 as it should be.

Appendix B: Calculation of Particle Radiation from the String
In this appendix, we calculate the spectrum of scalar and vector boson emission due to a coupling with a cosmic string of the linear or quadratic form. We also derive the spectrum of fermions emitted due to a direct coupling and an Aharonov-Bohm coupling. The results we obtain are not unique to the dark string model; they apply to any model that has couplings of the form considered here.
We use the matrix element formalism to perform these calculations [22]. Since the linear coupling gives rise to a classical source for the scalar or vector field, the radiation in these cases can also be calculated by solving the classical field equation [24,25]. We have verified that both approaches give identical spectra.
We also retain all factors of 2 and π, which were neglected in the previous calculations.

Scalar Radiation via Linear Coupling
Consider a real scalar field φ(x) of mass m that is coupled to the string worldsheet X µ (τ, σ) through the effective interaction where A is an arbitrary real parameter with mass dimension one. We calculate the amplitude for particle production by making a perturbative expansion in A. Then to leading order we have 1 where k is a one-particle state of momentum k. The action of the field operator on the one-particle state is simply where k µ = ω, k with ω = m 2 + |k| 2 . Then upon inserting Eq. (B.1) into Eq. (B.2) we obtain In Appendix D we calculate this integral for various string configurations, as specified by X µ (τ, σ).
For a given X we calculate the number of scalar bosons emitted into a phase space volume d 3 k = |k| 2 d|k| dΩ as Using Eq. (B.4) in Eq. (B.6) we obtain the final spectrum where the dimensionful coefficient is equal to A = g H str η for the dark string.

Scalar Radiation via Quadratic Coupling
Consider a real scalar field φ(x) of mass m that is coupled to the string worldsheet X µ (τ, σ) through the effective interaction where C is an arbitrary real parameter with mass dimension zero. We can calculate the radiation of scalar boson pairs using perturbation theory provided that C 1. Consider the radiation of a boson pair with momenta k andk. We can introduce the 4-vectors k µ = ω = √ k 2 + m 2 , k andk µ = ω = k2 + m 2 ,k . To leading order in C the amplitude for this process is Using Eq. (B.10) this becomes where C = g HH str for the dark string.

Vector Radiation via Linear Coupling
Consider a vector field A µ (x) of mass m that couples to the string worldsheet X µ (τ, σ), via the linear interaction where F µν = ∂ µ A ν − ∂ ν A µ is the field strength tensor and C is a real parameter of mass dimension zero.
Recall that the worldsheet area element was defined in Eq. (A.2). Since the radiation will be relativistic, we can treat the gauge boson as transversely polarized with two allowed helicties λ = ±1. We calculate the amplitude to radiate a vector boson with momentum k and helicity λ as 14) The action of the field operator on the one-particle state is where k µ = ω = m 2 + |k| 2 , k . Inserting Eq. (B.13) into Eq. (B.14) and using Eq. (B.15) gives Then the number of vector bosons emitted into the phase space volume d 3 k = |k| 2 d|k| dΩ is calculated as where we sum over the two polarization states. Using Eq. (B.16) this becomes We perform the spin sum using the completeness relationship Doing so we find the spectrum to be where the (positive, real) function has dimensions of length 4 and carries the dependence on the string worldsheet. By choosing C = g Z str (η/σ) 2 we obtain the spectrum of Z boson radiation from the dark string.

Vector Radiation via Quadratic Coupling
Consider a vector field A µ of mass m that couples to the string worldsheet via the quadratic inter- where C is a real parameter of mass dimension zero. The amplitude to radiate a pair of vector bosons with momenta k andk and helicities λ andλ is calculated as We can introduce the 4-vectors k µ = ω = √ k 2 + m 2 , k andk µ = ω = k2 + m 2 ,k . Upon inserting Eq. (B.23) into Eq. (B.24) and using Eq. (B.15) we obtain where I(k) was defined in Eq. (B.5). Then the number of vector bosons emitted into the phase space volume where we sum over the transverse polarization states λ,λ = ±1. (Since the radiation is highly boosted, we can neglect the longitudinal polarization states.) Using Eq. (B.25) this becomes We evaluate the spin sums using the completeness relation in Eq. (B.20) to find For the dark string model we take C = g ZZ str (η/σ) 4 .

Dirac Spinor Radiation -Direct Coupling
Consider a Dirac field Ψ(x) of mass m that is coupled to the string worldsheet X µ (τ, σ) through the effective interaction where C is an arbitrary real parameter with mass dimension zero, and M is the string mass scale. Consider the radiation of a particle / anti-particle pair with momenta k andk and spins s ands. We can introduce the 4-vectors k µ = ω = √ k 2 + m 2 , k andk µ = ω = k2 + m 2 ,k . To leading order the amplitude for this process is The action of the field operator on the one-particle state is given by Using the familiar Dirac gamma trace relations, we obtain where C = g ψψ str (η/σ) 2 for the dark string.

Dirac Spinor Radiation -AB Coupling
Consider a Dirac field Ψ(x) of mass m that is coupled to the string worldsheet X µ (τ, σ) through the effective interaction where C 1 is an arbitrary real parameter with mass dimension zero, and I µν (k) was defined in Eq. (B.17). In the momentum integral, the integration contour is extended above both poles at p 0 = ± |p|.
Following Sec. B 5 we calculate the amplitude for the radiation of a particle/anti-particle pair: where q ≡ k +k. The number of particle pairs emitted into the phase space volume d 3 k d 3k = |k| 2 d|k| dΩ| k| 2 d| k| dΩ is calculated as in Eq. (B.26) where now the sum is over spin states s,s = ±1/2.
Using Eq. (B.36) we obtain Then using the antisymmetry of I µν we find where Π(q) is defined in Eq. (B.22) and In general, the evaluation of Eq. (B.38) is very involved and must be done numerically for some choice of loops as in [29]. However, to extract the radiation spectrum, it is sufficient to note that dN > 0, and so the term containing Υ is never larger than the term containing Π [see also Eq. (D. 10)]. Hence, to extract scalings, we will take 2 where for the dark string C = −(2πθ q )/2.

Appendix C: Calculation of the Worldsheet Integrals
In Appendix B we encountered the two integrals while calculating the radiation spectra. In this appendix and the next, we will analytically calculate these integrals for the cusp, kink, and kink-kink collision string configurations.
It is convenient to define the integrals There is a danger that there can be cancellations between the Π and |Υ| 2 terms but we find that our scalings agree with the behavior that was numerically obtained in [29] for similar loops.
where I + is a functional of b µ (σ + ) with parameter k µ , and similarly I − is a functional of a µ (σ − ).
The problem is now reduced to calculating the two integrals, I µ + and I µ − , for a given loop configuration, specified by a µ and b µ .
These integrals cannot be performed analytically for general configurations. We, therefore, focus on the configurations that we expect to maximize the integrals, since this corresponds to maximum particle radiation. It turns out that for these optimum configurations, the saddle point and the discontinuity, the integrals are analytically tractable.

Saddle Point Integral
The integrals in Eq. (C.3) become analytically tractable if there is a saddle point at which the phase is stationary [33]. For the sake of discussion consider the integral I + . Its phase can be expanded about σ + = σ s as Subscripts are used to denote evaluation of the function at a particular point, e.g., b s = b (σ s ). We say that σ s is a saddle point if the stationary phase criterion, is satisfied. Using Eq. (A.7) this can be written as where θ is the angle between k and b s . If the particle being radiated is massless, ω = |k|, then the saddle point criterion is satisfied by choosing k = |k| b s (i.e., θ = 0). Then it follows from the identity in Eq. (A.8) that k · b s = 0 as well, and the leading term in Eq. (C.7) is cubic.
For massive particle radiation the saddle point criterion cannot be satisfied exactly. Instead, we have instead a quasi-saddle point, σ + = σ qsp , at which the phase is approximately stationary: where we have used Eq. (A.8). It will be convenient to write where the hatted quantities are unit vectors. The dimensionless parameters α asp and β asp are related to the acceleration or curvature of the loop at the quasi-saddle point (recall Eq. (A.6)). The stationary phase approximation is still applicable as long as Suppose that we are given a configuration b µ (σ + ) and a k µ such that there exists some point can be approximated by expanding in ∆σ = σ + − σ s , which gives The phase is also expanded in powers of ∆σ/L as φ(∆σ) = φ 1 (∆σ) + φ 3 (∆σ) + . . . where Here we have introduced the dimensionless parameter Θ ≡ (6/πβ 2 s ) 1/3 , and the shape parameter is β s = L |b s | /(2π) as per Eq. (C.11).
As long as φ 1 is negligible, the integral is in the stationary phase regime, and it can be evaluated directly with the saddle point approximation. Since the integral is dominated by the saddle point, we can extend the limits of integration to infinity. Doing so we obtain where 15) and Θ = (6/πβ 2 s ) 1/3 was defined in the paragraph above.
The linear phase, φ 1 , must be negligible if the saddle point approximation is to be valid. We define the "width of the saddle point" by the condition φ 3 (∆σ max ) = 2π, which gives The left-hand side vanishes in the relativistic limit, and the bound becomes saturated as the momentum is lowered. Approximating ω ≈ |k| + m 2 /2|k| we obtain a lower bound on the momentum [22] ψ m We can also translate ∆σ max into an upper bound on the angle between k and b s : For the I − integral, the analysis is similar, but the saddle point criterion is replaced with k · a s = −ω − k · a s = 0 implying that k = −|k| a s at the quasi-saddle point. Consequently, in the equations analogous to Eq. (C.10) all the signs on the right hand side are flipped. The results for both integrals can be summarized as where θ kb s and θ ka s are the angles between k and b s or a s , respectively. The dimensionless parameters are defined as terms.

Discontinuity Integral
In this appendix we will evaluate the integrals in Eq. (C.3) for the case in which the gradient of the string worldsheet, ∂ σ X µ , has a discontinuity [33]. We first suppose that b µ (σ + ) has N k discontinuities corresponding to N k kinks on the string loop. The typical distance between the kinks will be D = L/N k .
To calculate the contribution to I + coming from a single discontinuity located at σ + = σ d we parametrize wherem ± are unit vectors and b ± = σ + , (σ + − σ d )m ± . Inserting Eq. (C.22) into Eq. (C.3) we approximate the worldsheet integral as wherek µ ≡ k µ /ω = 1 , k/ω . Upon integrating over the entire loop, the second term cancels among the contributions from different discontinuities (summing all kinks). Then we can drop both this second term and the overall phase to write the contribution from a single discontinuity as To calculate the integral I − we parametrize a (σ − ) in terms of a ± in analogy with Eq. (C.22). We can summarize the results of both calculations as follows The dimensionless coefficients are bounded as 1 ≤ β ± , α ± . In the limit that k coincides with one of the discontinuity vectors, b ± or a ± , one finds that β ± or α ± → ∞. This apparent divergence is an artifact of neglecting the second set of terms in Eq. (C.23), and upon retaining these terms one can see that I + ∼ D 1/ω in the limit that (k · b + )ωD 1. Therefore we must restrict ourselves to the regime ω > D −1 ∼ N k L −1 and where k · b ± is away from zero; it follows that 1 ≤ β ± , α ± few. To properly treat the case k · b + = 0 in which the phase is stationary, one should use the saddle point approximation, as described in Sec. C 1.

Appendix D: Scalar and Tensor Integrals for Cusps, Kinks, and Kink Collisions
Here we evaluate the scalar and tensor integrals, I and I µν given by Eqs. (C.5) and (C.6), for the cusp, kink, and kink-kink collision string configurations. For the scalar integral, we will only be interested in the modulus |I| 2 . For the tensor integral, we will only be interested in the (positive, real) scalar combinations Π(q) = −g νβ q µ q α q 2 I µν (q)I αβ (q) * (D.1) which were originally defined in Eqs. (B.22) and (B.39).
We can simply the expression for Π(q) as follows. Using Eq. (C.6) and the identity (−g νβ ) µνγδ αβρσ = g µα g γρ g δσ + g µρ g γσ g δα + g µσ g γα g δρ − g µα g γσ g δρ − g µρ g γα g δσ − g µσ g γρ g δα Making this simplification we finally obtain We can simplify the expression for Υ(k,k) as follows. Let q = k +k and p = k −k. Using Eq. (C.6) we can express Υ as Using the identities q · p = q · I + = q · I − = 0, this can also be written as [29] Υ(k,k) = 1 2q 0 p · (I + × I − ) . (D.7) To compare Υ with Π, it is convenient to move to the frame in which q µ = q 0 , 0 . Then the identities q · p = q · I + = q · I − = 0 require p, I + , and I − to have vanishing time-like components. In this frame, we can write where θ +− is the angle between I + and I − . Further denoting θ p+− as the angle between p and I + × I − we have The two expressions are related by where the quantity is square brackets is always ≤ 1. The inequality is saturated when p is aligned with I + × I − (i.e., θ p+− ≈ 0) and either |k| | k| or| k| |k|. Then, the squared integral evaluates to

Scalar Integral -Cusp
where S (cusp) ≡ π 4 α 2 c β 2 c B 2 − B 2 + cos 2 θ ab with B ± defined in Eq. (C.21), and where θ ab is the angle between a c and b c . The angle between k and b c = −a c is bounded above by the saddle point criterion, and therefore k falls within a cone of opening angle Θ(|q|L) −1/3 centered on the cusp.
The dimensionless prefactor, S (cusp) , may be estimated using the expressions for B ± in Eq. (C.21).
The shape parameters, α c and β c , are expected to be O(1), but their precise values cannot be determined without greater knowledge of the nature of the cusp. In order to track how this uncertainty in the magnitude of the shape parameter feeds into the particle production calculation, we will consider a fiducial range of values for α c and β c . Estimating 1/5 α c , β c 5 and cos θ ab ≈ 1 we find 0.2 S (cusp) 10. The dimensionless parameters ψ and Θ, given by Eqs. (C. 18) and (C. 19), are less sensitive to the uncertainty in the shape parameters. Typically ψ ≈ 0.01 and Θ ≈ 0.1.

Scalar Integral -Kink
A kink occurs when the derivative of one of the functions b µ (σ + ) or a µ (σ − ) appearing in I + (b; q) or I − (a; q) has a discontinuity, and the other integral has a saddle point. For the sake of discussion we suppose that I + contains the saddle point and I − the discontinuity. We calculate I by inserting Eqs compared to the terms that we keep. 3 The calculation described above yields I (kink) (q) = −i B + 4 L 4/3 |q| 5/3 α + (b s · a + ) − α − (b s · a − ) (D.13) Here we have used q 0 ≈ |q| since the saddle point condition requires m √ mL < |q| and mL 1 for typical size loops. For the same reason, the bound on the discontinuity integral, L −1 < |q|, is subsumed by the bound on the saddle point integral, m √ mL < |q|. The squared integral becomes I (kink) (q) 2 = S (kink) L 2/3 |q| 10/3 , ψ m √ mL < |q| , θ < Θ (|q|L) −1/3 (band) (D.14) where S (kink) ≡ (2π) 2 16 B 2 + β 2 s α + (b s · a + ) − α − (b s · a − ) 2 and we have used the shape parameters, introduced in Eq. (C.11). The saddle point criterion requires q to be aligned with b s . Consequently, the radiation is emitted into a band (whose orientation is determined by b s (σ + )) of angular width Θ(|q|L) −1/3 and angular length ∼ 2π.
We can estimate a range of uncertainty for S (kink) as we did in Appendix D 1. Recall that B + was given by Eq. (C.21). Following the convention established in Appendix D 1, we estimate the shape parameter as 1/5 β s 5. We also take 1 α ± 5, as per the discussion below Eq. (C.25). Together this lets us estimate 0.1 S (kink) 20

Scalar Integral -Kink Collision
For the case of a kink-kink collision both integrals, I + and I − , have discontinuities and are given by Eq. (C.25). The scalar integral is evaluated from Eq. (C.5) to be 15) where ω = q 0 . The square is We have defined S (k−k) ≡ 1 16 ±(1 + b ± · a ± )β ± α ± 2 where the sum runs over all possible combinations of + and − as given by Eq. (D. 15). For the case of a discontinuity, the worldsheet integrals, I ± , are insensitive to the orientation of k (see Sec. C 2) and the corresponding radiation is emitted approximately isotropically.

Tensor Integral -Cusp
If both I ± contain a saddle point, then we evaluate the tensor integral by inserting Eq. where T (cusp) ≡ (2π) 4 4 (B + B − ) 2 α 2 c β 2 c sin 2 θ ab and θ ab is the angle between a c and b c . The angle between q and b c = −a c is bounded above by Θ(|q|L) −1/3 , and consequently q is oriented within a cone centered at the cusp.
We can estimate the dimensionless coefficient by making the same estimates as in Appendix D 1.
Following the conventions from the previous sections, we estimate 1/5 β s 5 and determine B + from Eq. (C.21). We estimate the parenthetical factor as simply |α + α − | and take 1 α ± 5 as before.