2017/11/27-12/01 JGRG27@Hiroshima

Y-junction intercommutationsof current carrying strings
D.Steer, M.Lilley, DY, T.Hiramatsu, arXiv:1710.07475

Daisuke Yamauchi
Kanagawa UniversityCosmic stringnetwork➢ Evolves according to scaling.
➢ Eventually loses its energy
through loop formation due topartner exchange :reconnection.
➢ Although the reconnection
almost always happens,
strings with Y-typed junctions
occurs in many models!
Hiramatsu+Sendouda+Takahashi+DY+Yoo, 1307.0308

Y-junction formation?
When two strings collide, bound states andY-junctions would be produced.

• Nearly parallel• Low relative velocity

[Copeland+Kibble+Steer(2006)(2007), Salmi+Achucarro+Copeland+de Putter+Steer(2008)]

Y-junction formation : condition
tension : μ3 tension : μ3αIntercommutation

tension : μ1

Y-junction
formation[ 2μ3≧μ1 is needed. ][Copeland+Kibble+Steer(2006),(2007)]

Strategy [ Nambu-Goto case ]① Consider effective action

Choose gauge conditions

Junction positionat σ = sj(τ)② Extend ① to include multiple strings with junctions③ Solve junction conditions

4-dim embeddingof a junctionQuestions we are interested in here are:• What happens to the currents when
two current-carrying strings collide?
• Can junctions actually form?

We would like to look for the condition for the junction
formation in the analytic way (if possible).

Differences between Nambu-Gotoand current-carrying strings
Conformal and temporal gauges➢ Worldsheet gauge choices generally made to study NG
DO NOT apply to general elastic strings
We need to develop a fully covariant formalism.
➢ Equations of motion are generally NOT integrable[exception : chiral strings]
We need to solve junction conditions for string
position and current simultaneously.

Step 1 : effective action➢ Lagrangian depends on its internal degree of freedom φ :[Carter(1989a,b)]✓ L is constant  Nambu-Goto strings✓ L is a function of state parameter w  superconducting strings[Witten(1985)][For superconducting strings, φ characterizes
a phase of scalar field living on a string.]

Energy-momentum tensor➢ Nambu-Goto strings

Conformal and temporal gauges

Tμν = U uμuν – T vμvν

U=Tuμ ∝ xμ,τ[For simplicity, w>0]

Energy-momentum tensor➢ Current-carrying strings

Conformal and temporal gauges

Tμν = U uμuν – T vμvν

U=Tuμ ∝ φ,a xμ,a

✓ NOTICE : One can use the residual freedom of Lorentz
rotation on the worldsheet!

φ(τ,σ)  φ(σ)
Preferred rest frame

[Nambu-Goto case：Copeland+Kibble+Steer(2006)(2007)]

Step 2 : Effective action with a junction
Junction positionat σ = sj(τ)

Current conservation
at a junction

4-dim embeddingof a junction
String 2
String 3

String 1

[Carter+Steer+Lilley+DY+Hiramatsu in progress]

Step 3 : Covariant form of junction conditions

Energy-momentum

Current-carrying strings

Outward-directed
unit normal vector[w>0]
Current

Nambu-Goto strings[w<0]

2 more constraintequations!

Step 4 : Apply to string collision
Consider a string collision between 2 incoming and identicalstrings at angle ±α with velocity ±vz :kinks

junctionsStep 4.1 : Nambu-Goto case
➢ Junction conditions➢ Unknowns

All unknowns can be determined by the junction conditions!
The solution makes sense only if s1>0 :the connecting string cannot get shorter.

Step 4.2 : Current-carrying case
➢ Junction conditions➢ Unknowns

2 more constraint eqs!
In the presence of the current, s3=s2 is determined
NOT by the junction conditions but by the configuration.

Step 4.2 : Current-carrying case
➢ Junction conditions

2 more constraint eqs!

(# of eqs to solve)➢ Unknowns>(# of unknowns)

The system is OVERDETERMINED!

Summary➢ We have extended the analysis of the subsequent formationof Y-junction to the elastic models characterizing currentcarrying strings.
➢ There are big differences between NG and elastic strings:✓ Gauge issues✓ Presence of internal DoF

In the case of the string collision, the joining string
can NOT be described by the elastic model.

Future prospects➢ The treatment of such a collision may generally require
the use of a non-conservative model.
➢ It is of great interest to compare the results from thenumerical investigation of a collision of current-carrying
strings.

Thank you!