Theoretical investigation of possibility to suppress FSR in specific - PowerPoint PPT Presentation

Theoretical investigation of possibility to suppress FSR in specific dark matter models explaining cosmic positron data Airat Kamaletdinov, Ekaterina Shlepkina, Konstantin Belotsky NRNU MEPhI 10 July 2020 1 / 23

Positron Anomaly Positron Anomaly 2 / 23

Positron Anomaly Contradiction with IGRB 3 / 23

Interaction vertex parametrization Interaction vertex parametrization We started by choosing the simplest decay vertices: L “ X ¯ Ψ p a ` b γ 5 q Ψ L “ X µ ¯ Ψ γ µ p a ` b γ 5 q Ψ and Two-body decay Three-body decay Suppression of the photon yield is achieved by σ p X Ñ e ´ e ` γ q σ p X Ñ e ´ e ` q Ñ min where a and b are fixed parameters. 4 / 23

Interaction vertex parametrization Decay into identical positrons Double charged Dark Matter particles model was also considered L C “ X ψ C p a ` b γ 5 q ψ ` X ˚ ψ p a ` b γ 5 q ψ C . X ˚ Ñ e ´ e ´ X Ñ e ` e ` We assume that there are no particles X ˚ in the DM sector. Similar models of heavy double charged DM particles are proposed, for example, in arXiv:1411.365 and arXiv:astro-ph/0511789 5 / 23

Interaction vertex parametrization Independence of photon yield on model parameters For X Ñ e ´ e ` p γ q L “ X ¯ Ψ p a ` b γ 5 q Ψ L “ X µ ¯ Ψ γ µ p a ` b γ 5 q Ψ 2 p a 2 ` b 2 q m 2 4 p a 2 ` b 2 q m 2 |M| 2 2 body X X p a 2 ` b 2 q F p k 1 , k 2 , l q (a 2 ` b 2 q G p k 1 , k 2 , l q |M| 2 p 3 body q σ p e ´ e ` γ q F p k 1 , k 2 , l q G p k 1 , k 2 , l q σ p e ´ e ` q 2 m 2 4 m 2 X X For X Ñ e ` e ` p γ q L “ X Ψ C ˆ O Ψ ` X ˚ Ψ ˆ | in y ” ˆ O Ψ C X | 0 y L “ X ¯ L “ X µ ¯ Ψ C p a ` b γ 5 q Ψ Ψ C γ µ p a ` b γ 5 q Ψ 8 p a 2 ` b 2 q m 2 |M| 2 16 b 2 m 2 p 2 body q X X p a 2 ` b 2 q F p k 1 , k 2 , l q |M| 2 b 2 G p k 1 , k 2 , l q p 3 body q σ p e ` e ` γ q F p k 1 , k 2 , l q G p k 1 , k 2 , l q σ p e ` e ` q 8 m 2 16 m 2 X X 6 / 23

Interaction vertex parametrization Difference of scalar coupling L “ X Ψ p a ` b γ 5 q Ψ in comparison with vector one L “ X µ Ψ γ µ p a ` b γ 5 q Ψ E ` “ ´ e 2 p m 2 ´ 2 m ω ` 2 ω 2 q log p| m ´ 2 E 1 ˇ m ´ 2 p E 1 ` ω q |q ˇ B σ p e ´ e ` γ q{B ω 1 ˇ ˇ ˇ ˇ σ p e ´ e ` q ˇ 4 π 2 m 2 ω ˇ ˇ ˇ E ´ scalar 1 E ` “ ´ e 2 p m 2 ´ 2 m ω ` 2 ω 2 q log p| m ´ 2 E 1 ˇ m ´ 2 p E 1 ` ω q |q ´ 4 E 1 ω ˇ B σ p e ´ e ` γ q{B ω 1 ˇ ˇ ˇ ˇ σ p e ´ e ` q 4 π 2 m 2 ω ˇ ˇ ˇ ˇ E ´ vector 1 7 / 23

Derivative in the interaction vertex Derivative in the interaction vertex Class of interaction vertices which depend on the decaying particle momentum was considered. L “ Ψ γ µ p a ` b γ ν B ν L “ Ψ γ µ p a ` b p γ ν B ν qp γ ρ B ρ q ... q X µ Ψ q X µ Ψ m m n L “ Ψ γ µ p a γ 5 ` b p γ ν B ν q q X µ Ψ ... m Such approach makes it possible to achieve an effect on the photon yield by the parametrization of interaction Lagrangian. p 1 ` � � a ` b ˆ p 2 ˆ a ` b ˆ p 1 u p p 1 q γ µ ´ ¯ u p p 1 q γ µ ´ ¯ p X Ñ e ` e ´ q ñ ¯ v p p 2 q “ ¯ v p p 2 q m m p 2 ` ˆ p 2 ` ˆ a ` b ˆ p 1 ` ˆ l ¯„ ˆ l  u p p 1 q γ µ ´ p X Ñ e ` e ´ γ q ñ ¯ ǫ p l q v p p 2 q p p 2 ` l q 2 ˆ m 8 / 23

Derivative in the interaction vertex For example, for vertex L “ Ψ γ µ p a ` b p γ ν B ν q q X µ Ψ : m E ` “ ´ e 2 p 2 a 2 ` b 2 q m p m 2 ´ 2 m ω ` 2 ω 2 q log p| m ´ 2 E 1 m ´ 2 p E 1 ` ω q |q ´ 8 E 1 ω p a 2 m ` 2 b 2 ω q ˇ B Br p e ` e ´ γ q 1 ˇ ˇ 4 π 2 m 3 ω p 2 a 2 ` b 2 q B ω ˇ E ´ ˇ 1 However, this vertex does not lead to a significant result. Moreover, the class of such vertices is bounded and their extension to arbitrary polynomials f p ˆ p q is impossible since: p ” p 2 “ m 2 p ˆ ˆ p q “ a ` b ˆ m ` c ˆ p m 2 ` d ˆ p ˆ p p ˆ m 3 ` ... ` A γ 5 ` B γ 5 ˆ p ˆ p m ` C γ 5 ˆ p m 2 ` D γ 5 ˆ p ˆ p p ˆ p ˆ p f p ˆ m 3 ` ... “ “ a ` b ˆ m ` c ` d ˆ p m ` ... ` A γ 5 ` B γ 5 ˆ p m ` C γ 5 ` D γ 5 ˆ p p m “ “ p a ` c ` ... q ` p b ` d ` ... q ˆ m ` p A ` C ` ... q γ 5 ` p B ` D ` ... q γ 5 ˆ p p m Thus f p ˆ p q can only be a linear function of ˆ p . 9 / 23

Consideration of loop contributions Consideration of loop contributions The dependences of the coefficients a and b on the decay energies can also be achieved by considering the loop processes. a Ñ F 1 p? s q , b Ñ F 2 p? s q The following processes were considered Corresponding interaction Lagrangians of such models are follows: L � “ X ¯ θ p a ` i b γ 5 q θ ` η ¯ θ p c ` i d γ 5 q θ ` η ¯ ΨΨ θ p c ` i d γ 5 q Ψ ` η ˚ ¯ L △ “ X ¯ θ p a ` i b γ 5 q θ ` η ¯ Ψ p c ` i d γ 5 q θ 10 / 23

Consideration of loop contributions The Passarino and Veltman reduction procedure was used for one-loop integrals, described in detail in https://arxiv.org/abs/1105.4319 . such procedure This procedure consists in reducing single-loop integrals to a linear combination of standard scalar integrals: d D q ż 1 A 0 p m q “ p q 2 ´ m 2 q p 2 π q D d D q ż 1 B 0 p p ; m 1 , m 2 q “ p q 2 ´ m 2 1 qpp q ` p q 2 ´ m 2 p 2 π q D 2 q d D q 1 ż C 0 p p 1 , p 2 ; m 1 , m 2 , m 3 q “ p 2 π q D d 1 d 2 d 3 i ´ 1 ´ p k q 2 ´ m 2 ¯ ÿ d i ” p q ` i k “ 1 11 / 23

Consideration of loop contributions Bubble vertex The functions A 0 , B 0 , C 0 depend quadratically on their arguments. Hence the loop contribution to p X Ñ e ` e ´ q and p X Ñ e ` e ´ γ q turns out to be the same. ´ ¯ p a ` ib γ 5 qp ˆ q ` m qp c ` id γ 5 qp ˆ Tr q ` ˆ p 1 ` m q d D q ż ˆ O “ “ p q 2 ´ m 2 qpp q ` p 1 q 2 ´ m 2 q p 2 π q D 4 p q 2 ´ p 1 ¨ q qp ac ` bd qq d D q 4 m 2 p ac ´ bd q d D q ż ż “ p q 2 ´ m 2 qpp q ` p 1 q 2 ´ m 2 q ` p q 2 ´ m 2 qpp q ` p 1 q 2 ´ m 2 q “ p 2 π q D p 2 π q D A 0 p m q ` m 2 B 0 p p 1 , m , m q ´ p 2 ´ ¯ “ 4 m 2 p ac ´ bd q B 0 p p 1 , m , m q ` 4 p ac ` bd q 1 2 B 0 p p 1 , m , m q 12 / 23

Consideration of loop contributions Triangle diagram (two-body decay) d D q q ` m 1 qp a ` ib γ 5 q i p ˆ „ ż i p ˆ q ´ ˆ p 1 ´ ˆ p 2 ` m 3 qp´ i q  p c ` id γ 5 q 3 q p c ` id γ 5 q i M “ ¯ u v “ p q 2 ´ m 2 1 qpp q ´ p 1 q 2 ´ m 2 2 qpp q ´ p 1 ´ p 2 q 2 ´ m 2 p 2 π q D i ´ 1 d D q ˆ f 1 p q q ´ i ˆ f 2 p q q γ 5 „ ż  p k q 2 ´ m 2 ÿ “ i ¯ u p p 1 q v p p 2 q ; d i ” p q ´ i p 2 π q D d 1 d 2 d 3 k “ 1 f 1 p q q “ a p c 2 ` d 2 q ´ ¯ ` a p c 2 ´ d 2 q ´ ¯ ˆ m 1 p ˆ q ´ ˆ p 1 ´ ˆ p 2 q ` m 3 ˆ q q p ˆ ˆ q ´ ˆ p 1 ´ ˆ p 2 q ` m 1 m 3 ` ´ ¯ ` 2 bcd q p ˆ ˆ q ´ ˆ p 1 ´ ˆ p 2 q ´ m 1 m 3 f 2 p q q “ b p c 2 ` d 2 q ´ ¯ ` b p c 2 ´ d 2 q ´ ¯ ˆ m 1 p ˆ q ´ ˆ p 1 ´ ˆ p 2 q ´ m 3 ˆ q p ˆ q ´ ˆ p 1 ´ ˆ p 2 q ´ m 1 m 3 ´ q ˆ ´ ¯ ´ 2 acd ˆ q p ˆ q ´ ˆ p 1 ´ ˆ p 2 q ` m 1 m 3 13 / 23

Consideration of loop contributions Calculations ( X Ñ e ` e ´ ) The following vertex factors should be integrated: f ˘ p q q “ H p˘q p c 2 ` d 2 q ` H p˘q p c 2 ´ d 2 q ´ ¯ ´ ¯ ˆ m 1 p ˆ q ´ ˆ p 1 ´ ˆ p 2 q ˘ m 3 ˆ q q p ˆ ˆ q ´ ˆ p 1 ´ ˆ p 2 q ˘ m 1 m 3 ˘ ´ ¯ ˘ 2 H p¯q cd ˆ q p ˆ q ´ ˆ p 1 ´ ˆ p 2 q ¯ m 1 m 3 где t H p`q , H p´q u ” t a , b u Let’s define the following vector integral d D q q µ ż C µ p p 1 , p 2 ; m 1 , m 2 , m 3 q “ p q 2 ´ m 2 1 qpp q ` p 1 q 2 ´ m 2 2 qpp q ` p 1 ` p 2 q 2 ´ m 2 p 2 π q D 3 q From Lorentz-invariance of this integral it follows that C µ p p 1 , p 2 ; m 1 , m 2 , m 3 q “ p µ 1 C 1 p p 1 , p 2 ; m 1 , m 2 , m 3 q ` p µ 2 C 2 p p 1 , p 2 ; m 1 , m 2 , m 3 q d D q ż ˆ q “ γ µ C µ “ ´ ˆ u p p 1 q γ µ C µ v p p 2 q “ 0 ñ p 1 C 1 ´ ˆ p 2 C 2 ñ ¯ p 2 π q D d 1 d 2 d 3 Thus the first term of vertex factors does not contribute to the two-body decay 14 / 23

Consideration of loop contributions f ˘ p q q “ H p˘q p c 2 ´ d 2 q ´ ¯ ´ ¯ ˆ q p ˆ ˆ q ´ ˆ p 1 ´ ˆ p 2 q ˘ m 1 m 3 ˘ 2 H p¯q cd q p ˆ ˆ q ´ ˆ p 1 ´ ˆ p 2 q ¯ m 1 m 3 q ” q 2 q 2 d D q d D q ¯ ż q ˆ ˆ ż “ “ d 1 C 0 p p 1 , p 2 ; m 1 , m 2 , m 3 q` 1 p 2 π q D d 1 d 2 d 3 p 2 π q D d 1 d 2 d 3 ` m 2 1 C 0 p p 1 , p 2 ; m 1 , m 2 , m 3 q “ B 0 p p 2 ; m 2 , m 3 q ` m 2 1 C 0 p p 1 , p 2 ; m 1 , m 2 , m 3 q d D q ˆ 1 ˙ ż q Ñ q ` p 1 d 1 C 0 p p 1 , p 2 ; m 1 , m 2 , m 3 q “ “ “ “ B 0 p p 2 ; m 2 , m 3 q p 2 π q D d 2 d 3 � d D q p 1 ´ � ż q p ˆ ˆ q ´ ˆ p 2 q ˆ ¯ M M ´ p 2 u p p 1 q v p p 2 q “ 1 , 2 “ 0 “ ¯ u p p 1 q B 0 p p 2 ; m 2 , m 3 q` 2 ¯ p 2 π q D d 1 d 2 d 3 ¯ ` m 2 1 C 0 p p 1 , p 2 ; m 1 , m 2 , m 3 q ` 2 p p 1 ¨ p 2 q C 2 p p 1 , p 2 ; m 1 . m 2 . m 3 q v p p 2 q “ ´ ¯ B 0 p p 1 ` p 2 ; m 1 , m 3 q ` m 2 “ ¯ u p p 1 q 2 C 0 p p 1 , p 2 ; m 1 , m 2 , m 3 q v p p 2 q 1 1 ´ ✓ ¯ ´ ✓ ¯ p m 2 2 ´ m 2 p 2 3 C 2 “ 1 q C 0 ` B 0 p p 1 ` p 2 ; m 1 , m 3 q ´ B 0 p p 2 ; m 2 , m 3 q 2 p p 1 ¨ p 2 q B 0 p? s ; m 1 , m 3 q ` p m 2 ñ F ˘ “ H p˘q p c 2 ´ d 2 q ´ ¯ 2 ˘ m 1 m 3 q C 0 p p 1 , p 2 ; m 1 , m 2 , m 3 q ˘ B 0 p? s ; m 1 , m 3 q ` p m 2 ´ ¯ ˘ 2 H p¯q 2 ¯ m 1 m 3 q C 0 p p 1 , p 2 ; m 1 , m 2 , m 3 q 15 / 23