Finite Pulse Effects
onFermion Pair Creationf r o m St r o n g E l e c t r i cFields
based on PRD 90, 014039 (2014)

Hidetoshi TAYA (Tokyo U.)

in collaboration w/ H.Fujii (Tokyo U.) and K.Itakura (KEK)INTRO > PROBLEM > GUESS > THEORY > RESULT > SUMMARY

Q. What is Fermion Pair Creation from Strong Electric Fields ?

INTRO > PROBLEM > GUESS > THEORY > RESULT > SUMMARY

Q. What is Fermion Pair Creation from Strong Electric Fields ?

Naïve understanding
QFT vacuum

𝑬𝑬𝑬𝑬

𝑬𝑬Strong Electric Fields
time<vac;in|𝑛𝑛� out |vac;in> = 0

𝑬𝑬

𝑬𝑬… … 𝑬𝑬
……

𝑬𝑬

time

<vac;in|𝑛𝑛� out |vac;in> ≠ 0▶ well-formulated for static homogeneous electric field as a non-perturbative physics

𝑚𝑚2 + 𝑝𝑝𝑇𝑇2�
<vac;in|𝑛𝑛� out |vac;in> = exp −𝜋𝜋𝑔𝑔𝑔𝑔[Schwinger’s formula, J.Schwinger 1951]

INTRO > PROBLEM > GUESS > THEORY > RESULT > SUMMARY

Q. What is Fermion Pair Creation from Strong Electric Fields ?

Naïve understanding
QFT vacuum

𝑬𝑬

𝑬𝑬… … 𝑬𝑬

𝑬𝑬

Strong Electric Fields
𝑬𝑬

time

……

𝑬𝑬

𝑬𝑬time<vac;in|𝑛𝑛� out |vac;in> ≠ 0<vac;in|𝑛𝑛� out |vac;in> = 0▶ well-formulated for static homogeneous electric field as a non-perturbative physics

𝑚𝑚2 + 𝑝𝑝𝑇𝑇2�
<vac;in|𝑛𝑛� out |vac;in> = exp −𝜋𝜋𝑔𝑔𝑔𝑔[Schwinger’s formula, J.Schwinger 1951]

Why interesting ?

Because it gives us a deeper understanding of QFT cf.) vacuum structure, non-perturbative aspects …
Because it is phenomenologically importantcf.) formation of QGP (particle creation from glasma) …

Because it is becoming relevant to experiment and very timely to studycf.) Laser:

𝒆𝒆𝒆𝒆 ∼ 𝟏𝟏𝟏𝟏𝟏𝟏 𝐤𝐤𝐤𝐤𝐤𝐤 ∼ 𝒎𝒎𝐞𝐞𝐞𝐞𝐞𝐞𝐞𝐞𝐞𝐞𝐞𝐞𝐞𝐞𝐞𝐞

HIC:

𝒈𝒈𝒈𝒈 ∼ 𝟏𝟏 𝐆𝐆𝐆𝐆𝐆𝐆 ≫ 𝒎𝒎𝐪𝐪𝐪𝐪𝐪𝐪𝐪𝐪𝐪𝐪

INTRO > PROBLEM > GUESS > THEORY > RESULT > SUMMARY
Fermion pair creation is well studied for static homogeneous field- Non-perturbative pair creation occurs- Schwinger’s formula is known to compute <vac;in|𝑛𝑛� out |vac;in>INTRO > PROBLEM > GUESS > THEORY > RESULT > SUMMARY
Fermion pair creation is well studied for static homogeneous field- Non-perturbative pair creation occurs- Schwinger’s formula is known to compute <vac;in|𝑛𝑛� out |vac;in>

PROBLEM: Strong fields are always finite in timecf.) Laser: 𝝉𝝉 ∼ 𝟏𝟏𝟎𝟎−𝟏𝟏𝟏𝟏 𝐬𝐬𝐬𝐬𝐬𝐬
HIC:𝝉𝝉 ∼ 𝟏𝟏𝟎𝟎−𝟐𝟐𝟐𝟐 𝐬𝐬𝐬𝐬𝐬𝐬

Go beyond static homogeneous fieldand consider FINITE PULSE EFFECTINTRO > PROBLEM > GUESS > THEORY > RESULT > SUMMARY

Q. What is expected if Finite pulse effect is considered ?

INTRO > PROBLEM > GUESS > THEORY > RESULT > SUMMARY

Q. What is expected if Finite pulse effect is considered ?

Naïve understanding(static homogeneous field)
𝑬𝑬

𝑬𝑬

𝑬𝑬𝑬𝑬𝑬𝑬… … 𝑬𝑬……

𝑬𝑬

Finite pulse effectFinitePulse
Effect

time

𝑬𝑬

𝑬𝑬

𝑬𝑬

𝑬𝑬

𝑬𝑬

… … 𝑬𝑬
……

𝑬𝑬

time

Infinite # of scattering

Finite # of scattering

Non-perturbative pair creation

Perturbative pair creation

𝑚𝑚2 + 𝑝𝑝𝑇𝑇2�
<vac;in|𝑛𝑛� out |vac;in> = exp −𝜋𝜋
𝑔𝑔𝑔𝑔

𝑔𝑔𝑔𝑔
<vac;in|𝑛𝑛� out |vac;in > = # × 2𝑚𝑚[Schwinger’s formula, J.Schwinger 1951]

A? “Transition” from non-perturbative to perturbative pair crea.

2

INTRO > PROBLEM > GUESS > THEORY > RESULT > SUMMARY

0.

Setting:- QED Lagrangian w/ external electric field:/ − 𝑚𝑚 − 𝑔𝑔𝑔𝑔/ 𝜓𝜓ℒ = 𝜓𝜓� 𝑖𝑖𝑖𝑖

- analytical model: Sauter-type pulsed electric field

1.
2.

𝑬𝑬𝐒𝐒𝐒𝐒𝐒𝐒𝐒𝐒𝐒𝐒𝐒𝐒 (𝒕𝒕) = 𝑬𝑬𝟎𝟎 𝐜𝐜𝐜𝐜𝐜𝐜𝐜𝐜−𝟐𝟐 [𝒕𝒕/𝝉𝝉]

𝑬𝑬𝟎𝟎

𝝉𝝉

[F.Sauter, 1932]
time

Compute full formula for <vac;in|�
𝒏𝒏out |vac;in >Compare the full formula with the formula obtained from the lowestorder perturbation and the non-perturbative Schwinger formulaINTRO > PROBLEM > GUESS > THEORY > RESULT > SUMMARY

0.

Setting:- QED Lagrangian w/ external electric field:/ − 𝑚𝑚 − 𝑔𝑔𝑔𝑔/ 𝜓𝜓ℒ = 𝜓𝜓� 𝑖𝑖𝑖𝑖

- analytical model: Sauter-type pulsed electric field

1.
2.

𝑬𝑬𝟎𝟎

𝑬𝑬𝐒𝐒𝐒𝐒𝐒𝐒𝐒𝐒𝐒𝐒𝐒𝐒 (𝒕𝒕) = 𝑬𝑬𝟎𝟎 𝐜𝐜𝐜𝐜𝐜𝐜𝐜𝐜−𝟐𝟐 [𝒕𝒕/𝝉𝝉]𝝉𝝉[F.Sauter, 1932]
time

Compute full formula for <vac;in|�
𝒏𝒏out |vac;in >- canonical quantization under external fields- deriving <vac;in|𝑛𝑛� out |vac;in >full⇔solving Dirac eq. under external fields:

0 = 𝑖𝑖𝑖𝑖/ − 𝑚𝑚 − 𝑔𝑔𝑔𝑔
/ 𝜓𝜓Compare the full formula with the formula obtained from the lowestorder perturbation and the non-perturbative Schwinger formulaINTRO > PROBLEM > GUESS > THEORY > RESULT > SUMMARY

0.

Setting:- QED Lagrangian w/ external electric field:/ − 𝑚𝑚 − 𝑔𝑔𝑔𝑔/ 𝜓𝜓ℒ = 𝜓𝜓� 𝑖𝑖𝑖𝑖

- analytical model: Sauter-type pulsed electric field

1.
2.

𝑬𝑬𝟎𝟎

𝑬𝑬𝐒𝐒𝐒𝐒𝐒𝐒𝐒𝐒𝐒𝐒𝐒𝐒 (𝒕𝒕) = 𝑬𝑬𝟎𝟎 𝐜𝐜𝐜𝐜𝐜𝐜𝐜𝐜−𝟐𝟐 [𝒕𝒕/𝝉𝝉]𝝉𝝉[F.Sauter, 1932]
time

Compute full formula for <vac;in|�
𝒏𝒏out |vac;in >- canonical quantization under external fields- deriving <vac;in|𝑛𝑛� out |vac;in >full⇔solving Dirac eq. under external fields:

0 = 𝑖𝑖𝑖𝑖/ − 𝑚𝑚 − 𝑔𝑔𝑔𝑔
/ 𝜓𝜓Compare the full formula with the formula obtained from the lowestorder perturbation and the non-perturbative Schwinger formula<vac;in|𝑛𝑛� out |vac;in >LOPT =2𝑬𝑬𝑝𝑝𝑧𝑧2= 1− 2𝑝𝑝0

�𝑔𝑔𝐸𝐸(2𝑝𝑝0)
4𝑝𝑝02

2

<vac;in|𝑛𝑛� out |vac;in >Schwinger

𝑚𝑚2 + 𝑝𝑝𝑇𝑇2�
= exp −𝜋𝜋𝑔𝑔𝑔𝑔

INTRO > PROBLEM > GUESS > THEORY > RESULT > SUMMARY
𝟏𝟏 𝒅𝒅𝟑𝟑 𝑵𝑵# of fermions
𝑽𝑽 𝒅𝒅𝒑𝒑𝟑𝟑

Result 1:

Result 2:𝒎𝒎𝒎𝒎

Pulse duration

INTRO > PROBLEM > GUESS > THEORY > RESULT > SUMMARY
𝟏𝟏 𝒅𝒅𝟑𝟑 𝑵𝑵# of fermions
𝑽𝑽 𝒅𝒅𝒑𝒑𝟑𝟑

Result 1:- Pair crea. is (non-)perturbative for small (large) 𝝉𝝉.
Result 2:

𝒎𝒎𝒎𝒎

Pulse duration

INTRO > PROBLEM > GUESS > THEORY > RESULT > SUMMARY”Short 𝝉𝝉” or “Large 𝝉𝝉” ?

3 dim. param. in the Lagrangian: 𝝉𝝉, 𝒈𝒈𝑬𝑬𝟎𝟎 , 𝒎𝒎.
𝜸𝜸 ≪ 𝟏𝟏, 𝝂𝝂 ≪ 𝟏𝟏

lowest-order perturbation

2 expansion param. : 𝜸𝜸 = 𝒈𝒈𝑬𝑬𝟎𝟎 𝝉𝝉/𝒎𝒎, 𝝂𝝂 = 𝒈𝒈𝑬𝑬𝟎𝟎 𝝉𝝉𝟐𝟐 .

𝒎𝒎𝒎𝒎 (Pulse duration)

FullSchwinger’s formula𝜸𝜸 ≫ 𝟏𝟏, 𝝂𝝂 ≫ 𝟏𝟏

Result 1:- Pair crea. is (non-)perturbative for small (large) 𝝉𝝉.

- 2 dim.-less param. 𝜸𝜸 = 𝒈𝒈𝑬𝑬𝟎𝟎 𝝉𝝉/𝒎𝒎, 𝝂𝝂 = 𝒈𝒈𝑬𝑬𝟎𝟎 𝝉𝝉𝟐𝟐 control the “transition.”

Result 2:𝒈𝒈𝒈𝒈/𝒎𝒎𝟐𝟐(Field strength)

INTRO > PROBLEM > GUESS > THEORY > RESULT > SUMMARY
𝟏𝟏 𝒅𝒅𝟑𝟑 𝑵𝑵# of fermions
𝑽𝑽 𝒅𝒅𝒑𝒑𝟑𝟑

Result 1:- Pair crea. is (non-)perturbative for small (large) 𝝉𝝉.

𝒎𝒎𝒎𝒎- 2 dim.-less param. 𝜸𝜸 = 𝒈𝒈𝑬𝑬𝟎𝟎 𝝉𝝉/𝒎𝒎, 𝝂𝝂 = 𝒈𝒈𝑬𝑬𝟎𝟎 𝝉𝝉𝟐𝟐 control the “transition.”

Result 2:

Pulse duration

INTRO > PROBLEM > GUESS > THEORY > RESULT > SUMMARY
𝟏𝟏 𝒅𝒅𝟑𝟑 𝑵𝑵# of fermions
𝑽𝑽 𝒅𝒅𝒑𝒑𝟑𝟑

Result 1:- Pair crea. is (non-)perturbative for small (large) 𝝉𝝉.

𝒎𝒎𝒎𝒎- 2 dim.-less param. 𝜸𝜸 = 𝒈𝒈𝑬𝑬𝟎𝟎 𝝉𝝉/𝒎𝒎, 𝝂𝝂 = 𝒈𝒈𝑬𝑬𝟎𝟎 𝝉𝝉𝟐𝟐 control the “transition.”

Result 2:- Pair creation is enhanced for 𝒈𝒈𝑬𝑬𝟎𝟎 /𝒎𝒎𝟐𝟐 ≲ 𝟏𝟏, 𝒎𝒎𝒎𝒎 ≲ 𝟏𝟏.
- This enhancement is perturbative phenomena.

Pulse durationcf.) P.Levai & V. Skokov, 2010

INTRO > PROBLEM > GUESS > THEORY > RESULT > SUMMARY■ Finite pulse effects on fermion pair creation from strong electric fields is studied.

▶ A “transition” from non-perturbative to perturbative pair creation occurs.

This “transition” is controlled by the 2 dimensionless parameters: 𝜸𝜸 = 𝒈𝒈𝑬𝑬𝟎𝟎 𝝉𝝉/𝒎𝒎, 𝝂𝝂 = 𝒈𝒈𝑬𝑬𝟎𝟎 𝝉𝝉𝟐𝟐 .

▶ Pair creation is enhanced compared to Schwinger's formula for 𝒈𝒈𝑬𝑬𝟎𝟎 /𝒎𝒎𝟐𝟐 ≲ 𝟏𝟏, 𝒎𝒎𝒎𝒎 ≲ 𝟏𝟏.

This enhancement is predominantly described by the lowest-order perturbation.

■ Implications to HIC phenomenology:
GlasmaStrong Coulomb field

Lifetime of strong Coulomb field
W.T.Deng & X.G.Huang, 2012

Finite pulse effect may be important: 𝜸𝜸 ≫ 𝟏𝟏, 𝝂𝝂 ≲ 𝟏𝟏.

There is a room for the non-perturbative pair crea.

The matter created in the HIC could let 𝝂𝝂 be large 𝝂𝝂 ≳ 𝟏𝟏

■ For more results and discussions, see H.Taya et al., PRD 90, 014039 (2014)

BACKUPDevelopment of strong electric field
pair creation requires strong electric fields of the order of 𝒈𝒈𝒈𝒈 ∼ 𝒎𝒎𝟐𝟐

BEFORE 21th century:

・ only weak lasers are available▶ always 𝒈𝒈𝒈𝒈 ≪ 𝒎𝒎𝟐𝟐 , so pair crea. is impossible

AFTER 21th century:・ laser strength dramatically increases
・ HIC creates strong (color-)electric fieldscf.)

GlasmaStrong Coulomb field▶ can be 𝒈𝒈𝒈𝒈 ≳ 𝒎𝒎𝟐𝟐 , so pair crea. can be possible▶ Now is the best time to study fermion pair creation from strong electric fields

RESULT
Comment on Result 2
Perturbative pair creation
𝑔𝑔𝑔𝑔<vac;in|𝑛𝑛� out |vac;in > = # × 2
𝑚𝑚2

Non-perturbative pair creation
𝑚𝑚2 + 𝑝𝑝𝑇𝑇2�
<vac;in|𝑛𝑛� out |vac;in> = exp −𝜋𝜋𝑔𝑔𝑔𝑔

Result 1:- Pair crea. is (non-)perturbative for small (large) 𝝉𝝉.

- 2 dim.-less param. 𝜸𝜸 = 𝒈𝒈𝑬𝑬𝟎𝟎 𝝉𝝉/𝒎𝒎, 𝝂𝝂 = 𝒈𝒈𝑬𝑬𝟎𝟎 𝝉𝝉𝟐𝟐 control the “transition.”

Result 2:- Pair creation is enhanced for 𝒈𝒈𝑬𝑬𝟎𝟎 /𝒎𝒎𝟐𝟐 ≲ 𝟏𝟏, 𝒎𝒎𝒎𝒎 ≲ 𝟏𝟏.
- This enhancement is perturbative phenomena.

cf.) P.Levai & V. Skokov, 2010

SUMMARY* For more results and discussions, see H.Taya et al., PRD 90, 014039 (2014)■ Finite pulse effects on fermion pair creation from strong electric fields is studied.

▶ A “transition” from non-perturbative to perturbative pair creation occurs and is controlled by the 2dimensionless parameters: 𝜸𝜸 = 𝒈𝒈𝑬𝑬𝟎𝟎 𝝉𝝉/𝒎𝒎, 𝝂𝝂 = 𝒈𝒈𝑬𝑬𝟎𝟎 𝝉𝝉𝟐𝟐.

▶ Pair creation is enhanced compared to Schwinger's formula for 𝒈𝒈𝑬𝑬𝟎𝟎 /𝒎𝒎𝟐𝟐 ≲ 𝟏𝟏, 𝒎𝒎𝒎𝒎 ≲ 𝟏𝟏.
This enhancement is predominantly described by the lowest-order perturbation.

■ Implications to HIC phenomenology:
GlasmaStrong Coulomb field

Finite pulse effect may be important: 𝜸𝜸 ≫ 𝟏𝟏, 𝝂𝝂 ≲ 𝟏𝟏.

There is a room for the non-perturbative pair crea. :𝜸𝜸 ≫ 𝟏𝟏. The matter created in the HIC could let 𝝂𝝂 belarge 𝝂𝝂 ≳ 𝟏𝟏

Lifetime of strong Coulomb field
W.T.Deng & X.G.Huang, 2012